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geometry of physics -- homotopy types

This entry contains one chapter of geometry of physics, see there for context and background.

previous chapter: smooth sets,

next chapter: smooth homotopy types

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Homotopy types

Traditionally, mathematics and physics have been founded on sets, “bags of points”. But fundamental physics is all governed by the gauge principle which says that fundamentally the points in these bags are connected, possibly, by various gauge transformation equivalences. In fact fundamental physics involves higher gauge theory, which asserts that in general there are gauge-of-gauge transformations between these gauge transformations, and so ever on. This means that what naively may have looked like spaces made of points are really in general like orbifolds, “orbispaces”. Since these are generalizations of groups of symmetries to a situation where symmetries act on and between various objects, one speaks of groupoids, and if the ever higher gauge-of-gauge equivalences are taken into account one speaks of infinity-groupoids. The field of mathematics that studies these structures is called homotopy theory. Here “gauge equivalences” are also called homotopies and n-groupoids (structures with nn-fold gauge-of-gauge transformations) are called homotopy n-types.

Here we discuss the basics of homotopy theory irrespective of any geometry. In the next chapter geometry of physics -- smooth homotopy types we then combine the concept of smooth sets of the previous chapter with the idea of generalizing sets to smooth homotopy types. These subsume the objects of actual interest in physics, such as notably the moduli stacks of fields in any gauge theory and higher gauge theory.

Motivation

The basic principle of homotopy theory happens to be well familiar in physics, just in slight disguise: it is the (higher) gauge principle. This we explain in

As it goes, this profound and yet simple fact is less widely appreciated than a rather sophisticated phenomenon which is but a special case of this: the appearance of derived categories of branes in topological string theory

The historical route in mathematics that lead to modern homotopy theory so happens to run via the study of ordinary cohomology and ordinary homology (singular homology) of topological spaces, structures that happen to recognize of a topological space only the homotopy type that it represents via the singular simplicial complex construction. This traditional route we recall in

While standard and traditional, this story hides a bit how utmost fundamental the principles of homotopy theory are and tends to misleadingly suggest that homotopy theory is a sub-topic of topology. Contrary to that, inspection of the foundations of mathematics reveals the foundational nature of homotopy theory. This we indicate in

Motivation from physics

The gauge principle

One of the fundamental principles of modern physics is the gauge principle. It says that every field configuration in physics – hence absolutely everything in physics – is, in general, a gauge field configuration. This in turn means that given two field configurations Φ 1\Phi_1 and Φ 2\Phi_2, then it makes no sense to ask whether they are equal or not. Instead what makes sense to ask for is a gauge transformation gg that, if it exists, exhibits Φ 1\Phi_1 as being gauge equivalent to Φ 2\Phi_2 via gg:

Φ 1gΦ 2. \Phi_1 \underoverset{\simeq}{g}{\longrightarrow} \Phi_2 \,.

This satisfies obvious rules, so obvious that physics textbooks usually don’t bother to mention this. First of all, if there is yet another field configuration Φ 3\Phi_3 and a gauge transformation gg' from Φ 2\Phi_2 to Φ 3\Phi_3, then there is also the composite gauge transformation

ggΦ 1gΦ 2gΦ 3 g' \circ g \Phi_1 \underoverset{\simeq}{g}{\longrightarrow} \Phi_2 \underoverset{\simeq}{g'}{\longrightarrow} \Phi_3

and this composition is associative.

Moreover, these being equivalences means that they have inverses,

Φ 2g 1Φ 1 \Phi_2 \underoverset{\simeq}{g^{-1}}{\longrightarrow} \Phi_1

such that the compositions

Φ 1gΦ 2g 1Φ 1 \Phi_1 \underoverset{\simeq}{g}{\longrightarrow} \Phi_2 \underoverset{\simeq}{g^{-1}}{\longrightarrow} \Phi_1

and

Φ 1gΦ 2g 1Φ 1 \Phi_1 \underoverset{\simeq}{g}{\longrightarrow} \Phi_2 \underoverset{\simeq}{g^{-1}}{\longrightarrow} \Phi_1

equal the identity transformation.

Obvious as this may be, in mathematics such structure gets a name: this is a groupoid or homotopy 1-type whose objects are field configurations and whose morphisms are gauge transformations.

But notice that in the last statement above about inverses, we were actually violating the gauge principle: we asked for a gauge transformation of the form g 1gg^{-1}\circ g (transforming one way and then just transforming back) to be equal to the identity transformation idid.

But the gauge principle applies also to gauge transformations themselves. This is the content of higher gauge theory. For instance a 2-form gauge field such as the Kalb-Ramond field has gauge-of-gauge transformations. In the physics literature these are best known in their infinitesimal approximation, which are called ghost-of-ghost fields (for some historical reasons). In fact physicists know the infinitesimalLie algebroid” version of Lie groupoids and their higher versions as BRST complexes.

This means that in general it makes no sense to ask whether two gauge transformations are equal or not. What makes sense is to ask for a gauge-of-gauge transformation that turns one into the other

Φ 1 g Φ 2 Φ 1 g Φ 2. \array{ \Phi_1 & \stackrel{g}{\longrightarrow} & \Phi_2 \\ & \Downarrow^{\mathrlap{\simeq}} \\ \Phi_1 & \underset{g'}{\longrightarrow} & \Phi_2 } \,.

Now it is clear that gauge-of-gauge transformations may be composed with each other, and that, being equivalences, they have inverses under this composition. Moreover, this composition of gauge-of-gauge transformations is to be compatible with the already existing composition of the first order gauge transformations themselves. This structure, when made explicit, is in mathematics called a 2-groupoid or homotopy 2-type.

But now it is clear that this pattern continues: next we may have a yet higher gauge theory, for instance that of a 3-form C-field, and then there are third order gauge transformations which we must use to identify, when possible, second order gauge transformations. They may in turn be composed and have inverses under this composition, and the resulting structure, when made explicit, is called a 3-groupoid or homotopy 3-type.

This logic of the gauge principle keeps applying, and hence we obtain an infinite sequence of concepts, which at stage nn \in \mathbb{N} are called n-groupoids or homotopy n-types. The limiting case where we never assume that some high order gauge-of-gauge transformation has no yet higher order transformations between them, the structure in this limiting case accordingly goes by the name of infinity-groupoid or just homotopy type.

The mathematics theory of these systems of higher-order gauge-of-gauge transformations is called homotopy theory or higher category theory in the flavor of (infinity,1)-category theory.

More motivation and exposition along these lines is at

Derived categories of branes

topological stringTCFThomological mirror symmetry

Motivation from topology: Singular homology

This section recalls how the “abelianization” of a topological space by singular chains gives rise to the notion of chain complexes and their homology.

This proceeds in three steps: given a topological space, first one passes to the collection of simplices in it (the curves, triangles, tetrahedra, …) which together form a simplicial set. Then one “linearizes” this by forming the free abelian groups on the simplices to obtain a simplicial abelian group. Finally one turns the resulting simplicial abelian group into a chain complex.

Below in Dold-Kan correspondence we see that this last step is an equivalent reformulation, and that from any chain complex (in non-negative degree) one may re-obtain the simplicial abelian group that it corresponds to. Further below in Kan complexes we see that (forgetting the group structure on these), these are Kan complexes and as such objects in simplicial homotopy theory. This we then turn to further below in Simplicial homotopy theory.

Singular simplicial set
Definition

For nn \in \mathbb{N}, the topological n-simplex is, up to homeomorphism, the topological space whose underlying set is the subset

Δ n{x n+1| i=0 nx i=1andi.x i0} n+1 \Delta^n \coloneqq \{ \vec x \in \mathbb{R}^{n+1} | \sum_{i = 0 }^n x_i = 1 \; and \; \forall i . x_i \geq 0 \} \subset \mathbb{R}^{n+1}

of the Cartesian space n+1\mathbb{R}^{n+1}, and whose topology is the subspace topology induces from the canonical topology in n+1\mathbb{R}^{n+1}.

Remark

The coordinate expression in def. 1 – also known as barycentric coordinates – is evidently just one of many possible ways to present topological nn-simplices. Another common choice are what are called Cartesian coordinates. Of course nothing of relevance will depend on which choice of coordinate presentation is used, but some are more convenient in some situations than others.

Example

For n=0n = 0 this is the point, Δ 0=*\Delta^0 = *.

For n=1n = 1 this is the standard interval object Δ 1=[0,1]\Delta^1 = [0,1].

For n=2n = 2 this is the filled triangle.

For n=3n = 3 this is the filled tetrahedron.

Definition

For nn \in \mathbb{N}, n1\n \geq 1 and 0kn0 \leq k \leq n, the kkth (n1)(n-1)-face (inclusion) of the topological nn-simplex, def. 1, is the subspace inclusion

δ k:Δ n1Δ n \delta_k : \Delta^{n-1} \hookrightarrow \Delta^n

induced under the coordinate presentation of def. 1, by the inclusion

n n+1 \mathbb{R}^n \hookrightarrow \mathbb{R}^{n+1}

which “omits” the kkth canonical coordinate:

(x 0,,x n1)(x 0,,x k1,0,x k,,x n1). (x_0, \cdots , x_{n-1}) \mapsto (x_0, \cdots, x_{k-1} , 0 , x_{k}, \cdots, x_{n-1}) \,.
Example

The inclusion

δ 0:Δ 0Δ 1 \delta_0 : \Delta^0 \to \Delta^1

is the inclusion

{1}[0,1] \{1\} \hookrightarrow [0,1]

of the “right” end of the standard interval. The other inclusion

δ 1:Δ 0Δ 1 \delta_1 : \Delta^0 \to \Delta^1

is that of the “left” end {0}[0,1]\{0\} \hookrightarrow [0,1].

Definition

For nn \in \mathbb{N} and 0k<n0 \leq k \lt n the kkth degenerate (n)(n)-simplex (projection) is the surjective map

σ k:Δ nΔ n1 \sigma_k : \Delta^{n} \to \Delta^{n-1}

induced under the barycentric coordinates of def. 1 under the surjection

n+1 n \mathbb{R}^{n+1} \to \mathbb{R}^n

which sends

(x 0,,x n)(x 0,,x k+x k+1,,x n). (x_0, \cdots, x_n) \mapsto (x_0, \cdots, x_{k} + x_{k+1}, \cdots, x_n) \,.
Definition

For XX \in Top a topological space and nn \in \mathbb{N} a natural number, a singular nn-simplex in XX is a continuous map

σ:Δ nX \sigma : \Delta^n \to X

from the topological nn-simplex, def. 1, to XX.

Write

(SingX) nHom Top(Δ n,X) (Sing X)_n \coloneqq Hom_{Top}(\Delta^n , X)

for the set of singular nn-simplices of XX.

So to a topological space XX is associated a sequence of sets

(SingX) nHom Top(Δ n,X) (Sing X)_n \coloneqq Hom_{Top}(\Delta^n, X)

of singular simplices. Since the topological nn-simplices Δ n\Delta^n from def. 1 sit inside each other by the face inclusions of def. 2

δ k:Δ n1Δ n \delta_k : \Delta^{n-1} \to \Delta^{n}

and project onto each other by the degeneracy maps, def. 3

σ k:Δ n+1Δ n \sigma_k : \Delta^{n+1} \to \Delta^n

we dually have functions

d kHom Top(δ k,X):(SingX) n(SingX) n1 d_k \coloneqq Hom_{Top}(\delta_k, X) : (Sing X)_n \to (Sing X)_{n-1}

that send each singular nn-simplex to its kk-face and functions

s kHom Top(σ k,X):(SingX) n(SingX) n+1 s_k \coloneqq Hom_{Top}(\sigma_k,X) : (Sing X)_{n} \to (Sing X)_{n+1}

that regard an nn-simplex as beign a degenerate (“thin”) (n+1)(n+1)-simplex. All these sets of simplicies and face and degeneracy maps between them form the following structure.

Definition

A simplicial set SsSetS \in sSet is

  • for each nn \in \mathbb{N} a set S nSetS_n \in Set – the set of nn-simplices;

  • for each injective map δ i:n1¯n¯\delta_i : \overline{n-1} \to \overline{n} of totally ordered sets n¯{0<1<<n}\bar n \coloneqq \{ 0 \lt 1 \lt \cdots \lt n \}

    a function d i:S nS n1d_i : S_{n} \to S_{n-1} – the iith face map on nn-simplices;

  • for each surjective map σ i:n+1¯n¯\sigma_i : \overline{n+1} \to \bar n of totally ordered sets

    a function σ i:S nS n+1\sigma_i : S_{n} \to S_{n+1} – the iith degeneracy map on nn-simplices;

such that these functions satisfy the simplicial identities.

Proposition

These face and degeneracy maps satisfy the following simplicial identities (whenever the maps are composable as indicated):

  1. d id j=d j1d i d_i \circ d_j = d_{j-1} \circ d_i if i<ji \lt j,

  2. s is j=s js i1s_i \circ s_j = s_j \circ s_{i-1} if i>ji \gt j.

  3. d is j={s j1d i ifi<j id ifi=jori=j+1 s jd i1 ifi>j+1d_i \circ s_j = \left\{ \array{ s_{j-1} \circ d_i & if \; i \lt j \\ id & if \; i = j \; or \; i = j+1 \\ s_j \circ d_{i-1} & if i \gt j+1 } \right.

It is straightforward to check by explicit inspection that the evident injection and restriction maps between the sets of singular simplices make (SingX) (Sing X)_\bullet into a simplicial set. We now briefly indicate a systematic way to see this using basic category theory, but the reader already satisfied with this statement should skip ahead to the Singular chain complex.

Definition

The simplex category Δ\Delta is the full subcategory of Cat on the free categories of the form

[0] {0} [1] {01} [2] {012} . \begin{aligned} [0] & \coloneqq \{0\} \\ [1] & \coloneqq \{0 \to 1\} \\ [2] & \coloneqq \{0 \to 1 \to 2\} \\ \vdots \end{aligned} \,.
Remark

This is called the “simplex category” because we are to think of the object [n][n] as being the “spine” of the nn-simplex. For instance for n=2n = 2 we think of 0120 \to 1 \to 2 as the “spine” of the triangle. This becomes clear if we don’t just draw the morphisms that generate the category [n][n], but draw also all their composites. For instance for n=2n = 2 we have_

[2]={ 1 0 2}. [2] = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&\to&& 2 } \right\} \,.
Proposition

A functor

S:Δ opSet S : \Delta^{op} \to Set

from the opposite category of the simplex category to the category Set of sets is canonically identified with a simplicial set, def. 5.

Proof

One checks by inspection that the simplicial identities characterize precisely the behaviour of the morphisms in Δ op([n],[n+1])\Delta^{op}([n],[n+1]) and Δ op([n],[n1])\Delta^{op}([n],[n-1]).

This makes the following evident:

Example

The topological simplices from def. 1 arrange into a cosimplicial object in Top, namely a functor

Δ :ΔTop. \Delta^\bullet : \Delta \to Top \,.

With this now the structure of a simplicial set on (SingX) (Sing X)_\bullet is manifest: it is just the nerve of XX with respect to Δ \Delta^\bullet, namely:

Definition

For XX a topological space its simplicial set of singular simplicies (often called the singular simplicial complex)

(SingX) :Δ opSet (Sing X)_\bullet : \Delta^{op} \to Set

is given by composition of the functor from example 3 with the hom functor of Top:

(SingX):[n]Hom Top(Δ n,X). (Sing X) : [n] \mapsto Hom_{Top}( \Delta^n , X ) \,.
Remark (aside)

It turns out that that homotopy type of the topological space XX is entirely captured by its singular simplicial complex SingXSing X (this is the content of the homotopy hypothesis-theorem).

Singular chain complex

Now we abelianize the singular simplicial complex (SingX) (Sing X)_\bullet in order to make it simpler and hence more tractable.

Definition

A formal linear combination of elements of a set SS \in Set is a function

a:S a : S \to \mathbb{Z}

such that only finitely many of the values a sa_s \in \mathbb{Z} are non-zero.

Identifying an element sSs \in S with the function SS \to \mathbb{Z}, which sends ss to 11 \in \mathbb{Z} and all other elements to 0, this is written as

a= sSa ss. a = \sum_{s \in S} a_s \cdot s \,.

In this expression one calls a sa_s \in \mathbb{Z} the coefficient of ss in the formal linear combination.

Remark

For SS \in Set, the group of formal linear combinations [S]\mathbb{Z}[S] is the group whose underlying set is that of formal linear combinations, def. 8, and whose group operation is the pointwise addition in \mathbb{Z}:

( sSa ss)+( sSb ss)= sS(a s+b s)s. (\sum_{s \in S} a_s \cdot s) + (\sum_{s \in S} b_s \cdot s) = \sum_{s \in S} (a_s + b_s) \cdot s \,.

For the present purpose the following statement may be regarded as just introducing different terminology for the group of formal linear combinations:

Proposition

The group [S]\mathbb{Z}[S] is the free abelian group on SS.

Definition

For S S_\bullet a simplicial set, def. 5, the free abelian group [S n]\mathbb{Z}[S_n] is called the group of (simplicial) nn-chains on SS.

Definition

For XX a topological space, an nn-chain on the singular simplicial complex SingXSing X is called a singular nn-chain on XX.

This construction makes the sets of simplices into abelian groups. But this allows to formally add the different face maps in the simplicial set to one single boundary map:

Definition

For SS a simplicial set, its alternating face map differential in degree nn is the linear map

:[S n][S n1] \partial : \mathbb{Z}[S_n] \to \mathbb{Z}[S_{n-1}]

defined on basis elements σS n\sigma \in S_n to be the alternating sum of the simplicial face maps:

(1)σ k=0 n(1) kd kσ. \partial \sigma \coloneqq \sum_{k = 0}^n (-1)^k d_k \sigma \,.
Proposition

The simplicial identity, prop. 1 part 1), implies that the alternating sum boundary map of def. 12 squares to 0:

=0. \partial \circ \partial = 0 \,.
Proof

By linearity, it is sufficient to check this on a basis element σS n\sigma \in S_n. There we compute as follows:

σ =( j=0 n(1) jd jσ) = j=0 n i=0 n1(1) i+jd id jσ = 0i<jn(1) i+jd id jσ+ 0ji<n(1) i+jd id jσ = 0i<jn(1) i+jd j1d iσ+ 0ji<n(1) i+jd id jσ = 0ij<n(1) i+jd jd iσ+ 0ji<n(1) i+jd id jσ =0. \begin{aligned} \partial \partial \sigma & = \partial \left( \sum_{j = 0}^n (-1)^j d_j \sigma \right) \\ & = \sum_{j=0}^n \sum_{i = 0}^{n-1} (-1)^{i+j} d_i d_j \sigma \\ & = \sum_{0 \leq i \lt j \leq n} (-1)^{i+j} d_i d_j \sigma + \sum_{0 \leq j \leq i \lt n} (-1)^{i + j} d_i d_j \sigma \\ & = \sum_{0 \leq i \lt j \leq n} (-1)^{i+j} d_{j-1} d_i \sigma + \sum_{0 \leq j \leq i \lt n} (-1)^{i + j} d_i d_j \sigma \\ & = - \sum_{0 \leq i \leq j \lt n} (-1)^{i+j} d_{j} d_i \sigma + \sum_{0 \leq j \leq i \lt n} (-1)^{i + j} d_i d_j \sigma \\ & = 0 \end{aligned} \,.

Here

  1. the first equality is (1);

  2. the second is (1) together with the linearity of dd;

  3. the third is obtained by decomposing the sum into two summands;

  4. the fourth finally uses the simplicial identity prop. 1 (1) in the first summand;

  5. the fifth relabels the summation index jj by j+1j +1;

  6. the last one observes that the resulting two summands are negatives of each other.

Example

Let XX be a topological space. Let σ 1:Δ 1X\sigma^1 : \Delta^1 \to X be a singular 1-simplex, regarded as a 1-chain

σ 1C 1(X). \sigma^1 \in C_1(X) \,.

Then its boundary σH 0(X)\partial \sigma \in H_0(X) is

σ 1=σ(0)σ(1) \partial \sigma^1 = \sigma(0) -\sigma(1)

or graphically (using notation as for orientals)

(σ(0)σσ(1))=(σ(0))(σ(1)). \partial \left( \sigma(0) \stackrel{\sigma}{\to} \sigma(1) \right) = (\sigma(0)) - (\sigma(1)) \,.

In particular σ\sigma is a 1-cycle precisely if σ(0)=σ(1)\sigma(0) = \sigma(1), hence precisely if σ\sigma is a loop.

Let σ 2:Δ 2X\sigma^2 : \Delta^2 \to X be a singular 2-chain. The boundary is

( σ(1) σ(0,1) σ σ 1,2 σ(0) σ(0,2) σ(2))=( σ(1) σ(0,1) σ(0))( σ(0) σ(0,2) σ(2))+( σ(1) σ 1,2 σ(2)). \partial \left( \array{ && \sigma(1) \\ & {}^{\mathllap{\sigma(0,1)}}\nearrow & \Downarrow^{\mathrlap{\sigma}}& \searrow^{\mathrlap{\sigma^{1,2}}} \\ \sigma(0) &&\underset{\sigma(0,2)}{\to}&& \sigma(2) } \right) = \left( \array{ && \sigma(1) \\ & {}^{\mathllap{\sigma(0,1)}}\nearrow & & \\ \sigma(0) } \right) - \left( \array{ && \\ & & & \\ \sigma(0) &\underset{\sigma(0,2)}{\to}& \sigma(2) } \right) + \left( \array{ && \sigma(1) \\ & & & \searrow^{\mathrlap{\sigma^{1,2}}} \\ && && \sigma(2) } \right) \,.

Hence the boundary of the boundary is:

σ =(( σ(1) σ(0,1) σ(0))( σ(0) σ(0,2) σ(2))+( σ(1) σ 1,2 σ(2))) =( σ(0))( σ(1) )( σ(0) )+( σ(2))+( σ(1) )( σ(2)) =0 \begin{aligned} \partial \partial \sigma &= \partial \left( \left( \array{ && \sigma(1) \\ & {}^{\mathllap{\sigma(0,1)}}\nearrow & & \\ \sigma(0) } \right) - \left( \array{ && \\ & & & \\ \sigma(0) &\underset{\sigma(0,2)}{\to}& \sigma(2) } \right) + \left( \array{ && \sigma(1) \\ & & & \searrow^{\mathrlap{\sigma^{1,2}}} \\ && && \sigma(2) } \right) \right) \\ & = \left( \array{ && \\ & & & \\ \sigma(0) } \right) - \left( \array{ && \sigma(1) \\ & & & \\ } \right) - \left( \array{ && \\ & & & \\ \sigma(0) && } \right) + \left( \array{ && \\ & & & \\ && \sigma(2) } \right) + \left( \array{ && \sigma(1) \\ & & & \\ && && } \right) - \left( \array{ && \\ & & & \\ && && \sigma(2) } \right) \\ & = 0 \end{aligned}
Definition

For SS a simplicial set, we call the collection

  1. of abelian groups of chains C n(S)[S n]C_n(S) \coloneqq \mathbb{Z}[S_n], prop. 3;

  2. and boundary homomorphisms n:C n+1(S)C n(X)\partial_n : C_{n+1}(S) \to C_n(X), def. 12

(for all nn \in \mathbb{N}) the alternating face map chain complex of SS:

C (S)=[ 2[S 2] 1[S 1] 0[S 0]]. C_\bullet(S) = [ \cdots \stackrel{\partial_2}{\to} \mathbb{Z}[S_2] \stackrel{\partial_1}{\to} \mathbb{Z}[S_1] \stackrel{\partial_0}{\to} \mathbb{Z}[S_0] ] \,.

Specifically for S=SingXS = Sing X we call this the singular chain complex of XX.

This motivates the general definition:

Definition

A chain complex of abelian groups C C_\bullet is a collection {C nAb} n\{C_n \in Ab\}_{n} of abelian groups together with group homomorphisms { n:C n+1C n}\{\partial_n : C_{n+1} \to C_n\} such that =0\partial \circ \partial = 0.

We turn to this definition in more detail in the below. The thrust of this construction lies in the fact that the chain complex C (SingX)C_\bullet(Sing X) remembers the abelianized fundamental group of XX, as well as aspects of the higher homotopy groups: in its chain homology.

Definition

For C (S)C_\bullet(S) a chain complex as in def. 13, and for nn \in \mathbb{N} we say

  • an nn-chain of the form σC(S) n\partial \sigma \in C(S)_n is an nn-boundary;

  • a chain σC n(S)\sigma \in C_n(S) is an nn-cycle if σ=0\partial \sigma = 0

    (every 0-chain is a 0-cycle).

By linearity of \partial the boundaries and cycles form abelian sub-groups of the group of chains, and we write

B nim( n)C n(S) B_n \coloneqq im(\partial_n) \subset C_n(S)

for the group of nn-boundaries, and

Z nker( n)C (S) Z_n \coloneqq ker(\partial_n) \subset C_(S)

for the group of nn-cycles.

Remark

This means that a singular chain is a cycle if the formal linear combination of the oriented boundaries of all its constituent singular simplices sums to 0.

Remark

More generally, for RR any unital ring one can form the degreewise free module R[SingX]R[Sing X] over RR. The corresponding homology is the singular homology with coefficients in RR, denoted H n(X,R)H_n(X,R). This generality we come to below in the next section.

Definition

For C (S)C_\bullet(S) a chain complex as in def. 13 and for nn \in \mathbb{N}, the degree-nn chain homology group H n(C(S))AbH_n(C(S)) \in Ab is the quotient group

H n(C(S))ker( n1)im( n)=Z nB n H_n(C(S)) \coloneqq \frac{ker(\partial_{n-1})}{im(\partial_n)} = \frac{Z_n}{B_n}

of the nn-cycles by the nn-boundaries – where for n=0n = 0 we declare that 10\partial_{-1} \coloneqq 0 and hence Z 0C 0Z_0 \coloneqq C_0.

Specifically, the chain homology of C (SingX)C_\bullet(Sing X) is called the singular homology of the topological space XX.

One usually writes H n(X,)H_n(X, \mathbb{Z}) or just H n(X)H_n(X) for the singular homology of XX in degree nn.

Remark

So H 0(C (S))=C 0(S)/im( 0)H_0(C_\bullet(S)) = C_0(S)/im(\partial_0).

Example

For XX a topological space we have that the degree-0 singular homology

H 0(X)[π 0(X)] H_0(X) \simeq \mathbb{Z}[\pi_0(X)]

is the free abelian group on the set of connected components of XX.

Example

For XX a connected, orientable manifold of dimension nn we have

H n(X). H_n(X) \simeq \mathbb{Z} \,.

The precise choice of this isomorphism is a choice of orientation on XX. With a choice of orientation, the element 11 \in \mathbb{Z} under this identification is called the fundamental class

[X]H n(X) [X] \in H_n(X)

of the manifold XX.

Definition

Given a continuous map f:XYf : X \to Y between topological spaces, and given nn \in \mathbb{N}, every singular nn-simplex σ:Δ nX\sigma : \Delta^n \to X in XX is sent to a singular nn-simplex

f *σ:Δ nσXfY f_* \sigma : \Delta^n \stackrel{\sigma}{\to} X \stackrel{f}{\to} Y

in YY. This is called the push-forward of σ\sigma along ff. Accordingly there is a push-forward map on groups of singular chains

(f *) n:C n(X)C n(Y). (f_*)_n : C_n(X) \to C_n(Y) \,.
Proposition

These push-forward maps make all diagrams of the form

C n+1(X) (f *) n+1 C n+1(Y) n X n Y C n(X) (f *) n C n(Y) \array{ C_{n+1}(X) &\stackrel{(f_*)_{n+1}}{\to}& C_{n+1}(Y) \\ \downarrow^{\mathrlap{\partial^X_n}} && \downarrow^{\mathrlap{\partial^Y_n}} \\ C_n(X) &\stackrel{(f_*)_n}{\to}& C_n(Y) }

commute.

Proof

It is in fact evident that push-forward yields a functor of singular simplicial complexes

f *:SingXSingY. f_* : Sing X \to Sing Y \,.

From this the statement follows since []:sSetsAb\mathbb{Z}[-] : sSet \to sAb is a functor.

Therefore we have an “abelianized analog” of the notion of topological space:

Definition

For C ,D C_\bullet, D_\bullet two chain complexes, def. 14, a homomorphism between them – called a chain map f :C D f_\bullet : C_\bullet \to D_\bullet – is for each nn \in \mathbb{N} a homomorphism f n:C nD nf_n : C_n \to D_n of abelian groups, such that f n n C= n Df n+1f_n \circ \partial^C_n = \partial^D_n \circ f_{n+1}:

n+1 C n+1 D C n+1 f n+1 D n+1 n C n D C n f n D n n1 C n1 D . \array{ \vdots && \vdots \\ \downarrow^{\mathrlap{\partial^C_{n+1}}} && \downarrow^{\mathrlap{\partial^D_{n+1}}} \\ C_{n+1} &\stackrel{f_{n+1}}{\to}& D_{n+1} \\ \downarrow^{\mathrlap{\partial^C_n}} && \downarrow^{\mathrlap{\partial^D_n}} \\ C_{n} &\stackrel{f_{n}}{\to}& D_{n} \\ \downarrow^{\mathrlap{\partial^C_{n-1}}} && \downarrow^{\mathrlap{\partial^D_{n-1}}} \\ \vdots && \vdots } \,.

Composition of such chain maps is given by degreewise composition of their components. Clearly, chain complexes with chain maps between them hence form a category – the category of chain complexes in abelian groups, – which we write

Ch (Ab))Cat. Ch_\bullet(Ab)) \in Cat \,.

Accordingly we have:

Proposition

Sending a topological space to its singular chain complex C (X)C_\bullet(X), def. 13, and a continuous map to its push-forward chain map, prop. 5, constitutes a functor

C ():TopCh (Ab) C_\bullet(-) : Top \to Ch_\bullet(Ab)

from the category Top of topological spaces and continuous maps, to the category of chain complexes.

In particular for each nn \in \mathbb{N} singular homology extends to a functor

H n():TopAb. H_n(-) : Top \to Ab \,.

We close this section by stating the basic properties of singular homology, which make precise the sense in which it is an abelian approximation to the homotopy type of XX. The proof of these statements requires some of the tools of homological algebra that we develop in the later chapters, as well as some tools in algebraic topology.

Proposition

If f:XYf : X \to Y is a continuous map between topological spaces which is a weak homotopy equivalence, then the induced morphism on singular homology groups

H n(f):H n(X)H n(Y) H_n(f) : H_n(X) \to H_n(Y)

is an isomorphism.

(A proof (via CW approximations) is spelled out for instance in (Hatcher, prop. 4.21)).

We therefore also have an “abelian analog” of weak homotopy equivalences:

Definition

For C ,D C_\bullet, D_\bullet two chain complexes, a chain map f :C D f_\bullet : C_\bullet \to D_\bullet is called a quasi-isomorphism if it induces isomorphisms on all homology groups:

f n:H n(C)H n(D). f_n : H_n(C) \stackrel{\simeq}{\to} H_n(D) \,.

In summary: chain homology sends weak homotopy equivalences to quasi-isomorphisms. Quasi-isomorphisms of chain complexes are the abelianized analog of weak homotopy equivalences of topological spaces.

In particular we have the analog of prop. 8:

Proposition

The relation “There exists a quasi-isomorphism from C C_\bullet to D D_\bullet.” is a reflexive and transitive relation, but it is not a symmetric relation.

Proof

Reflexivity and transitivity are evident. An explicit counter-example showing the non-symmetry is the chain map

0 2 0 0 0 /2 0 \array{ \cdots &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z} &\to& 0 &\to& \cdots \\ \cdots && \downarrow && \downarrow && \downarrow && \downarrow && \cdots \\ \cdots &\to& 0 &\to& 0 &\to& \mathbb{Z}/2\mathbb{Z} &\to& 0 &\to& \cdots }

from the chain complex concentrated on the morphism of multiplication by 2 on integers, to the chain complex concentrated on the cyclic group of order 2.

This clearly induces an isomorphism on all homology groups. But there is not even a non-zero chain map in the other direction, since there is no non-zero group homomorphism /2\mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}.

Accordingly, as for homotopy types of topological spaces, in homological algebra one regards two chain complexes C C_\bullet, D D_\bullet as essentially equivalent – “of the same weak homology type” – if there is a zigzag of quasi-isomorphisms

C D C_\bullet \leftarrow \to \leftarrow \cdots \to D_\bullet

between them. This is made precise by the central notion of the derived category of chain complexes. We turn to this below in section Derived categories and derived functors.

But quasi-isomorphisms are a little coarser than weak homotopy equivalences. The singular chain functor C ()C_\bullet(-) forgets some of the information in the homotopy types of topological spaces. The following series of statements characterizes to some extent what exactly is lost when passing to singular homology, and which information is in fact retained.

First we need a comparison map:

Definition

(Hurewicz homomorphism)

For (X,x)(X,x) a pointed topological space, the Hurewicz homomorphism is the function

Φ:π k(X,x)H k(X) \Phi : \pi_k(X,x) \to H_k(X)

from the kkth homotopy group of (X,x)(X,x) to the kkth singular homology group defined by sending

Φ:(f:S kX) f *[S k] \Phi : (f : S^k \to X)_{\sim} \mapsto f_*[S_k]

a representative singular kk-sphere ff in XX to the push-forward along ff of the fundamental class [S k]H k(S k)[S_k] \in H_k(S^k), example 6.

Proposition

For XX a topological space the Hurewicz homomorphism in degree 0 exhibits an isomorphism between the free abelian group [π 0(X)]\mathbb{Z}[\pi_0(X)] on the set of path connected components of XX and the degree-0 singular homlogy:

[π 0(X)]H 0(X). \mathbb{Z}[\pi_0(X)] \simeq H_0(X) \,.

Since a homotopy group in positive degree depends on the homotopy type of the connected component of the base point, while the singular homology does not depend on a basepoint, it is interesting to compare these groups only for the case that XX is connected.

Proposition

For XX a path-connected topological space the Hurewicz homomorphism in degree 1

Φ:π 1(X,x)H 1(X) \Phi : \pi_1(X,x) \to H_1(X)

is surjective. Its kernel is the commutator subgroup of π 1(X,x)\pi_1(X,x). Therefore it induces an isomorphism from the abelianization π 1(X,x) abπ 1(X,x)/[π 1,π 1]\pi_1(X,x)^{ab} \coloneqq \pi_1(X,x)/[\pi_1,\pi_1]:

π 1(X,x) abH 1(X). \pi_1(X,x)^{ab} \stackrel{\simeq}{\to} H_1(X) \,.

For higher connected XX we have the

Theorem

If XX is (n-1)-connected for n2n \geq 2 then

Φ:π n(X,x)H n(X) \Phi : \pi_n(X,x) \to H_n(X)

is an isomorphism.

This is known as the Hurewicz theorem.

This gives plenty of motivation for studying

  1. chain complexes

  2. chain homology

  3. quasi-isomorphism

of chain complexes. This is essentially what homological algebra is about. In the next section we start to develop these notions more systematically.

Motivation from first principles: Homotopy type-theory

(…type theoryidentity typehomotopy type theory…)

Model layer

A homotopy 0-type is equivalently just a set (an h-set). A homotopy 1-type is equivalently a groupoid. These we introduce and discuss below in

Another model for general homotopy types are simplicial sets and in particular the Kan complexes among them. These we introduce below in

But a simpler and more familiar structure turns out to be a model for the important subsector of “abelian” homotopy types, namely chain complexes. We recall fundamentals of these in

As a model for abelian homotopy theory, this is nothing but homological algebra. We present the key constructions of homological algebra from the “derived” perspective that makes them fit well into homotopy theory below in

The construction that embeds chain complexes into simplicial homotopy theory is the Dold-Kan correspondence and the Eilenberg-Zilber theorem. This we discuss below in

A standard model for general homotopy types is given by topological spaces. This model and its formalization via model category theory we introduce in

Similarly there is a model category that reflects the homotopy theory of Kan complexes, called simplicial homotopy theory. This we look into below in

Groupoids – Homotopy 1-types

Definition

A (small) groupoid 𝒢 \mathcal{G}_\bullet is

  • a pair of sets 𝒢 0Set\mathcal{G}_0 \in Set (the set of objects) and 𝒢 1Set\mathcal{G}_1 \in Set (the set of morphisms)

  • equipped with functions

    𝒢 1× 𝒢 0𝒢 1 𝒢 1 sit 𝒢 0, \array{ \mathcal{G}_1 \times_{\mathcal{G}_0} \mathcal{G}_1 &\stackrel{\circ}{\longrightarrow}& \mathcal{G}_1 & \stackrel{\overset{t}{\longrightarrow}}{\stackrel{\overset{i}{\leftarrow}}{\underset{s}{\longrightarrow}}}& \mathcal{G}_0 }\,,

    where the fiber product on the left is that over 𝒢 1t𝒢 0s𝒢 1\mathcal{G}_1 \stackrel{t}{\to} \mathcal{G}_0 \stackrel{s}{\leftarrow} \mathcal{G}_1,

such that

  • ii takes values in endomorphisms;

    ti=si=id 𝒢 0, t \circ i = s \circ i = id_{\mathcal{G}_0}, \;\;\;
  • \circ defines a partial composition operation which is associative and unital for i(𝒢 0)i(\mathcal{G}_0) the identities; in particular

    s(gf)=s(f)s (g \circ f) = s(f) and t(gf)=t(g)t (g \circ f) = t(g);

  • every morphism has an inverse under this composition.

Remark

This data is visualized as follows. The set of morphisms is

𝒢 1={ϕ 0kϕ 1} \mathcal{G}_1 = \left\{ \phi_0 \stackrel{k}{\to} \phi_1 \right\}

and the set of pairs of composable morphisms is

𝒢 2𝒢 1×𝒢 0𝒢 1={ ϕ 1 k 1 k 2 ϕ 0 k 2k 1 ϕ 2}. \mathcal{G}_2 \coloneqq \mathcal{G}_1 \underset{\mathcal{G}_0}{\times} \mathcal{G}_1 = \left\{ \array{ && \phi_1 \\ & {}^{\mathllap{k_1}}\nearrow && \searrow^{\mathrlap{k_2}} \\ \phi_0 && \stackrel{k_2 \circ k_1}{\to} && \phi_2 } \right\} \,.

The functions p 1,p 2,:𝒢 2𝒢 1p_1, p_2, \circ \colon \mathcal{G}_2 \to \mathcal{G}_1 are those which send, respectively, these triangular diagrams to the left morphism, or the right morphism, or the bottom morphism.

Example

For XX a set, it becomes a groupoid by taking XX to be the set of objects and adding only precisely the identity morphism from each object to itself

(XidididX). \left( X \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longrightarrow}} { \overset{id}{\longleftarrow} } X \right) \,.
Example

For GG a group, its delooping groupoid (BG) (\mathbf{B}G)_\bullet has

  • (BG) 0=*(\mathbf{B}G)_0 = \ast;

  • (BG) 1=G(\mathbf{B}G)_1 = G.

For GG and KK two groups, group homomorphisms f:GKf \colon G \to K are in natural bijection with groupoid homomorphisms

(Bf) :(BG) (BK) . (\mathbf{B}f)_\bullet \;\colon\; (\mathbf{B}G)_\bullet \to (\mathbf{B}K)_\bullet \,.

In particular a group character c:GU(1)c \colon G \to U(1) is equivalently a groupoid homomorphism

(Bc) :(BG) (BU(1)) . (\mathbf{B}c)_\bullet \;\colon\; (\mathbf{B}G)_\bullet \to (\mathbf{B}U(1))_\bullet \,.

Here, for the time being, all groups are discrete groups. Since the circle group U(1)U(1) also has a standard structure of a Lie group, and since later for the discussion of Chern-Simons type theories this will be relevant, we will write from now on

U(1)Grp \flat U(1) \in Grp

to mean explicitly the discrete group underlying the circle group. (Here “\flat” denotes the “flat modality”.)

Example

For XX a set, GG a discrete group and ρ:X×GX\rho \colon X \times G \to X an action of GG on XX (a permutation representation), the action groupoid or homotopy quotient of XX by GG is the groupoid

X// ρG=(X×Gp 1ρX) X//_\rho G = \left( X \times G \stackrel{\overset{\rho}{\longrightarrow}}{\underset{p_1}{\longrightarrow}} X \right)

with composition induced by the product in GG. Hence this is the groupoid whose objects are the elements of XX, and where morphisms are of the form

x 1gx 2=ρ(x 1)(g) x_1 \stackrel{g}{\to} x_2 = \rho(x_1)(g)

for x 1,x 2Xx_1, x_2 \in X, gGg \in G.

As an important special case we have:

Example

For GG a discrete group and ρ\rho the trivial action of GG on the point *\ast (the singleton set), the corresponding action groupoid according to def. 9 is the delooping groupoid of GG according to def. 8:

(*//G) =(BG) . (\ast //G)_\bullet = (\mathbf{B}G)_\bullet \,.

Another canonical action is the action of GG on itself by right multiplication. The corresponding action groupoid we write

(EG) G//G. (\mathbf{E}G)_\bullet \coloneqq G//G \,.

The constant map G*G \to \ast induces a canonical morphism

G//G EG *//G BG. \array{ G//G & \simeq & \mathbf{E}G \\ \downarrow && \downarrow \\ \ast //G & \simeq & \mathbf{B}G } \,.

This is known as the GG-universal principal bundle. See below in 16 for more on this.

Example

The interval II is the groupoid with

  • I 0={a,b}I_0 = \{a,b\};
  • I 1={id a,id b,ab}I_1 = \{\mathrm{id}_a, \mathrm{id}_b, a \to b \}.
Example

For Σ\Sigma a topological space, its fundamental groupoid Π 1(Σ)\Pi_1(\Sigma) is

  • Π 1(Σ) 0=\Pi_1(\Sigma)_0 = points in XX;
  • Π 1(Σ) 1=\Pi_1(\Sigma)_1 = continuous paths in XX modulo homotopy that leaves the endpoints fixed.
Example

For 𝒢 \mathcal{G}_\bullet any groupoid, there is the path space groupoid 𝒢 I\mathcal{G}^I_\bullet with

  • 𝒢 0 I=𝒢 1={ϕ 0 k ϕ 1}\mathcal{G}^I_0 = \mathcal{G}_1 = \left\{ \array{ \phi_0 \\ \downarrow^{\mathrlap{k}} \\ \phi_1 } \right\};

  • 𝒢 1 I=\mathcal{G}^I_1 = commuting squares in 𝒢 \mathcal{G}_\bullet = {ϕ 0 h 0 ϕ˜ 0 k k˜ ϕ 1 h 1 ϕ˜ 1}. \left\{ \array{ \phi_0 &\stackrel{h_0}{\to}& \tilde \phi_0 \\ {}^{\mathllap{k}}\downarrow && \downarrow^{\mathrlap{\tilde k}} \\ \phi_1 &\stackrel{h_1}{\to}& \tilde \phi_1 } \right\} \,.

This comes with two canonical homomorphisms

𝒢 Iev 0ev 1𝒢 \mathcal{G}^I_\bullet \stackrel{\overset{ev_1}{\longrightarrow}}{\underset{ev_0}{\longrightarrow}} \mathcal{G}_\bullet

which are given by endpoint evaluation, hence which send such a commuting square to either its top or its bottom hirizontal component.

Definition

For f ,g :𝒢 𝒦 f_\bullet, g_\bullet : \mathcal{G}_\bullet \to \mathcal{K}_\bullet two morphisms between groupoids, a homotopy fgf \Rightarrow g (a natural transformation) is a homomorphism of the form η :𝒢 𝒦 I\eta_\bullet : \mathcal{G}_\bullet \to \mathcal{K}^I_\bullet (with codomain the path space object of 𝒦 \mathcal{K}_\bullet as in example 13) such that it fits into the diagram as depicted here on the right:

f 𝒢 η 𝒦 g 𝒦 f (ev 1) 𝒢 η 𝒦 I g (ev 0) 𝒦. \array{ & \nearrow \searrow^{\mathrlap{f_\bullet}} \\ \mathcal{G} &\Downarrow^{\mathrlap{\eta}}& \mathcal{K} \\ & \searrow \nearrow_{\mathrlap{g_\bullet}} } \;\;\;\; \coloneqq \;\;\;\; \array{ && \mathcal{K}_\bullet \\ & {}^{\mathllap{f_\bullet}}\nearrow & \uparrow^{\mathrlap{(ev_1)_\bullet}} \\ \mathcal{G}_\bullet &\stackrel{\eta_\bullet}{\to}& \mathcal{K}^I_\bullet \\ & {}_{\mathllap{g_\bullet}}\searrow & \downarrow^{\mathrlap{(ev_0)_\bullet}} \\ && \mathcal{K} } \,.
Definition (Notation)

Here and in the following, the convention is that we write

  • 𝒢 \mathcal{G}_\bullet (with the subscript decoration) when we regard groupoids with just homomorphisms (functors) between them,

  • 𝒢\mathcal{G} (without the subscript decoration) when we regard groupoids with homomorphisms (functors) between them and homotopies (natural transformations) between these

    f X Y g. \array{ & \nearrow \searrow^{\mathrlap{f}} \\ X &\Downarrow& Y \\ & \searrow \nearrow_{g} } \,.

The unbulleted version of groupoids are also called homotopy 1-types (or often just their homotopy-equivalence classes are called this way.) Below we generalize this to arbitrary homotopy types (def. 107).

Example

For X,YX,Y two groupoids, the mapping groupoid [X,Y][X,Y] or Y XY^X is

  • [X,Y] 0=[X,Y]_0 = homomorphisms XYX \to Y;
  • [X,Y] 1=[X,Y]_1 = homotopies between such.
Definition

A (homotopy-) equivalence of groupoids is a morphism 𝒢𝒦\mathcal{G} \to \mathcal{K} which has a left and right inverse up to homotopy.

Example

The map

BΠ(S 1) \mathbf{B}\mathbb{Z} \stackrel{}{\to} \Pi(S^1)

which picks any point and sends nn \in \mathbb{Z} to the loop based at that point which winds around nn times, is an equivalence of groupoids.

Proposition

Assuming the axiom of choice in the ambient set theory, every groupoid is equivalent to a disjoint union of delooping groupoids, example 8 – a skeleton.

Remark

The statement of prop. 11 becomes false as when we pass to groupoids that are equipped with geometric structure. This is the reason why for discrete geometry all Chern-Simons-type field theories (namely Dijkgraaf-Witten theory-type theories) fundamentally involve just groups (and higher groups), while for nontrivial geometry there are genuine groupoid theories, for instance the AKSZ sigma-models. But even so, Dijkgraaf-Witten theory is usefully discussed in terms of groupoid technology, in particular since the choice of equivalence in prop. 11 is not canonical.

Definition

Given two morphisms of groupoids XfBgYX \stackrel{f}{\leftarrow} B \stackrel{g}{\to} Y their homotopy fiber product

X×BY X f Y g B \array{ X \underset{B}{\times} Y &\stackrel{}{\to}& X \\ \downarrow &\swArrow& \downarrow^{\mathrlap{f}} \\ Y &\underset{g}{\to}& B }

is the limit cone

X ×B B I×B Y X f B I (ev 0) B (ev 1) Y g B , \array{ X_\bullet \underset{B_\bullet}{\times} B^I_\bullet \underset{B_\bullet}{\times} Y_\bullet &\to& &\to& X_\bullet \\ \downarrow && && \downarrow^{\mathrlap{f_\bullet}} \\ && B^I_\bullet &\underset{(ev_0)_\bullet}{\to}& B_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ Y_\bullet &\underset{g_\bullet}{\to}& B_\bullet } \,,

hence the ordinary iterated fiber product over the path space groupoid, as indicated.

Remark

An ordinary fiber product X ×B Y X_\bullet \underset{B_\bullet}{\times}Y_\bullet of groupoids is given simply by the fiber product of the underlying sets of objects and morphisms:

(X ×B Y ) i=X i×B iY i. (X_\bullet \underset{B_\bullet}{\times}Y_\bullet)_i = X_i \underset{B_i}{\times} Y_i \,.
Example

For XX a groupoid, GG a group and XBGX \to \mathbf{B}G a map into its delooping, the pullback PXP \to X of the GG-universal principal bundle of example 10 is equivalently the homotopy fiber product of XX with the point over matrhbfBG\matrhbf{B}G:

PX×BG*. P \simeq X \underset{\mathbf{B}G}{\times} \ast \,.

Namely both squares in the following diagram are pullback squares

P EG * (BG) I (ev 0) (BG) (ev 1) X (BG) . \array{ P &\to& \mathbf{E}G &\to& \ast_\bullet \\ \downarrow && && \downarrow^{\mathrlap{}} \\ && (\mathbf{B}G)^I_\bullet &\underset{(ev_0)_\bullet}{\to}& (\mathbf{B}G)_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ X_\bullet &\underset{}{\to}& (\mathbf{B}G)_\bullet } \,.

(This is the first example of the more general phenomenon of universal principal infinity-bundles.)

Example

For XX a groupoid and *X\ast \to X a point in it, we call

ΩX*×X* \Omega X \coloneqq \ast \underset{X}{\times} \ast

the loop space groupoid of XX.

For GG a group and BG\mathbf{B}G its delooping groupoid from example 8, we have

GΩBG=*×BG*. G \simeq \Omega \mathbf{B}G = \ast \underset{\mathbf{B}G}{\times} \ast \,.

Hence GG is the loop space object of its own delooping, as it should be.

Proof

We are to compute the ordinary limiting cone *×BG (BG I) ×BG *\ast \underset{\mathbf{B}G_\bullet}{\times} (\mathbf{B}G^I)_\bullet \underset{\mathbf{B}G_\bullet}{\times} \ast in

* (BG) I (ev 0) BG (ev 1) * BG , \array{ &\to& &\to& \ast \\ \downarrow && && \downarrow^{\mathrlap{}} \\ && (\mathbf{B}G)^I_\bullet &\underset{(ev_0)_\bullet}{\to}& \mathbf{B}G_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ \ast &\underset{}{\to}& \mathbf{B}G_\bullet } \,,

In the middle we have the groupoid (BG) I(\mathbf{B}G)^I_\bullet whose objects are elements of GG and whose morphisms starting at some element are labeled by pairs of elements h 1,h 2Gh_1, h_2 \in G and end at h 1gh 2h_1 \cdot g \cdot h_2. Using remark 9 the limiting cone is seen to precisely pick those morphisms in (BG ) I(\mathbf{B}G_\bullet)^I_\bullet such that these two elements are constant on the neutral element h 1=h 2=e=id *h_1 = h_2 = e = id_{\ast}, hence it produces just the elements of GG regarded as a groupoid with only identity morphisms, as in example 7.

Proposition

The free loop space object is

[Π(S 1),X]X×[Π(S 0),X]X [\Pi(S^1), X] \simeq X \underset{[\Pi(S^0), X]}{\times}X
Proof

Notice that Π 1(S 0)**\Pi_1(S^0) \simeq \ast \coprod \ast. Therefore the path space object [Π(S 0),X ] I[\Pi(S^0), X_\bullet]^I_\bullet has

  • objects are pairs of morphisms in X X_\bullet;

  • morphisms are commuting squares of such.

Now the fiber product in def. 25 picks in there those pairs of morphisms for which both start at the same object, and both end at the same object. Therefore X ×[Π(S 0),X ] [Π(S 0),X ] I×[Π(S 0),X ] XX_\bullet \underset{[\Pi(S^0), X_\bullet]_\bullet}{\times} [\Pi(S^0), X_\bullet]^I_\bullet \underset{[\Pi(S^0), X_\bullet]_\bullet}{\times} X is the groupoid whose

  • objects are diagrams in X X_\bullet of the form

    x 0 x 1 \array{ & \nearrow \searrow \\ x_0 && x_1 \\ & \searrow \nearrow }
  • morphism are cylinder-diagrams over these.

One finds along the lines of example 15 that this is equivalent to maps from Π 1(S 1)\Pi_1(S^1) into X X_\bullet and homotopies between these.

Remark

Even though all these models of the circle Π 1(S 1)\Pi_1(S^1) are equivalent, below the special appearance of the circle in the proof of prop. 12 as the combination of two semi-circles will be important for the following proofs. As we see in a moment, this is the natural way in which the circle appears as the composition of an evaluation map with a coevaluation map.

Example

For GG a discrete group, the free loop space object of its delooping BG\mathbf{B}G is G// adGG//_{ad} G, the action groupoid, def. 9, of the adjoint action of GG on itself:

[Π(S 1),BG]G// adG. [\Pi(S^1), \mathbf{B}G] \simeq G//_{ad} G \,.
Example

For an abelian group such as U(1)\flat U(1) we have

[Π(S 1),BU(1)]U(1)// adU(1)(U(1))×(BU(1)). [\Pi(S^1), \mathbf{B}\flat U(1)] \simeq \flat U(1)//_{ad} \flat U(1) \simeq (\flat U(1)) \times (\mathbf{B}\flat U(1)) \,.
Example

Let c:GU(1)c \colon G \to \flat U(1) be a group homomorphism, hence a group character. By example 8 this has a delooping to a groupoid homomorphism

Bc:BGBU(1). \mathbf{B}c \;\colon\; \mathbf{B}G \to \mathbf{B}\flat U(1) \,.

Under the free loop space object construction this becomes

[Π(S 1),Bc]:[Π(S 1),BG][Π(S 1),BU(1)] [\Pi(S^1), \mathbf{B}c] \;\colon\; [\Pi(S^1), \mathbf{B}G] \to [\Pi(S^1), \mathbf{B}\flat U(1)]

hence

[Π(S 1),Bc]:G// adGU(1)×BU(1). [\Pi(S^1), \mathbf{B}c] \;\colon\; G//_{ad}G \to \flat U(1) \times \mathbf{B}U(1) \,.

So by postcomposing with the projection on the first factor we recover from the general homotopy theory of groupoids the statement that a group character is a class function on conjugacy classes:

[Π(S 1),Bc]:G// adGU(1). [\Pi(S^1), \mathbf{B}c] \;\colon\; G//_{ad}G \to U(1) \,.

Chain complexes – Abelian homotopy types

Categories of chain complexes

From the traditional concept of singular cohomology the idea of the chain complex of formal linear combinations of simplices in a topological space is familar. Here we discuss such chain complexes in their own right in a bit more depth.

Often a singular chain is taken to be a formal sum of singular simplices with coefficients in the abelian group of integers \mathbb{Z}. It is just as straightforward, natural and useful to allow the coefficients to be an arbitrary abelian group AA, or in fact to be a module over a ring.

So we start by developing a bit of the theory of abelian groups, rings and modules.

Definition

Write Ab \in Cat for the category of abelian groups and group homomorphisms between them:

  • an object is a group AA such that for all elements a 1,a 2Aa_1, a_2 \in A we have that the group product of a 1a_1 with a 2a_2 is the same as that of a 2a_2 with a 1a_1, which we write a 1+a 2Aa_1 + a_2 \in A (and the neutral element is denoted by 0A0 \in A);

  • a morphism ϕ:A 1A 2\phi : A_1 \to A_2 is a group homomorphism, hence a function of the underlying sets, such that for all elements as above ϕ(a 1+a 2)=ϕ(a 1)+ϕ(a 2)\phi(a_1 + a_2) = \phi(a_1) + \phi(a_2).

Among the basic constructions that produce new abelian groups from given ones are the tensor product of abelian groups and the direct sum of abelian groups. These we discuss now.

Definition

For AA, BB and CC abelian groups and A×BA \times B the cartesian product group, a bilinear map

f:A×BC f : A \times B \to C

is a function of the underlying sets which is linear – hence is a group homomorphism – in each argument separately.

Remark

In terms of elements this means that a bilinear map f:A×BCf : A \times B \to C is a function of sets that satisfies for all elements a 1,a 2Aa_1, a_2 \in A and b 1,b 2Bb_1, b_2 \in B the two relations

f(a 1+a 2,b 1)=f(a 1,b 1)+f(a 2,b 1) f(a_1 + a_2, b_1) = f(a_1,b_1) + f(a_2, b_1)

and

f(a 1,b 1+b 2)=f(a 1,b 1)+f(a 1,b 2). f(a_1, b_1 + b_2) = f(a_1, b_1) + f(a_1, b_2) \,.

Notice that this is not a group homomorphism out of the product group. The product group A×BA \times B is the group whose elements are pairs (a,b)(a,b) with aAa \in A and bBb \in B, and whose group operation is

(a 1,b 1)+(a 2,b 2)=(a 1+a 2,b 1+b 2). (a_1, b_1) + (a_2, b_2) = (a_1 + a_2 \;,\; b_1 + b_2) \,.

A group homomorphism

ϕ:A×BC \phi : A \times B \to C

hence satisfies

ϕ(a 1+a 2,b 1+b 2)=ϕ(a 1,b 1)+ϕ(a 2,b 2) \phi( a_1+a_2, b_1 + b_2 ) = \phi(a_1,b_1) + \phi(a_2, b_2)

and hence in particular

ϕ(a 1+a 2,b 1)=ϕ(a 1,b 1)+ϕ(a 2,0) \phi( a_1+a_2, b_1 ) = \phi(a_1,b_1) + \phi(a_2, 0)
ϕ(a 1,b 1+b 2)=ϕ(a 1,b 1)+ϕ(0,b 2) \phi( a_1, b_1 + b_2 ) = \phi(a_1,b_1) + \phi(0, b_2)

which is (in general) different from the behaviour of a bilinear map.

Definition

For A,BA, B two abelian groups, their tensor product of abelian groups is the abelian group ABA \otimes B which is the quotient group of the free group on the product (direct sum) A×BA \times B by the relations

  • (a 1,b)+(a 2,b)(a 1+a 2,b)(a_1,b)+(a_2,b)\sim (a_1+a_2,b)

  • (a,b 1)+(a,b 2)(a,b 1+b 2)(a,b_1)+(a,b_2)\sim (a,b_1+b_2)

for all a,a 1,a 2Aa, a_1, a_2 \in A and b,b 1,b 2Bb, b_1, b_2 \in B.

In words: it is the group whose elements are presented by pairs of elements in AA and BB and such that the group operation for one argument fixed is that of the other group in the other argument.

Remark

There is a canonical function of the underlying sets

A×BAB. A \times B \stackrel{\otimes}{\to} A \otimes B \,.

On elements this sends (a,b)(a,b) to the equivalence class that it represents under the above equivalence relations.

Proposition

A function of underlying sets f:A×BCf : A \times B \to C is a bilinear function precisely if it factors by the morphism of 12 through a group homomorphism ϕ:ABC\phi : A \otimes B \to C out of the tensor product:

f:A×BABϕC. f : A \times B \stackrel{\otimes}{\to} A \otimes B \stackrel{\phi}{\to} C \,.
Proposition

Equipped with the tensor product \otimes of def. 28 Ab becomes a monoidal category.

The unit object in (Ab,)(Ab, \otimes) is the additive group of integers \mathbb{Z}.

This means:

  1. forming the tensor product is a functor in each argument

    A():AbAb, A \otimes (-) : Ab \to Ab \,,
  2. there is an associativity natural isomorphism (AB)CA(BC)(A \otimes B) \otimes C \stackrel{\simeq}{\to} A \otimes (B \otimes C) which is “coherent” in the sense that all possible ways of using it to rebracket a given expression are equal.

  3. There is a unit natural isomorphism AAA \otimes \mathbb{Z} \stackrel{\simeq}{\to} A which is compatible with the asscociativity isomorphism in the evident sense.

Proof

To see that \mathbb{Z} is the unit object, consider for any abelian group AA the map

AA A \otimes \mathbb{Z} \to A

which sends for nn \in \mathbb{N} \subset \mathbb{Z}

(a,n)naa+a++a nsummands. (a, n) \mapsto n \cdot a \coloneqq \underbrace{a + a + \cdots + a}_{n\;summands} \,.

Due to the quotient relation defining the tensor product, the element on the left is also equal to

(a,n)=(a,1+1+1 nsummands)=(a,1)+(a,1)++(a,1) nsummands. (a, n) = (a, \underbrace{1 + 1 \cdots + 1}_{n\; summands}) = \underbrace{ (a,1) + (a,1) + \cdots + (a,1) }_{n\; summands} \,.

This shows that AAA \otimes \mathbb{Z} \to A is in fact an isomorphism.

The other properties are similarly direct to check.

We see simple but useful examples of tensor products of abelian groups put to work below in the context of example 39 and then in many of the applications to follow. An elementary but not entirely trivial example that may help to illustrate the nature of the tensor product is the following.

Example

For a,ba,b \in \mathbb{N} and positive, we have

a b LCM(a,b), \mathbb{Z}_a \otimes \mathbb{Z}_b \simeq \mathbb{Z}_{LCM(a,b)} \,,

where LCM(,)LCM(-,-) denotes the least common multiple.

Definition

Let II \in Set be a set and {A i} iI\{A_i\}_{i \in I} an II-indexed family of abelian groups. The direct sum iIAb\oplus_{i \in I} \in Ab is the coproduct of these objects in Ab.

This means: the direct sum is an abelian group equipped with a collection of homomorphisms

A j A k ι j ι k iIA i, \array{ A_j &&\cdots && A_k \\ & {}_{\mathllap{\iota_j} }\searrow &\cdots& \swarrow_{\mathrlap{\iota_{k}}} \\ && \oplus_{i \in I} A_i } \,,

which is characterized (up to unique isomorphism) by the following universal property: for every other abelian group KK equipped with maps

A j A k f j f k K \array{ A_j &&\cdots && A_k \\ & {}_{\mathllap{f_j} }\searrow &\cdots& \swarrow_{\mathrlap{f_{k}}} \\ && K }

there is a unique homomorphism ϕ: iIA iK\phi : \oplus_{i \in I} A_i \to K such that f i=ϕι i f_i = \phi \circ \iota_i for all iIi \in I.

Explicitly in terms of elements we have:

Proposition

The direct sum iIA i\oplus_{i \in I} A_i is the abelian group whose ements are formal sums

a 1+a 2++a k a_1 + a_2 + \cdots + a_k

of finitely many elements of the {A i}\{A_i\}, with addition given by componentwise addition in the corresponding A iA_i.

Example

If each A i=A_i = \mathbb{Z}, then the direct sum is again the free abelian group on II

iI[I]. \oplus_{i \in I} \mathbb{Z} \simeq \mathbb{Z}[I] \,.
Proposition

The tensor product of abelian groups distributes over arbitrary direct sums:

A( iIB i) iIAB o. A \otimes (\oplus_{i \in I} B_i) \simeq \oplus_{i \in I} A \otimes B_o \,.
Example

For ISetI \in Set and AAbA \in Ab, the direct sum of |I|{\vert I\vert} copies of AA with itself is equivalently the tensor product of abelian groups of the free abelian group on II with AA:

iIA( iI)A([I])A. \oplus_{i \in I} A \simeq (\oplus_{i \in I} \mathbb{Z}) \otimes A \simeq (\mathbb{Z}[I]) \otimes A \,.
Remark

Together, tensor product and direct sum of abelian groups make Ab into what is called a bimonoidal category.

This now gives us enough structure to define rings and consider basic examples of their modules.

Definition

A ring (unital and not-necessarily commutative) is an abelian group RR equipped with

  1. an element 1R1 \in R

  2. a bilinear operation, hence a group homomorphism

    :RRR \cdot : R \otimes R \to R

    out of the tensor product of abelian groups,

such that this is associative and unital with respect to 1.

Remark

The fact that the product is a bilinear map is the distributivity law: for all r,r 1,r 2Rr, r_1, r_2 \in R we have

r(r 1+r 2)=rr 1+rr 2 r \cdot (r_1 + r_2) = r \cdot r_1 + r \cdot r_2

and

(r 1+r 2)r=(r 1+r 2)r. (r_1 + r_2) \cdot r = (r_1 + r_2) \cdot r \,.
Example
  • The integers \mathbb{Z} are a ring under the standard addition and multiplication operation.

  • For each nn, this induces a ring structure on the cyclic group n\mathbb{Z}_n, given by operations in \mathbb{Z} modulo nn.

  • The rational numbers \mathbb{Q}, real numbers \mathbb{R} and complex numbers are rings under their standard operations (in fact these are even fields).

Example

For RR a ring, the polynomials

r 0+r 1x+r 2x 2++r nx n r_0 + r_1 x + r_2 x^2 + \cdots + r_n x^n

(for arbitrary nn \in\mathbb{N}) in a variable xx with coefficients in RR form another ring, the polynomial ring denoted R[x]R[x]. This is the free RR-associative algebra on a single generator xx.

Example

For RR a ring and nn \in \mathbb{N}, the set M(n,R)M(n,R) of n×nn \times n-matrices with coefficients in RR is a ring under elementwise addition and matrix multiplication.

Example

For XX a topological space, the set of continuous functions C(X,)C(X,\mathbb{R}) or C(X,)C(X,\mathbb{C}) with values in the real numbers or complex numbers is a ring under pointwise (points in XX) addition and multiplication.

Just as an outlook and a suggestion for how to think geometrically of the objects appearing here, we mention the following.

Remark

The Gelfand duality theorem says that if one remembers certain extra structure on the rings of functions C(X,)C(X, \mathbb{C}) in example 26 – called the structure of a C-star algebra, then this construction

C(,):TopC *Alg opforgetRing op C(-,\mathbb{C}) : Top \stackrel{\simeq}{\to} C^\ast Alg^{op} \stackrel{forget}{\to} Ring^op

is an equivalence of categories between that of topological spaces, and the opposite category of C *C^\ast-algebras. Together with remark 16 further below this provides a useful dual geometric way of thinking about the theory of modules.

From now on and throughout, we take RR to be a commutative ring.

Definition

A module NN over a ring RR is

  1. an object NN \in Ab, hence an abelian group;

  2. equipped with a morphism

    α:RNN \alpha : R \otimes N \to N

    in Ab; hence a function of the underlying sets that sends elements

    (r,n)rnα(r,n) (r,n) \mapsto r n \coloneqq \alpha(r,n)

    and which is a bilinear function in that it satisfies

    (r,n 1+n 2)rn 1+rn 2 (r, n_1 + n_2) \mapsto r n_1 + r n_2

    and

    (r 1+r 2,n)r 1n+r 2n (r_1 + r_2, n) \mapsto r_1 n + r_2 n

    for all r,r 1,r 2Rr, r_1, r_2 \in R and n,n 1,n 2Nn,n_1, n_2 \in N;

  3. such that the diagram

    RRN RId N RN Id Rα α RN N \array{ R \otimes R \otimes N &\stackrel{\cdot_R \otimes Id_N}{\to}& R \otimes N \\ {}^{\mathllap{Id_R \otimes \alpha}}\downarrow && \downarrow^{\mathrlap{\alpha}} \\ R \otimes N &\to& N }

    commutes in Ab, which means that for all elements as before we have

    (r 1r 2)n=r 1(r 2n). (r_1 \cdot r_2) n = r_1 (r_2 n) \,.
  4. such that the diagram

    1N 1id N RN α N \array{ 1 \otimes N &&\stackrel{1 \otimes id_N}{\to}&& R \otimes N \\ & \searrow && \swarrow_{\mathrlap{\alpha}} \\ && N }

    commutes, which means that on elements as above

    1n=n. 1 \cdot n = n \,.
Example

The ring RR is naturally a module over itself, by regarding its multiplication map RRRR \otimes R \to R as a module action RNNR \otimes N \to N with NRN \coloneqq R.

Example

More generally, for nn \in \mathbb{N} the nn-fold direct sum of the abelian group underlying RR is naturally a module over RR

R nR nRRR nsummands. R^n \coloneqq R^{\oplus_n} \coloneqq \underbrace{R \oplus R \oplus \cdots \oplus R}_{n\;summands} \,.

The module action is componentwise:

r(r 1,r 2,,r n)=(rr 1,rr 2,rr n). r \cdot (r_1, r_2, \cdots, r_n) = (r \cdot r_1, r\cdot r_2, \cdot r \cdot r_n) \,.
Example

Even more generally, for II \in Set any set, the direct sum iIR\oplus_{i \in I} R is an RR-module.

This is the free module (over RR) on the set SS.

The set II serves as the basis of a free module: a general element v iRv \in \oplus_i R is a formal linear combination of elements of II with coefficients in RR.

For special cases of the ring RR, the notion of RR-module is equivalent to other notions:

Example

For R=R = \mathbb{Z} the integers, an RR-module is equivalently just an abelian group.

Example

For R=kR = k a field, an RR-module is equivalently a vector space over kk.

Every finitely-generated free kk-module is a free module, hence every finite dimensional vector space has a basis. For infinite dimensions this is true if the axiom of choice holds.

Example

For NN a module and {n i} iI\{n_i\}_{i \in I} a set of elements, the linear span

n i iIN, \langle n_i\rangle_{i \in I} \hookrightarrow N \,,

(hence the completion of this set under addition in NN and multiplication by RR) is a submodule of NN.

Example

Consider example 32 for the case that the module is N=RN = R, the ring itself, as in example 27. Then a submodule is equivalently (called) an ideal of RR.

Definition

Write RRMod for the category or RR-modules and RR-linear maps between them.

Example

For R=R = \mathbb{Z} we have ModAb\mathbb{Z} Mod \simeq Ab.

Example

Let XX be a topological space and let

RC(X,) R \coloneqq C(X,\mathbb{C})

be the ring of continuous functions on XX with values in the complex numbers.

Given a complex vector bundle EXE \to X on XX, write Γ(E)\Gamma(E) for its set of continuous sections. Since for each point xXx \in X the fiber E xE_x of EE over xx is a \mathbb{C}-module (by example 31), Γ(X)\Gamma(X) is a C(X,)C(X,\mathbb{C})-module.

Just as an outlook and a suggestion for how to think of modules geometrically, we mention the following.

Remark

The Serre-Swan theorem says that if XX is Hausdorff and compact with ring of functions C(X,)C(X,\mathbb{C}) – as in remark 15 above – then Γ(X)\Gamma(X) is a projective C(X,)C(X,\mathbb{C})-module and indeed there is an equivalence of categories between projective C(X,)C(X,\mathbb{C})-modules and complex vector bundles over XX. (We introduce the notion of projective modules below in Derived categories and derived functors.)

We now discuss a bunch of properties of the category RRMod which together will show that there is a reasonable concept of chain complexes of RR-modules, in generalization of how there is a good concept of chain complexes of abelian groups. In a more abstract category theoretical context than we invoke here, all of the following properties are summarized in the following statement.

Theorem

Let RR be a commutative ring. Then RModR Mod is an abelian category.

But for the moment we ignore this further abstraction and just consider the following list of properties.

Definition

An object in a category which is both an initial object and a terminal object is called a zero object.

Remark

This means that 0𝒞0 \in \mathcal{C} is a zero object precisely if for every other object AA there is a unique morphism A0A \to 0 to the zero object as well as a unique morphism 0A0 \to A from the zero object.

Proposition

The trivial group is a zero object in Ab.

The trivial module is a zero object in RRMod.

Proof

Clearly the 0-module 00 is a terminal object, since every morphism N0N \to 0 has to send all elements of NN to the unique element of 00, and every such morphism is a homomorphism. Also, 0 is an initial object because a morphism 0N0 \to N always exists and is unique, as it has to send the unique element of 0, which is the neutral element, to the neutral element of NN.

Definition

In a category with an initial object 00 and pullbacks, the kernel ker(f)ker(f) of a morphism f:ABf: A \to B is the pullback ker(f)Aker(f) \to A along ff of the unique morphism 0B0 \to B

ker(f) 0 p A f B. \array{ ker(f) &\to& 0 \\ {}^{\mathllap{p}}\downarrow && \downarrow \\ A &\stackrel{f}{\to}& B } \,.
Remark

More explicitly, this characterizes the object ker(f)ker(f) as the object (unique up to unique isomorphism) that satisfies the following universal property:

for every object CC and every morphism h:CAh : C \to A such that fh=0f\circ h = 0 is the zero morphism, there is a unique morphism ϕ:Cker(f)\phi : C \to ker(f) such that h=pϕh = p\circ \phi.

Example

In the category Ab of abelian groups, the kernel of a group homomorphism f:ABf : A \to B is the subgroup of AA on the set f 1(0)f^{-1}(0) of elements of AA that are sent to the zero-element of BB.

Example

More generally, for RR any ring, this is true in RRMod: the kernel of a morphism of modules is the preimage of the zero-element at the level of the underlying sets, equipped with the unique sub-module structure on that set.

Definition

In a category with zero object, the cokernel of a morphism f:ABf : A \to B is the pushout coker(f)coker(f) in

A f B i 0 coker(f). \array{ A &\stackrel{f}{\to}& B \\ \downarrow && \downarrow^{\mathrlap{i}} \\ 0 &\to& coker(f) } \,.
Remark

More explicitly, this characterizes the object coker(f)coker(f) as the object (unique up to unique isomorphism) that satisfies the following universal property:

for every object CC and every morphism h:BCh : B \to C such that hf=0h \circ f = 0 is the zero morphism, there is a unique morphism ϕ:coker(f)C\phi : coker(f) \to C such that h=ϕih = \phi \circ i.

Example

In the category Ab of abelian groups the cokernel of a morphism f:ABf : A \to B is the quotient group of BB by the image (of the underlying morphism of sets) of ff.

Proposition

RModR Mod has all kernels. The kernel of a homomorphism f:N 1N 2f : N_1 \to N_2 is the set-theoretic preimage U(f) 1(0)U(f)^{-1}(0) equipped with the induced RR-module structure.

RModR Mod has all cokernels. The cokernel of a homomorphism f:N 1N 2f : N_1 \to N_2 is the quotient abelian group

cokerf=N 2im(f) coker f = \frac{N_2}{im(f)}

of N 2N_2 by the image of ff.

The reader unfamiliar with the general concept of monomorphism and epimorphism may take the following to define these in Ab to be simply the injections and surjections.

Proposition

U:RModSetU : R Mod \to Set preserves and reflects monomorphisms and epimorphisms:

A homomorphism f:N 1N 2f : N_1 \to N_2 in RModR Mod is a monomorphism / epimorphism precisely if U(f)U(f) is an injection / surjection.

Proof

Suppose that ff is a monomorphism, hence that f:N 1N 2f : N_1 \to N_2 is such that for all morphisms g 1,g 2:KN 1g_1, g_2 : K \to N_1 such that fg 1=fg 2f \circ g_1 = f \circ g_2 already g 1=g 2g_1 = g_2. Let then g 1g_1 and g 2g_2 be the inclusion of submodules generated by a single element k 1Kk_1 \in K and k 2Kk_2 \in K, respectively. It follows that if f(k 1)=f(k 2)f(k_1) = f(k_2) then already k 1=k 2k_1 = k_2 and so ff is an injection. Conversely, if ff is an injection then its image is a submodule and it follows directly that ff is a monomorphism.

Suppose now that ff is an epimorphism and hence that f:N 1N 2f : N_1 \to N_2 is such that for all morphisms g 1,g 2:N 2Kg_1, g_2 : N_2 \to K such that fg 1=fg 2f \circ g_1 = f \circ g_2 already g 1=g 2g_1 = g_2. Let then g 1:N 2N 2im(f)g_1 : N_2 \to \frac{N_2}{im(f)} be the natural projection. and let g 2:N 20g_2 : N_2 \to 0 be the zero morphism. Since by construction fg 1=0f \circ g_1 = 0 and fg 2=0f \circ g_2 = 0 we have that g 1=0g_1 = 0, which means that Nim(f)=0\frac{N}{im(f)} = 0 and hence that N=im(f)N = im(f) and so that ff is surjective. The other direction is evident on elements.

Definition

For N 1,N 2RModN_1, N_2 \in R Mod two modules, define on the hom set Hom RMod(N 1,N 2)Hom_{R Mod}(N_1,N_2) the structure of an abelian group whose addition is given by argumentwise addition in N 2N_2: (f 1+f 2):nf 1(n)+f 2(n)(f_1 + f_2) : n \mapsto f_1(n) + f_2(n).

Proposition

With def. 37 RModR Mod composition of morphisms

:Hom(N 1,N 2)×Hom(N 2,N 3)Hom(N 1,N 3) \circ : Hom(N_1, N_2) \times Hom(N_2, N_3) \to Hom(N_1,N_3)

is a bilinear map, hence is equivalently a morphism

Hom(N 1,N 2)Hom(N 2,N 3)Hom(N 1,N 3) Hom(N_1, N_2) \otimes Hom(N_2,N_3) \to Hom(N_1, N_3)

out of the tensor product of abelian groups.

This makes RModR Mod into an Ab-enriched category.

Proof

Linearity of composition in the second argument is immediate from the pointwise definition of the abelian group structure on morphisms. Linearity of the composition in the first argument comes down to linearity of the second module homomorphism.

Remark

In fact RModR Mod is even a closed category, but this we do not need for showing that it is abelian.

Prop. 17 and prop. 20 together say that:

Corollary

RModR Mod is an pre-additive category.

Proposition

RModR Mod has all products and coproducts, being direct products and direct sums.

The products are given by cartesian product of the underlying sets with componentwise addition and RR-action.

The direct sum is the subobject of the product consisting of tuples of elements such that only finitely many are non-zero.

Proof

The defining universal properties are directly checked. Notice that the direct product iIN i\prod_{i \in I} N_i consists of arbitrary tuples because it needs to have a projection map

p j: iIN iN j p_j : \prod_{i \in I} N_i \to N_j

to each of the modules in the product, reproducing all of a possibly infinite number of non-trivial maps {KN j}\{K \to N_j\}. On the other hand, the direct sum just needs to contain all the modules in the sum

ι j:N j iIN i \iota_j : N_j \to \oplus_{i \in I} N_i

and since, being a module, it needs to be closed only under addition of finitely many elements, so it consists only of linear combinations of the elements in the N jN_j, hence of finite formal sums of these.

Together cor. 2 and prop. 21 say that:

Corollary

RModR Mod is an additive category.

Proposition

In RModR Mod

Proof

Using prop. 18 this is directly checked on the underlying sets: given a monomorphism KNK \hookrightarrow N, its cokernel is NNKN \to \frac{N}{K}, The kernel of that morphism is evidently KNK \hookrightarrow N.

Now cor. 2 and prop. 22 imply theorem 2, by definition.

Now we finally have all the ingredients to talk about chain complexes of RR-modules. The following definitions are the direct analogs of the definitions of chain complexes of abelian groups in Simplicial and singular homology above.

Definition

A (\mathbb{Z}-graded) chain complex in RRMod is

  • a collection of objects {C n} n\{C_n\}_{n\in\mathbb{Z}},

  • and of morphisms n:C nC n1\partial_n : C_n \to C_{n-1}

3C 2 2C 1 1C 0 0C 1 1 \cdots \overset{\partial_3}{\to} C_2 \overset{\partial_2}{\to} C_1 \overset{\partial_1}{\to} C_0 \overset{\partial_0}{\to} C_{-1} \overset{\partial_{-1}}{\to} \cdots

such that

n n+1=0 \partial_n \circ \partial_{n+1} = 0

(the zero morphism) for all nn \in \mathbb{N}.

Definition

For C C_\bullet a chain complex and nn \in \mathbb{N}

  • the morphisms n\partial_n are called the differentials or boundary maps;

  • the elements of C nC_n are called the nn-chains;

  • for n1n \geq 1 the elements in the kernel

    Z nker( n1) Z_n \coloneqq ker(\partial_{n-1})

    of n1:C nC n1\partial_{n-1} : C_n \to C_{n-1} are called the nn-cycles

    and for n=0n = 0 we say that every 0-chain is a 0-cycle

    Z 0C 0 Z_0 \coloneqq C_0

    (equivalently we declare that 1=0\partial_{-1} = 0).

  • the elements in the image

    B nim( n) B_n \coloneqq im(\partial_n)

    of n:C n+1C n\partial_{n} : C_{n+1} \to C_{n} are called the nn-boundaries;

Notice that due to =0\partial \partial = 0 we have canonical inclusions

0B nZ nC n. 0 \hookrightarrow B_n \hookrightarrow Z_n \hookrightarrow C_n \,.
0B nZ nH n0. 0 \to B_n \to Z_n \to H_n \to 0 \,.
Definition

A chain map f:V W f : V_\bullet \to W_\bullet is a collection of morphism {f n:V nW n} n\{f_n : V_n \to W_n\}_{n \in \mathbb{Z}} in 𝒜\mathcal{A} such that all the diagrams

V n+1 d n V V n f n+1 f n W n+1 d n W W n \array{ V_{n+1} &\stackrel{d^V_n}{\to}& V_n \\ \downarrow^{\mathrlap{f_{n+1}}} && \downarrow^{\mathrlap{f_{n}}} \\ W_{n+1} &\stackrel{d^W_n}{\to} & W_n }

commute, hence such that all the equations

f nd n V=d n+1 Wf n+1 f_n \circ d^V_n = d^W_{n+1} \circ f_{n+1}

hold.

Proposition

For f:C D f : C_\bullet \to D_\bullet a chain map, it respects boundaries and cycles, so that for all nn \in \mathbb{Z} it restricts to a morphism

B n(f):B n(C )B n(D ) B_n(f) : B_n(C_\bullet) \to B_n(D_\bullet)

and

Z n(f):Z n(C )Z n(D ). Z_n(f) : Z_n(C_\bullet) \to Z_n(D_\bullet) \,.

In particular it also respects chain homology

H n(f):H n(C )H n(D ). H_n(f) : H_n(C_\bullet) \to H_n(D_\bullet) \,.
Corollary

Conversely this means that taking chain homology is a functor

H n():Ch (𝒜)𝒜 H_n(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A}

from the category of chain complexes in 𝒜\mathcal{A} to 𝒜\mathcal{A} itself.

This establishes the basic objects that we are concerned with in the following. But as before, we are not so much interested in chain complexes up to chain map isomorphism, rather, we are interested in them up to a notion of homotopy equivalence. This we begin to study in the next section Homology exact sequences and homotopy fiber sequences. But in order to formulate that neatly, it is useful to have the tensor product of chain complexes. We close this section with introducing that notion.

Definition

For X,YCh (𝒜)X, Y \in Ch_\bullet(\mathcal{A}) write XYCh (𝒜)X \otimes Y \in Ch_\bullet(\mathcal{A}) for the chain complex whose component in degree nn is given by the direct sum

(XY) n:= i+j=nX i RY j (X \otimes Y)_n := \oplus_{i + j = n} X_i \otimes_R Y_j

over all tensor products of components whose degrees sum to nn, and whose differential is given on elements (x,y)(x,y) of homogeneous degree by

XY(x,y)=( Xx,y)+(1) deg(x)(x, Yy). \partial^{X \otimes Y} (x, y) = (\partial^X x, y) + (-1)^{deg(x)} (x, \partial^Y y) \,.
Example

(square as tensor product of interval with itself)

For RR some ring, let I Ch (RMod)I_\bullet \in Ch_\bullet(R Mod) be the chain complex given by

I =[00R 0 IRR], I_\bullet = \left[ \cdots \to 0 \to 0 \to R \stackrel{\partial^{I}_0}{\to} R \oplus R \right] \,,

where 0 I=(id,id)\partial^I_0 = (-id, id).

This is the normalized chain complex of the simplicial chain complex of the standard simplicial interval, the 1-simplex Δ 1\Delta_1, which means: we may think of

I 0=RRR[{(0),(1)}] I_0 = R \oplus R \simeq R[ \{(0), (1)\} ]

as the RR-linear span of two basis elements labelled “(0)(0)” and “(1)(1)”, to be thought of as the two 0-chains on the endpoints of the interval. Similarly we may think of

I 1=RR[{(01)}] I_1 = R \simeq R[\{(0 \to 1)\}]

as the free RR-module on the single basis element which is the unique non-degenerate 1-simplex (01)(0 \to 1) in Δ 1\Delta^1.

Accordingly, the differential 0 I\partial^I_0 is the oriented boundary map of the interval, taking this basis element to

0 I:(01)(1)(0) \partial^I_0 : (0 \to 1) \mapsto (1) - (0)

and hence a general element r(01)r\cdot(0 \to 1) for some rRr \in R to

0 I:r(01)r(1)r(0). \partial^I_0 : r\cdot(0 \to 1) \mapsto r\cdot (1) - r\cdot(0) \,.

We now write out in full details the tensor product of chain complexes of I I_\bullet with itself, according to def. 41:

S I I . S_\bullet \coloneqq I_\bullet \otimes I_\bullet \,.

By definition and using the above choice of basis element, this is in low degree given as follows:

S 0 =I 0I 0 =(RR)(RR) RRRR ={r 00((0),(0))+r 01((0),(1))+r 10((1),(0))+r 11((1),(1))|r ,R}, \begin{aligned} S_0 &= I_0 \oplus I_0 \\ & = (R \oplus R) \otimes (R \oplus R) \\ & \simeq R \oplus R \oplus R \oplus R \\ & = \left\{ r_{00} \cdot ((0),(0)') + r_{01} \cdot ((0),(1)') + r_{10} \cdot ((1),(0)') + r_{11} \cdot ((1),(1)') | r_{\cdot, \cdot} \in R \right\} \end{aligned} \,,

where in the last line we express a general element as a linear combination of the canonical basis elements which are obtained as tensor products (a,b)RR(a,b) \in R\otimes R of the previous basis elements. Notice that by the definition of tensor product of modules we have relations like

r((0),(1))=(r(0),(1))=((0),r(1)) r ( (0), (1)') = (r(0), (1)') = ((0), r(1)')

etc.

Similarly then, in degree-1 the tensor product chain complex is

(II) 1 =(I 0I 1)(I 1I 0) R(RR)(RR)R RRRR {r 0((0),(01))+r 1((1),(01))+r¯ 0((01),(0))+r¯ 1((01),(1))|r ,r¯ R}. \begin{aligned} (I \otimes I)_1 & = (I_0 \otimes I_1) \oplus (I_1 \otimes I_0) \\ & \simeq R \otimes (R \oplus R) \oplus (R \oplus R) \otimes R \\ & \simeq R \oplus R \oplus R \oplus R \\ & \simeq \left\{ r_{0} \cdot ((0),(0\to 1)') + r_{1} \cdot ((1), (0 \to 1)') + \bar r_0 \cdot ((0\to 1), (0)') + \bar r_1 \cdot ((0 \to 1), (1)') | r_{\cdot}, \bar r_{\cdot} \in R \right\} \end{aligned} \,.

And finally in degree 2 it is

(II) 2 I 1I 1 RR R {r((01),(01))|rR}. \begin{aligned} (I \otimes I)_2 & \simeq I_1 \otimes I_1 \\ & \simeq R \otimes R \\ & \simeq R \\ & \simeq \left\{ r\cdot ((0 \to 1), (0 \to 1)') | r \in R \right\} \end{aligned} \,.

All other contributions that are potentially present in (II) (I \otimes I)_\bullet vanish (are the 0-module) because all higher terms in I I_\bullet are.

The tensor product basis elements appearing in the above expressions have a clear geometric interpretation: we can label a square with them as follows

((0),(1)) ((01),(0)) ((1),(1)) ((0),(01)) ((01),(01)) ((1),(01)) ((0),(0)) ((01),(0)) ((1),(0)). \array{ ((0),(1)') &&\underset{((0\to 1),(0))}{\to}&& ((1),(1)') \\ \\ {}^{\mathllap{((0),(0\to 1)')}}\uparrow &&\righttoleftarrow^{((0 \to 1), (0\to 1)')}&& \uparrow^{\mathrlap{((1),(0 \to 1)')}} \\ \\ ((0),(0)') &&\underset{((0\to 1),(0)')}{\to}&& ((1),(0)') } \,.

This diagram indicates a cellular square and identifies its canonical singular chains with the elements of (II) (I \otimes I)_\bullet. The arrows indicate the orientation. For instance the fact that

II((01),(0)) =( I(01),(0))+(1) 1((01), I(0)) =((1)(0),(0))0 =((1),(0))((0),(0)) \begin{aligned} \partial^{I \otimes I} ((0 \to 1), (0)') & = (\partial^I (0 \to 1), (0)') + (-1)^1 ((0\to 1), \partial^I (0)) \\ & = ( (1) - (0), \;(0)' ) - 0 \\ & = ((1), (0)') - ((0), (0)') \end{aligned}

says that the oriented boundary of the bottom morphism is the bottom right element (its target) minus the bottom left element (its source), as indicated. Here we used that the differential of a degree-0 element in I I_\bullet is 0, and hence so is any tensor product with it.

Similarly the oriented boundary of the square itself is computed to

II((01),(01)) =( I(01),(01))((01), I(01)) =((1)(0),(01))((01),(1)(0)) =((1),(01))((0),(01))((01),(1))+((01),(0)), \begin{aligned} \partial^{I \otimes I} ((0 \to 1), (0 \to 1)') &= (\partial^I (0 \to 1), (0 \to 1)') - ((0 \to 1), \partial^I(0 \to 1)) \\ & = ((1)- (0), (0 \to 1)') - ((0 \to 1), (1)' - (0)') \\ & = ((1), (0 \to 1)') - ((0), (0 \to 1)') - ((0 \to 1), (1)') + ((0 \to 1), (0)') \end{aligned} \,,

which can be read as saying that the boundary is the evident boundary thought of as oriented by drawing it counterclockwise into the plane, so that the right arrow (which points up) contributes with a +1 prefactor, while the left arrow (which also points up) contributes with a -1 prefactor.

Proposition

Equipped with the standard tensor product of chain complexes \otimes, def. 41 the category of chain complexes is a monoidal category (Ch (RMod),)(Ch_\bullet(R Mod), \otimes). The unit object is the chain complex concentrated in degree 0 on the tensor unit RR of RModR Mod.

Definition

We write Ch ubCh_\bullet^{ub} for the category of unbounded chain complexes.

Definition

For X,YCh ub(𝒜)X,Y \in Ch^{ub}_\bullet(\mathcal{A}) any two objects, define a chain complex [X,Y]Ch ub(𝒜)[X,Y] \in Ch^{ub}_\bullet(\mathcal{A}) to have components

[X,Y] n:= iHom RMod(X i,Y i+n) [X,Y]_n := \prod_{i \in \mathbb{Z}} Hom_{R Mod}(X_i, Y_{i+n})

(the collection of degree-nn maps between the underlying graded modules) and whose differential is defined on homogeneously graded elements f[X,Y] nf \in [X,Y]_n by

df:=d Yf(1) nfd X. d f := d_Y \circ f - (-1)^{n} f \circ d_X \,.

This defines a functor

[,]:Ch ub(𝒜) op×Ch ub(𝒜)Ch ub(𝒜). [-,-] : Ch^{ub}_\bullet(\mathcal{A})^{op} \times Ch^{ub}_\bullet(\mathcal{A}) \to Ch^{ub}_\bullet(\mathcal{A}) \,.
Proposition

This functor

[,]:Ch ub×Ch ubCh ub[-,-] : Ch^{ub}_\bullet \times Ch^{ub}_\bullet \to Ch^{ub}_\bullet

is the internal hom of the category of chain complexes.

Proposition

The collection of cycles of the internal hom [X,Y] [X,Y]_\bullet in degree 0 coincides with the external hom functor

Z 0([X,Y])Hom Ch ub(X,Y). Z_0([X,Y]) \simeq Hom_{Ch^{ub}_\bullet}(X,Y) \,.

The chain homology of the internal hom [X,Y][X,Y] in degree 0 coincides with the homotopy classes of chain maps.

Proof

By Definition 43 the 0-cycles in [X,Y][X,Y] are collections of morphisms {f k:X kY k}\{f_k : X_k \to Y_k\} such that

f k+1d X=d Yf k. f_{k+1} \circ d_X = d_Y \circ f_k \,.

This is precisely the condition for ff to be a chain map.

Similarly, the boundaries in degree 0 are precisely the collections of morphisms of the form

λ k+1d X+d Yλ k \lambda_{k+1} \circ d_X + d_Y \circ \lambda_k

for a collection of maps {λ k:X kY k+1}\{\lambda_k : X_k \to Y_{k+1}\}. This are precisely the null homotopies.

Proposition

The monoidal category (Ch ,)(Ch_\bullet, \otimes) is a closed monoidal category, the internal hom is the standard internal hom of chain complexes.

Homology exact sequences

With the basic definition of the category of chain complexes in hand, we now consider the first application, which is as simple as it is of ubiquituous use in mathematics: long exact sequences in homology. This is the “abelianization”, in the sense of the discussion in 2) above, of what in homotopy theory are long exact sequences of homotopy groups. But both concepts, in turn, are just the shadow on homology groups/homotopy groups, respectively of homotopy fiber sequences of the underlying chain complexes/topological spaces themselves. Since these are even more useful, in particular in chapter III) below, we discuss below in 5) how to construct these using chain homotopy and mapping cones.

First we need the fundamental notion of exact sequences. As before, we fix some commutative ring RR throughout and consider the category of modules over RR, which we will abbreviate

𝒜RMod. \mathcal{A} \coloneqq R Mod \,.
Definition

An exact sequence in 𝒜\mathcal{A} is a chain complex C C_\bullet in 𝒜\mathcal{A} with vanishing chain homology in each degree:

n.H n(C)=0. \forall n \in \mathbb{N} . H_n(C) = 0 \,.
Definition

A short exact sequence is an exact sequence, def. 44 of the form

00ABC00. \cdots \to 0 \to 0 \to A \to B \to C \to 0 \to 0 \to \cdots \,.

One usually writes this just “0ABC00 \to A \to B \to C \to 0” or even just “ABCA \to B \to C”.

Remark

A general exact sequence is sometimes called a long exact sequence, to distinguish from the special case of a short exact sequence.

Beware that there is a difference between ABCA \to B \to C being exact (at BB) and ABCA \to B \to C being a “short exact sequence” in that 0ABC00 \to A \to B \to C \to 0 is exact at AA, BB and CC. This is illustrated by the following proposition.

Proposition

Explicitly, a sequence of morphisms

0AiBpC0 0 \to A \stackrel{i}\to B \stackrel{p}\to C \to 0

in 𝒜\mathcal{A} is short exact, def. 45, precisely if

  1. ii is a monomorphism,

  2. pp is an epimorphism,

  3. and the image of ii equals the kernel of pp (equivalently, the coimage of pp equals the cokernel of ii).

Proof

The third condition is the definition of exactness at BB. So we need to show that the first two conditions are equivalent to exactness at AA and at CC.

This is easy to see by looking at elements when 𝒜R\mathcal{A} \simeq RMod, for some ring RR (and the general case can be reduced to this one using one of the embedding theorems):

The sequence being exact at

0AB 0 \to A \to B

means, since the image of 0A0 \to A is just the element 0A0 \in A, that the kernel of ABA \to B consists of just this element. But since ABA \to B is a group homomorphism, this means equivalently that ABA \to B is an injection.

Dually, the sequence being exact at

BC0 B \to C \to 0

means, since the kernel of C0C \to 0 is all of CC, that also the image of BCB \to C is all of CC, hence equivalently that BCB \to C is a surjection.

Example

Let 𝒜=\mathcal{A} = \mathbb{Z}Mod \simeq Ab. For nn \in \mathbb{N} with n1n \geq 1 let n\mathbb{Z} \stackrel{\cdot n}{\to} \mathbb{Z} be the linear map/homomorphism of abelian groups which acts by the ordinary multiplication of integers by nn. This is clearly an injection. The cokernel of this morphism is the projection to the quotient group, which is the cyclic group n/n\mathbb{Z}_n \coloneqq \mathbb{Z}/n\mathbb{Z}. Hence we have a short exact sequence

0n n. 0 \to \mathbb{Z} \stackrel{\cdot n}{\to} \mathbb{Z} \to \mathbb{Z}_n \,.

A typical use of a long exact sequence, notably of the homology long exact sequence to be discussed, is that it allows to determine some of its entries in terms of others.

The characterization of short exact sequences in prop. 28 is one example for this. Another is this:

Proposition

If part of an exact sequence looks like

0C n+1 nC n0, \cdots \to 0 \to C_{n+1} \stackrel{\partial_n}{\to} C_n \to 0 \to \cdots \,,

then n\partial_n is an isomorphism and hence

C n+1C n. C_{n+1} \simeq C_n \,.

Often it is useful to make the following strengthening of short exactness explicit.

Definition

A short exact sequence 0AiBpC00\to A \stackrel{i}{\to} B \stackrel{p}{\to} C\to 0 in 𝒜\mathcal{A} is called split if either of the following equivalent conditions hold

  1. There exists a section of pp, hence a homomorphism s:BCs \colon B\to C such that ps=id Cp \circ s = id_C.

  2. There exists a retract of ii, hence a homomorphism r:BAr \colon B\to A such that ri=id Ar \circ i = id_A.

  3. There exists an isomorphism of sequences with the sequence

    0AACC0 0\to A\to A\oplus C\to C\to 0

    given by the direct sum and its canonical injection/projection morphisms.

Proposition

(splitting lemma)

The three conditions in def. 46 are indeed equivalent.

Proof

It is clear that the third condition implies the first two: take the section/retract to be given by the canonical injection/projection maps that come with a direct sum.

Conversely, suppose we have a retract r:BAr \colon B \to A of i:ABi \colon A \to B. Write P:BrAiBP \colon B \stackrel{r}{\to} A \stackrel{i}{\to} B for the composite. Notice that by ri=idr\circ i = id this is an idempotent: PP=PP \circ P = P, hence a projector.

Then every element bBb \in B can be decomposed as b=(bP(b))+P(b)b = (b - P(b)) + P(b) hence with bP(b)ker(r)b - P(b) \in ker(r) and P(b)im(i)P(b) \in im(i). Moreover this decomposition is unique since if b=i(a)b = i(a) while at the same time r(b)=0r(b) = 0 then 0=r(i(a))=a0 = r(i(a)) = a. This shows that Bim(i)ker(r)B \simeq im(i) \oplus ker(r) is a direct sum and that i:ABi \colon A \to B is the canonical inclusion of im(i)im(i). By exactness it then follows that ker(r)ker(p)ker(r) \simeq ker(p) and hence that BACB \simeq A \oplus C with the canonical inclusion and projection.

The implication that the second condition also implies the third is formally dual to this argument.

Moreover, of particular interest are exact sequences of chain complexes. We consider this concept in full beauty below in section 5). In order to motivate the discussion there we here content ourselves with the following quick definition, which already admits discussion of some of its rich consequences.

Definition

A sequence of chain maps of chain complexes

0A B C 0 0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0

is a short exact sequence of chain complexes in 𝒜\mathcal{A} if for each nn the component

0A nB nC n0 0 \to A_n \to B_n \to C_n \to 0

is a short exact sequence in 𝒜\mathcal{A}, according to def. 45.

Definition

Consider a short exact sequence of chain complexes as in def. 47. For nn \in \mathbb{Z}, define a group homomorphism

δ n:H n(C)H n1(A), \delta_n : H_n(C) \to H_{n-1}(A) \,,

called the nnth connecting homomorphism of the short exact sequence, by sending

δ n:[c][ Bc^] A, \delta_n : [c] \mapsto [\partial^B \hat c]_A \,,

where

  1. cZ n(C)c \in Z_n(C) is a cycle representing the given homology group [c][c];

  2. c^C n(B)\hat c \in C_n(B) is any lift of that cycle to an element in B nB_n, which exists because pp is a surjection (but which no longer needs to be a cycle itself);

  3. [ Bc^] A[\partial^B \hat c]_A is the AA-homology class of Bc^\partial^B \hat c which is indeed in A n1B n1A_{n-1} \hookrightarrow B_{n-1} by exactness (since p( Bc^)= Cp(c^)= Cc=0p(\partial^B \hat c) = \partial^C p(\hat c) = \partial^C c = 0) and indeed in Z n1(A)A n1Z_{n-1}(A) \hookrightarrow A_{n-1} since A Bc^= B Bc^=0\partial^A \partial^B \hat c = \partial^B \partial^B \hat c = 0.

Proposition

Def. 48 is indeed well defined in that the given map is independent of the choice of lift c^\hat c involved and in that the group structure is respected.

Proof

To see that the construction is well-defined, let c˜B n\tilde c \in B_{n} be another lift. Then p(c^c˜)=0p(\hat c - \tilde c) = 0 and hence c^c˜A nB n\hat c - \tilde c \in A_n \hookrightarrow B_n. This exhibits a homology-equivalence [ Bc^] A[ Bc˜] A[\partial^B\hat c]_A \simeq [\partial^B \tilde c]_A since A(c^c˜)= Bc^ Bc˜ \partial^A(\hat c - \tilde c) = \partial^B \hat c - \partial^B \tilde c.

To see that δ n\delta_n is a group homomorphism, let [c]=[c 1]+[c 2][c] = [c_1] + [c_2] be a sum. Then c^c^ 1+c^ 2\hat c \coloneqq \hat c_1 + \hat c_2 is a lift and by linearity of \partial we have [ Bc^] A=[ Bc^ 1]+[ Bc^ 2][\partial^B \hat c]_A = [\partial^B \hat c_1] + [\partial^B \hat c_2].

Proposition

Under chain homology H ()H_\bullet(-) the morphisms in the short exact sequence together with the connecting homomorphisms yield the homology long exact sequence

H n(A)H n(B)H n(C)δ nH n1(A)H n1(B)H n1(C). \cdots \to H_n(A) \to H_n(B) \to H_n(C) \stackrel{\delta_n}{\to} H_{n-1}(A) \to H_{n-1}(B) \to H_{n-1}(C) \to \cdots \,.
Proof

Consider first the exactness of H n(A)H n(i)H n(B)H n(p)H n(C)H_n(A) \stackrel{H_n(i)}{\to} H_n(B) \stackrel{H_n(p)}{\to} H_n(C).

It is clear that if aZ n(A)Z n(B)a \in Z_n(A) \hookrightarrow Z_n(B) then the image of [a]H n(B)[a] \in H_n(B) is [p(a)]=0H n(C)[p(a)] = 0 \in H_n(C). Conversely, an element [b]H n(B)[b] \in H_n(B) is in the kernel of H n(p)H_n(p) if there is cC n+1c \in C_{n+1} with Cc=p(b)\partial^C c = p(b). Since pp is surjective let c^B n+1\hat c \in B_{n+1} be any lift, then [b]=[b Bc^][b] = [b - \partial^B \hat c] but p(b Bc)=0p(b - \partial^B c) = 0 hence by exactness b Bc^Z n(A)Z n(B)b - \partial^B \hat c \in Z_n(A) \hookrightarrow Z_n(B) and so [b][b] is in the image of H n(A)H n(B)H_n(A) \to H_n(B).

It remains to see that

  1. the image of H n(B)H n(C)H_n(B) \to H_n(C) is the kernel of δ n\delta_n;

  2. the kernel of H n1(A)H n1(B)H_{n-1}(A) \to H_{n-1}(B) is the image of δ n\delta_n.

This follows by inspection of the formula in def. 48. We spell out the first one:

If [c][c] is in the image of H n(B)H n(C)H_n(B) \to H_n(C) we have a lift c^\hat c with Bc^=0\partial^B \hat c = 0 and so δ n[c]=[ Bc^] A=0\delta_n[c] = [\partial^B \hat c]_A = 0. Conversely, if for a given lift c^\hat c we have that [ Bc^] A=0[\partial^B \hat c]_A = 0 this means there is aA na \in A_n such that Aa Ba= Bc^\partial^A a \coloneqq \partial^B a = \partial^B \hat c. But then c˜c^a\tilde c \coloneqq \hat c - a is another possible lift of cc for which Bc˜=0\partial^B \tilde c = 0 and so [c][c] is in the image of H n(B)H n(C)H_n(B) \to H_n(C).

Example

The connecting homomorphism of the long exact sequence in homology induced from short exact sequences of the form in example 40 is called a Bockstein homomorphism.

We now discuss a deeper, more conceptual way of understanding the origin of long exact sequences in homology and the nature of connecting homomorphisms. This will give first occasion to see some actual homotopy theory of chain complexes at work, and hence serves also as a motivating example for the discussions to follow in chapter III).

For this we need the notion of chain homotopy, which is the abelianized analog of the notion of homotopy of continuous maps above in def. \ref{LeftHomotopyContinousMaps}. We now first introduce this concept by straightforwardly mimicking the construction in def. \ref{LeftHomotopyContinousMaps} with topological spaces replaced by chain complexes. Then we use chain homotopies to construct mapping cones of chain maps. Finally we explain how these refine the above long exact sequences in homology groups to homotopy cofiber sequences of the chain complexes themselves.

A chain homotopy is a homotopy in Ch (𝒜)Ch_\bullet(\mathcal{A}). We first give the explicit definition, the more abstract characterization is below in prop. 33.

Definition

A chain homotopy ψ:fg\psi : f \Rightarrow g between two chain maps f,g:C D f,g : C_\bullet \to D_\bullet in Ch (𝒜)Ch_\bullet(\mathcal{A}) is a sequence of morphisms

{(ψ n:C nD n+1)𝒜|n} \{ (\psi_n : C_n \to D_{n+1}) \in \mathcal{A} | n \in \mathbb{N} \}

in 𝒜\mathcal{A} such that

f ng n= Dψ n+ψ n1 C. f_n - g_n = \partial^D \circ \psi_n + \psi_{n-1} \partial^C \,.
Remark

It may be useful to illustrate this with the following graphics, which however is not a commuting diagram:

C n+1 f n+1g n+1 D n+1 n C ψ n n D C n f ng n D n n1 C ψ n1 n1 D C n1 f n1g n1 D n1 . \array{ \vdots && \vdots \\ \downarrow && \downarrow \\ C_{n+1} &\stackrel{f_{n+1} - g_{n+1}}{\to}& D_{n+1} \\ \downarrow^{\mathrlap{\partial^C_{n}}} &\nearrow_{\mathrlap{\psi_{n}}}& \downarrow^{\mathrlap{\partial^D_{n}}} \\ C_n &\stackrel{f_n - g_n}{\to}& D_n \\ \downarrow^{\mathrlap{\partial^C_{n-1}}} &\nearrow_{\mathrlap{\psi_{n-1}}}& \downarrow^{\mathrlap{\partial^D_{n-1}}} \\ C_{n-1} &\stackrel{f_{n-1} - g_{n-1}}{\to}& D_{n-1} \\ \downarrow && \downarrow \\ \vdots && \vdots } \,.

Instead, a way to encode chain homotopies by genuine diagrammatics is below in prop. 33, for which we introduce the interval object for chain complexes:

Definition

Let

I N (C(Δ[1])) I_\bullet \coloneqq N_\bullet(C(\Delta[1]))

be the normalized chain complex in 𝒜\mathcal{A} of the simplicial chains on the simplicial 1-simplex:

I =[00R(id,id)RR]. I_\bullet = [ \cdots \to 0 \to 0 \to R \stackrel{(-id,id)}{\to} R \oplus R ] \,.
Remark

This is the standard interval in chain complexes. Indeed it is manifestly the “abelianization” of the standard interval object Δ 1\Delta^1 in sSet/Top: the 1-simplex.

Proposition

A chain homotopy ψ:fg\psi : f \Rightarrow g is equivalently a commuting diagram

C f I C (f,g,ψ) D g C \array{ C_\bullet \\ \downarrow & \searrow^{\mathrlap{f}} \\ I_\bullet \otimes C_\bullet &\stackrel{(f,g,\psi)}{\to}& D_\bullet \\ \uparrow & \nearrow_{\mathrlap{g}} \\ C_\bullet }

in Ch (𝒜)Ch_\bullet(\mathcal{A}), hence a genuine left homotopy with respect to the interval object in chain complexes.

Proof

For notational simplicity we discuss this in 𝒜=\mathcal{A} = Ab.

Observe that N ((Δ[1]))N_\bullet(\mathbb{Z}(\Delta[1])) is the chain complex

(00(id,id)00) ( \cdots \to 0 \to 0 \to \mathbb{Z} \stackrel{(-id,id)}{\to} \mathbb{Z} \oplus \mathbb{Z} \to 0 \to 0 \to \cdots)

where the term \mathbb{Z} \oplus \mathbb{Z} is in degree 0: this is the free abelian group on the set {(0),(1)}\{(0),(1)\} of 0-simplices in Δ[1]\Delta[1]. The other copy of \mathbb{Z} is the free abelian group on the single non-degenerate edge (01)(0 \to 1) in Δ[1]\Delta[1]. (All other simplices of Δ[1]\Delta[1] are degenerate and hence do not contribute to the normalized chain complex which we are discussing here.) The single nontrivial differential sends 11 \in \mathbb{Z} to (1,1)(-1,1) \in \mathbb{Z} \oplus \mathbb{Z}, reflecting the fact that one of the vertices is the 0-boundary the other the 1-boundary of the single nontrivial edge.

It follows that the tensor product of chain complexes I C I_\bullet \otimes C_\bullet is

(IC) 2 (IC) 1 (IC) 0 C 1C 2C 2 C 0C 1C 1 C 1C 0C 0 . \array{ && (I \otimes C)_2 &\to& (I \otimes C)_1 &\to& (I \otimes C)_0 &\to& \cdots \\ \cdots &\to& C_1 \oplus C_{2} \oplus C_2 &\to& C_0 \oplus C_{1} \oplus C_{1} &\to& C_{-1} \oplus C_0 \oplus C_0 &\to& \cdots } \,.

Therefore a chain map (f,g,ψ):I C D (f,g,\psi) : I_\bullet \otimes C_\bullet \to D_\bullet that restricted to the two copies of C C_\bullet is ff and gg, respectively, is characterized by a collection of commuting diagrams

C n+1C n+1C n (f n+1,g n+1,ψ n) D n IC D C nC nC n1 (f n,g n,ψ n1) D n1. \array{ C_{n+1}\oplus C_{n+1} \oplus C_{n} &\stackrel{(f_{n+1},g_{n+1}, \psi_n)}{\to}& D_n \\ {}^{\mathllap{\partial^{I \otimes C}}}\downarrow && \downarrow^{\mathrlap{\partial^D}} \\ C_{n} \oplus C_{n} \oplus C_{n-1} &\stackrel{(f_n,g_n,\psi_{n-1})}{\to} & D_{n-1} } \,.

On the elements (1,0,0)(1,0,0) and (0,1,0)(0,1,0) in the top left this reduces to the chain map condition for ff and gg, respectively. On the element (0,0,1)(0,0,1) this is the equation for the chain homotopy

f ng nψ n1d C=d Dψ n. f_n - g_n - \psi_{n-1} d_C = d_D \psi_{n} \,.

Let C ,D Ch (𝒜)C_\bullet, D_\bullet \in Ch_\bullet(\mathcal{A}) be two chain complexes.

Definition

Define the relation chain homotopic on Hom(C ,D )Hom(C_\bullet, D_\bullet) by

(fg)(ψ:fg). (f \sim g) \Leftrightarrow \exists (\psi : f \Rightarrow g) \,.
Proposition

Chain homotopy is an equivalence relation on Hom(C ,D )Hom(C_\bullet,D_\bullet).

Definition

Write Hom(C ,D ) Hom(C_\bullet,D_\bullet)_{\sim} for the quotient of the hom set Hom(C ,D )Hom(C_\bullet,D_\bullet) by chain homotopy.

Proposition

This quotient is compatible with composition of chain maps.

Accordingly the following category exists:

Definition

Write 𝒦 (𝒜)\mathcal{K}_\bullet(\mathcal{A}) for the category whose objects are those of Ch (𝒜)Ch_\bullet(\mathcal{A}), and whose morphisms are chain homotopy classes of chain maps:

Hom 𝒦 (𝒜)(C ,D )Hom Ch (𝒜)(C ,D ) . Hom_{\mathcal{K}_\bullet(\mathcal{A})}(C_\bullet, D_\bullet) \coloneqq Hom_{Ch_\bullet(\mathcal{A})}(C_\bullet, D_\bullet)_\sim \,.

This is usually called the (strong) homotopy category of chain complexes in 𝒜\mathcal{A}.

Remark

Beware, as we will discuss in detail below in 8), that another category that would deserve to carry this name instead is called the derived category of 𝒜\mathcal{A}. In the derived category one also quotients out chain homotopy, but one allows that first the domain of the two chain maps ff and gg is refined along a quasi-isomorphism.

Definition

A chain map f :C D f_\bullet : C_\bullet \to D_\bullet in Ch (𝒜)Ch_\bullet(\mathcal{A}) is called a quasi-isomorphism if for each nn \in \mathbb{N} the induced morphisms on chain homology groups

H n(f):H n(C)H n(D) H_n(f) \colon H_n(C) \to H_n(D)

is an isomorphism.

Remark

Quasi-isomorphisms are also called, more descriptively, homology isomorphisms or H H_\bullet-isomorphisms. See at homology localization for more on this.

With the homotopy theoretic notions of chain homotopy and quasi-isomorphism in hand, we can now give a deeper explanation of long exact sequences in homology. We first give now a heuristic discussion that means to serve as a guide through the constructions to follow. The reader wishing to skip this may directly jump ahead to definition 38.

While the notion of a short exact sequence of chain complexes is very useful for computations, it does not have invariant meaning if one considers chain complexes as objects in (abelian) homotopy theory, where one takes into account chain homotopies between chain maps and takes equivalence of chain complexes not to be given by isomorphism, but by quasi-isomorphism.

For if a chain map A B A_\bullet \to B_\bullet is the degreewise kernel of a chain map B C B_\bullet \to C_\bullet, then if A^ A \hat A_\bullet \stackrel{\simeq}{\to} A_\bullet is a quasi-isomorphism (for instance a projective resolution of A A_\bullet) then of course the composite chain map A^ B \hat A_\bullet \to B_\bullet is in general far from being the degreewise kernel of C C_\bullet. Hence the notion of degreewise kernels of chain maps and hence that of short exact sequences is not meaningful in the homotopy theory of chain complexes in 𝒜\mathcal{A} (for instance: not in the derived category of 𝒜\mathcal{A}).

That short exact sequences of chain complexes nevertheless play an important role in homological algebra is due to what might be called a “technical coincidence”:

Proposition

If A B C A_\bullet \to B_\bullet \to C_\bullet is a short exact sequence of chain complexes, then the commuting square

A 0 B C \array{ A_\bullet &\to& 0 \\ \downarrow && \downarrow \\ B_\bullet &\to& C_\bullet }

is not only a pullback square in Ch (𝒜)Ch_\bullet(\mathcal{A}), exhibiting A A_\bullet as the fiber of B C B_\bullet \to C_\bullet over 0C 0 \in C_\bullet, it is in fact also a homotopy pullback.

This means it is universal not just among commuting such squares, but also among such squares which commute possibly only up to a chain homotopy ϕ\phi:

Q 0 ϕ B C \array{ Q_\bullet &\to& 0 \\ \downarrow &\swArrow_{\phi}& \downarrow \\ B_\bullet &\to& C_\bullet }

and with morphisms between such squares being maps A A A_\bullet \to A'_\bullet correspondingly with further chain homotopies filling all diagrams in sight.

Equivalently, we have the formally dual result

Proposition

If A B C A_\bullet \to B_\bullet \to C_\bullet is a short exact sequence of chain complexes, then the commuting square

A 0 B C \array{ A_\bullet &\to& 0 \\ \downarrow && \downarrow \\ B_\bullet &\to& C_\bullet }

is not only a pushout square in Ch (𝒜)Ch_\bullet(\mathcal{A}), exhibiting C C_\bullet as the cofiber of A B A_\bullet \to B_\bullet over 0C 0 \in C_\bullet, it is in fact also a homotopy pushout.

But a central difference between fibers/cofibers on the one hand and homotopy fibers/homotopy cofibers on the other is that while the (co)fiber of a (co)fiber is necessarily trivial, the homotopy (co)fiber of a homotopy (co)fiber is in general far from trivial: it is instead the looping Ω()\Omega(-) or suspension Σ()\Sigma(-) of the codomain/domain of the original morphism: by the pasting law for homotopy pullbacks the pasting composite of successive homotopy cofibers of a given morphism f:A B f : A_\bullet \to B_\bullet looks like this:

A f B 0 ϕ 0 cone(f) A[1] 0 f[1] 0 B[1] cone(f)[1] \array{ A_\bullet &\stackrel{f}{\to}& B_\bullet &\to& 0 \\ \downarrow &\swArrow_{\mathrlap{\phi}}& \downarrow &\swArrow& \downarrow \\ 0 &\to& cone(f) &\to& A[1]_{\bullet} &\stackrel{}{\to}& 0 \\ && \downarrow &\swArrow& \downarrow^{\mathrlap{f[1]}} &\swArrow& \downarrow \\ && 0 &\to& B[1] &\to& cone(f)[1]_\bullet &\to& \cdots \\ && && \downarrow && \downarrow &\ddots& \\ && && \vdots && && }

here

  • cone(f)cone(f) is a specific representative of the homotopy cofiber of ff called the mapping cone of ff, whose construction comes with an explicit chain homotopy ϕ\phi as indicated, hence cone(f)cone(f) is homology-equivalence to C C_\bullet above, but is in general a “bigger” model of the homotopy cofiber;

  • A[1]A[1] etc. is the suspension of a chain complex of AA, hence the same chain complex but pushed up in degree by one.

In conclusion we get from every morphim of chain complexes a long homotopy cofiber sequence

A fB cone(f)A[1] f[1]B[1] cone(f)[1] . \cdots \to A_\bullet \stackrel{f}{\to}B_\bullet \stackrel{}{\to} cone(f) \stackrel{}{\to} A[1]_\bullet \stackrel{f[1]}{\to} B[1]_\bullet \stackrel{}{\to} cone(f)[1]_\bullet \to \cdots \,.

And applying the chain homology functor to this yields the long exact sequence in chain homology which is traditionally said to be associated to the short exact sequence A B C A_\bullet \to B_\bullet \to C_\bullet.

In conclusion this means that it is not really the passage to homology groups which “makes a short exact sequence become long”. It’s rather that passing to homology groups is a shadow of passing to chain complexes regarded up to quasi-isomorphism, and this is what makes every short exact sequence be realized as but a special presentation of a stage in a long homotopy fiber sequence.

We give a precise account of this story in the next section.

Homotopy fiber sequences and mapping cones

We have seen in 4) the long exact sequence in homology implied by a short exact sequence of chain complexes, constructed by an elementary if somewhat un-illuminating formula for the connecting homomorphism. We ended 4) by sketching how this formula arises as the shadow under the homology functor of a homotopy fiber sequence of chain complexes, constructed using mapping cones. This we now discuss in precise detail.

In the following we repeatedly mention that certain chain complexes are colimits of certain diagrams of chain complexes. The reader unfamiliar with colimits may simply ignore them and regard the given chain complex as arising by definition. However, even a vague intuitive understanding of the indicated colimits as formalizations of “gluing” of chain complexes along certain maps should help to motivate why these definitions are what they are. The reader unhappy even with this can jump ahead to prop. 40 and take this and the following propositions up to and including prop. 43 as definitions.

The notion of a mapping cone that we introduce now is something that makes sense whenever

  1. there is a notion of cylinder object, such as the topological cylinder [0,1]×X[0,1] \times X over a topological space, or the chain complex cylinder I X I_\bullet \otimes X_\bullet of a chain complex from def. 50.

  2. there is a way to glue objects along maps between them, a notion of colimit.

Definition

For f:XYf : X \to Y a morphism in a category with cylinder objects cyl()cyl(-), the mapping cone or homotopy cofiber of ff is the colimit in the following diagram

X f Y i 1 X i 0 cyl(X) * cone(f) \array{ && X &\stackrel{f}{\to}& Y \\ && \downarrow^{\mathrlap{i_1}} && \downarrow \\ X &\stackrel{i_0}{\to}& cyl(X) \\ \downarrow && &\searrow & \downarrow \\ {*} &\to& &\to& cone(f) }

in CC using any cylinder object cyl(X)cyl(X) for XX.

Remark

Heuristically this says that cone(f)cone(f) is the object obtained by

  1. forming the cylinder over XX;

  2. gluing to one end of that the object YY as specified by the map ff.

  3. shrinking the other end of the cylinder to the point.

Heuristically it is clear that this way every cycle in YY that happens to be in the image of XX can be “continuously” translated in the cylinder-direction, keeping it constant in YY, to the other end of the cylinder, where it becomes the point. This means that every homotopy group of YY in the image of ff vanishes in the mapping cone. Hence in the mapping cone the image of XX under ff in YY is removed up to homotopy. This makes it clear how cone(f)cone(f) is a homotopy-version of the cokernel of ff. And therefore the name “mapping cone”.

Another interpretation of the mapping cone is just as important:

Remark

A morphism η:cyl(X)Y\eta : cyl(X) \to Y out of a cylinder object is a left homotopy η:gh\eta : g \Rightarrow h between its restrictions gη(0)g\coloneqq \eta(0) and hη(1)h \coloneqq \eta(1) to the cylinder boundaries

X i 0 g cyl(X) η Y i 1 h X. \array{ X \\ \downarrow^{\mathrlap{i_0}} & \searrow^{\mathrlap{g}} \\ cyl(X) &\stackrel{\eta}{\to}& Y \\ \uparrow^{\mathrlap{i_1}} & \nearrow_{\mathrlap{h}} \\ X } \,.

Therefore prop. 38 says that the mapping cone is the universal object with a morphism ii from YY and a left homotopy from ifi \circ f to the zero morphism.

X f Y η * cone(f) \array{ X &\stackrel{f}{\to}& Y \\ \downarrow &\swArrow_{\eta}& \downarrow \\ * &\to& cone(f) }

The interested reader can find more on the conceptual background of this construction at factorization lemma and at homotopy pullback.

Proposition

This colimit, in turn, may be computed in two stages by two consecutive pushouts in CC, and in two ways by the following pasting diagram:

X f Y i 1 X i 0 cyl(X) cyl(f) * cone(X) cone(f). \array{ && X &\stackrel{f}{\to}& Y \\ && \downarrow^{i_1} && \downarrow \\ X &\stackrel{i_0}{\to}& cyl(X) &\to & cyl(f) \\ \downarrow && \downarrow && \downarrow \\ {*} &\to& cone(X) &\to& cone(f) } \,.

Here every square is a pushout, (and so by the pasting law is every rectangular pasting composite).

This now is a basic fact in ordinary category theory. The pushouts appearing here go by the following names:

Definition

The pushout

X i 0 cyl(X) * cone(X) \array{ X &\stackrel{i_0}{\to}& cyl(X) \\ \downarrow && \downarrow \\ {*} &\to& cone(X) }

defines the cone cone(X)cone(X) over XX (with respect to the chosen cylinder object): the result of taking the cylinder over XX and identifying one XX-shaped end with the point.

The pushout

X f Y cyl(X) cyl(f) \array{ X &\stackrel{f}{\to}& Y \\ \downarrow && \downarrow \\ cyl(X) &\to& cyl(f) }

defines the mapping cylinder cyl(f)cyl(f) of ff, the result of identifying one end of the cylinder over XX with YY, using ff as the gluing map.

The pushout

cyl(x) cyl(f) cone(X) cone(f) \array{ cyl(x) &\to& cyl(f) \\ \downarrow && \downarrow \\ cone(X) &\to& cone(f) }

defines the mapping cone cone(f)cone(f) of ff: the result of forming the cyclinder over XX and then identifying one end with the point and the other with YY, via ff.

Remark

As in remark 26 all these step have evident heuristic geometric interpretations:

  1. cone(X)cone(X) is obtained from the cylinder over XX by contracting one end of the cylinder to the point;

  2. cyl(f)cyl(f) is obtained from the cylinder over XX by gluing YY to one end of the cylinder, as specified by the map ff;

We discuss now this general construction of the mapping cone cone(f)cone(f) for a chain map ff between chain complexes. The end result is prop. 43 below, reproducing the classical formula for the mapping cone.

Definition

Write * Ch (𝒜)*_\bullet \in Ch_\bullet(\mathcal{A}) for the chain complex concentrated on RR in degree 0

* 0=[00R]. *_\bullet 0 = [\cdots \to 0 \to 0 \to R] \,.
Remark

This may be understood as the normalized chain complex of chains of simplices on the terminal simplicial set Δ 0\Delta^0, the 0-simplex.

Definition

Let I Ch (𝒜)I_\bullet \in Ch_{\bullet}(\mathcal{A}) be given by

I =(00R(id,id)RR). I_\bullet = (\cdots 0 \to 0 \to R \stackrel{(-id,id)}{\to} R \oplus R) \,.

Denote by

i 0:* I i_0 : *_\bullet \to I_\bullet

the chain map which in degree 0 is the canonical inclusion into the second summand of a direct sum and by

i 1:* I i_1 : *_\bullet \to I_\bullet

correspondingly the canonical inclusion into the first summand.

Remark

This is the standard interval object in chain complexes.

It is in fact the normalized chain complex of chains on a simplicial set for the canonical simplicial interval, the 1-simplex:

I =C (Δ[1]). I_\bullet = C_\bullet(\Delta[1]) \,.

The differential I=(id,id)\partial^I = (-id, id) here expresses the alternating face map complex boundary operator, which in terms of the three non-degenerate basis elements is given by

(01)=(1)(0). \partial ( 0 \to 1 ) = (1) - (0) \,.

We decompose the proof of this statement is a sequence of substatements.

Proposition

For X Ch X_\bullet \in Ch_\bullet the tensor product of chain complexes

(IX) Ch (I \otimes X)_\bullet \in Ch_\bullet

is a cylinder object of X X_\bullet for the structure of a category of cofibrant objects on Ch Ch_\bullet whose cofibrations are the monomorphisms and whose weak equivalences are the quasi-isomorphisms (the substructure of the standard injective model structure on chain complexes).

Example

In example 39 above we saw the cyclinder over the interval itself: the square.

Proposition

The complex (IX) (I \otimes X)_\bullet has components

(IX) n=X nX nX n1 (I \otimes X)_n = X_n \oplus X_n \oplus X_{n-1}

and the differential is given by

X n+1X n+1 X X X nX n (id,id) X n X X n1, \array{ X_{n+1} \oplus X_{n+1} &\stackrel{\partial^X \oplus \partial^X}{\to}& X_n \oplus X_n \\ \oplus &\nearrow_{(-id,id)}& \oplus \\ X_{n} &\underset{-\partial^X}{\to}& X_{n-1} } \,,

hence in matrix calculus by

IX=( X X (id,id) 0 X):(X n+1X n+1)X n(X nX n)X n1. \partial^{I \otimes X} = \left( \array{ \partial^X \oplus \partial^X & (-id, id) \\ 0 & -\partial^X } \right) : (X_{n+1} \oplus X_{n+1}) \oplus X_{n} \to (X_{n} \oplus X_{n}) \oplus X_{n-1} \,.
Proof

By the formula discussed at tensor product of chain complexes the components arise as the direct sum

(IX) n=(R (0)X n)(R (1)X n)(R (01)X (n1)) (I \otimes X )_n = (R_{(0)} \otimes X_n ) \oplus (R_{(1)} \otimes X_n ) \oplus (R_{(0 \to 1)} \otimes X_{(n-1)} )

and the differential picks up a sign when passed past the degree-1 term R (01)R_{(0 \to 1)}:

IX((01),x) =(( I(01)),x)((01), Xx) =((0)+(1),x)((01), Xx) =((0),x)+((1),x)((01), Xx). \begin{aligned} \partial^{I \otimes X} ( (0 \to 1), x ) &= ( (\partial^I (0 \to 1)), x ) - ( (0\to 1), \partial^X x ) \\ & = ( - (0) + (1), x ) - ( (0 \to 1), \partial^X x ) \\ & = -((0), x) + ((1), x) - ( (0 \to 1), \partial^X x ) \end{aligned} \,.
Remark

The two boundary inclusions of X X_\bullet into the cylinder are given in terms of def. 57 by

i 0 X:X * X i 0id X(IX) i^X_0 : X_\bullet \simeq *_\bullet \otimes X_\bullet \stackrel{i_0 \otimes id_X}{\to} (I\otimes X)_\bullet

and

i 1 X:X * X i 1id X(IX) i^X_1 : X_\bullet \simeq *_\bullet \otimes X_\bullet \stackrel{i_1 \otimes id_X}{\to} (I\otimes X)_\bullet

which in components is the inclusion of the second or first direct summand, respectively

X nX nX nX n1. X_n \hookrightarrow X_n \oplus X_n \oplus X_{n-1} \,.

One part of definition 55 now reads:

Definition

For f :X Y f_\bullet : X_\bullet \to Y_\bullet a chain map, the mapping cylinder cyl(f)cyl(f) is the pushout

cyl(f) Y f I X i 0 X . \array{ cyl(f)_\bullet &\leftarrow& Y_\bullet \\ \uparrow && \uparrow^{\mathrlap{f}} \\ I_\bullet \otimes X_\bullet &\stackrel{i_0}{\leftarrow}& X_\bullet } \,.
Proposition

The components of cyl(f)cyl(f) are

cyl(f) n=X nY nX n1 cyl(f)_n = X_n \oplus Y_n \oplus X_{n-1}

and the differential is given by

X n+1Y n+1 X Y X nY n (id,f) X n X X n1, \array{ X_{n+1} \oplus Y_{n+1} &\stackrel{\partial^X \oplus \partial^Y}{\to}& X_n \oplus Y_n \\ \oplus &\nearrow_{(-id,f)}& \oplus \\ X_{n} &\underset{-\partial^X}{\to}& X_{n-1} } \,,

hence in matrix calculus by

cyl(f)=( X Y (id,f n) 0 X):(X n+1Y n+1)X n(X nY n)X n1. \partial^{cyl(f)} = \left( \array{ \partial^X \oplus \partial^Y & (-id, f_n) \\ 0 & -\partial^X } \right) : (X_{n+1} \oplus Y_{n+1}) \oplus X_{n} \to (X_{n} \oplus Y_{n}) \oplus X_{n-1} \,.
Proof

The colimits in a category of chain complexes Ch (𝒜)Ch_\bullet(\mathcal{A}) are computed in the underlying presheaf category of towers in 𝒜\mathcal{A}. There they are computed degreewise in 𝒜\mathcal{A} (see at limits in presheaf categories). Here the statement is evident:

the pushout identifies one direct summand X nX_n with Y nY_n along f nf_n and so where previously a id X nid_{X_n} appeared on the diagonl, there is now f nf_n.

The last part of definition 55 now reads:

Definition

For f :X Y f_\bullet : X_\bullet \to Y_\bullet a chain map, the mapping cone cone(f)cone(f) is the pushout

cone(f) cyl(f) cone(X) XI i 1 0 X \array{ cone(f) &\leftarrow& cyl(f) \\ \uparrow && \uparrow \\ cone(X) &\leftarrow& X \otimes I \\ \uparrow && \uparrow^{\mathrlap{i_1}} \\ 0 &\leftarrow& X }
Proposition

The components of the mapping cone cone(f)cone(f) are

cone(f) n=Y nX n1 cone(f)_n = Y_n \oplus X_{n-1}

with differential given by

Y n+1 Y Y n f n X n X X n1, \array{ Y_{n+1} &\stackrel{\partial^Y}{\to}& Y_n \\ \oplus &\nearrow_{f_n}& \oplus \\ X_{n} &\underset{-\partial^X}{\to}& X_{n-1} } \,,

and hence in matrix calculus by

cone(f)=( n Y f n 0 n X):Y n+1X nY nX n1. \partial^{cone(f)} = \left( \array{ \partial^Y_n & f_n \\ 0 & -\partial^X_n } \right) : Y_{n+1} \oplus X_{n} \to Y_{n} \oplus X_{n-1} \,.
Proof

As before the pushout is computed degreewise. This identifies the remaining unshifted copy of XX with 0.

Proposition

For f:X Y f : X_\bullet \to Y_\bullet a chain map, the canonical inclusion i:Y cone(f) i : Y_\bullet \to cone(f)_\bullet of Y Y_\bullet into the mapping cone of ff is given in components

i n:Y ncone(f) n=Y nX n1 i_n : Y_n \to cone(f)_n = Y_n \oplus X_{n-1}

by the canonical inclusion of a summand into a direct sum.

Proof

This follows by starting with remark 31 and then following these inclusions through the formation of the two colimits as discussed above.

Using these mapping cones of chain maps, we now explain how the long exact sequences of homology groups, prop. 32, are a shadow under homology of genuine homotopy cofiber sequences of the chain complexes themselves.

Let f:X Y f : X_\bullet \to Y_\bullet be a chain map and write cone(f)Ch (𝒜)cone(f) \in Ch_\bullet(\mathcal{A}) for its mapping cone as explicitly given in prop. 43.

Definition

Write X[1] Ch (𝒜)X[1]_\bullet \in Ch_\bullet(\mathcal{A}) for the suspension of a chain complex of XX. Write

p:cone(f)X[1] p : cone(f) \to X[1]_\bullet

for the chain map which in components

p n:cone(f) nX[1] n p_n : cone(f)_n \to X[1]_n

is given, via prop. 43, by the canonical projection out of a direct sum

p n:Y nX n1X n1. p_n : Y_\n \oplus X_{n-1} \to X_{n-1} \,.

This defines the mapping cone construction on chain complex. Its definition as a universal left homotopy should make the following proposition at least plausible, which we cannot prove yet at this point, but which we state nevertheless to highlight the meaning of the mapping cone construction. The tools for the proof of propositions like this are discussed further below in 7) Derived categories and derived functors.

Proposition

The chain map p:cone(f) X[1] p : cone(f)_\bullet \to X[1]_\bullet represents the homotopy cofiber of the canonical map i:Y cone(f) i : Y_\bullet \to cone(f)_\bullet.

Proof

By prop. 44 and def. 60 the sequence

Y icone(f) pX[1] Y_\bullet \stackrel{i}{\to} cone(f)_\bullet \stackrel{p}{\to} X[1]_\bullet

is a short exact sequence of chain complexes (since it is so degreewise, in fact degreewise it is even a split exact sequence, def. 46). In particular we have a cofiber pushout diagram

Y i cone(f) 0 X[1] . \array{ Y_\bullet &\stackrel{i}{\hookrightarrow}& cone(f)_\bullet \\ \downarrow && \downarrow \\ 0 &\to& X[1]_\bullet } \,.

Now, in the injective model structure on chain complexes all chain complexes are cofibrant objects and an inclusion such as i:Y cone(f) i : Y_\bullet \hookrightarrow cone(f)_\bullet is a cofibration. By the detailed discussion at homotopy limit this means that the ordinary colimit here is in fact a homotopy colimit, hence exhibits pp as the homotopy cofiber of ii.

Accordingly one says:

Corollary

For f :X Y f_\bullet : X_\bullet \to Y_\bullet a chain map, there is a homotopy cofiber sequence of the form

X f Y i cone(f) p X[1] f[1] Y i[1] cone(f) p[1] X[2] X_\bullet \stackrel{f_\bullet}{\to} Y_\bullet \stackrel{i_\bullet}{\to} cone(f)_\bullet \stackrel{p_\bullet}{\to} X[1]_\bullet \stackrel{f[1]_\bullet}{\to} Y_\bullet \stackrel{i[1]_\bullet}{\to} cone(f)_\bullet \stackrel{p[1]_\bullet}{\to} X[2]_\bullet \to \cdots

In order to compare this to the discussion of connecting homomorphisms, we now turn attention to the case that f f_\bullet happens to be a monomorphism. Notice that this we can always assume, up to quasi-isomorphism, for instance by prolonging ff by the map into its mapping cylinder

X Y cyl(f). X_\bullet \to Y_\bullet \stackrel{\simeq}{\to} cyl(f) \,.

By the axioms on an abelian category in this case we have a short exact sequence

0X f Y p Z 0 0 \to X_\bullet \stackrel{f_\bullet}{\to} Y_\bullet \stackrel{p_\bullet}{\to} Z_\bullet \to 0

of chain complexes. The following discussion revolves around the fact that now cone(f) cone(f)_\bullet as well as Z Z_\bullet are both models for the homotopy cofiber of ff.

Lemma

Let

X f Y p Z X_\bullet \stackrel{f_\bullet}{\to} Y_\bullet \stackrel{p_\bullet}{\to} Z_\bullet

be a short exact sequence of chain complexes.

The collection of linear maps

h n:Y nX n1Y nZ n h_n : Y_n \oplus X_{n-1} \to Y_n \stackrel{}{\to} Z_n

constitutes a chain map

h :cone(f) Z . h_\bullet : cone(f)_\bullet \to Z_\bullet \,.

This is a quasi-isomorphism. The inverse of H n(h )H_n(h_\bullet) is given by sending a representing cycle zZ nz \in Z_n to

(z^ n, Yz^ n)Y nX n+1, (\hat z_n, \partial^Y \hat z_n) \in Y_n \oplus X_{n+1} \,,

where z^ n\hat z_n is any choice of lift through p np_n and where Yz^ n\partial^Y \hat z_n is the formula expressing the connecting homomorphism in terms of elements, as discussed at Connecting homomorphism – In terms of elements.

Finally, the morphism i :Y cone(f) i_\bullet : Y_\bullet \to cone(f)_\bullet is eqivalent in the homotopy category (the derived category) to the zigzag

cone(f) h Y Z . \array{ && cone(f)_\bullet \\ && \downarrow^{\mathrlap{h}}_{\mathrlap{\simeq}} \\ Y_\bullet &\to& Z_\bullet } \,.
Proof

To see that h h_\bullet defines a chain map recall the differential cone(f)\partial^{cone(f)} from prop. 43, which acts by

cone(f)(x n1,z^ n)=( Xx n1, Yz^ n+x n1) \partial^{cone(f)} (x_{n-1}, \hat z_n) = ( -\partial^X x_{n-1} , \partial^Y \hat z_n + x_{n-1} )

and use that x n1x_{n-1} is in the kernel of p np_n by exactness, hence

h n1 cone(f)(x n1,z^ n) =h n1( Xx n1, Yz^ n+x n1) =p n1( Yz^ n+x n1) =p n1( Yz^ n) = Zp nz^ n = Zh n(x n1,z^ n). \begin{aligned} h_{n-1}\partial^{cone(f)}(x_{n-1}, \hat z_n) &= h_{n-1}( -\partial^X x_{n-1}, \partial^Y \hat z_n + x_{n-1} ) \\ & = p_{n-1}( \partial^Y \hat z_n + x_{n-1}) \\ & = p_{n-1}( \partial^Y \hat z_n ) \\ & = \partial^Z p_n \hat z_n \\ & = \partial^Z h_n(x_{n-1}, \hat z_n) \end{aligned} \,.

It is immediate to see that we have a commuting diagram of the form

cone(f) i h Y Z \array{ && cone(f)_\bullet \\ & {}^{\mathllap{i_\bullet}}\nearrow& \downarrow^{\mathrlap{h}}_{\mathrlap{\simeq}} \\ Y_\bullet &\to& Z_\bullet }

since the composite morphism is the inclusion of YY followed by the bottom morphism on YY.

Abstractly, this already implies that cone(f) Z cone(f)_\bullet \to Z_\bullet is a quasi-isomorphism, for this diagram gives a morphism of cocones under the diagram defining cone(f)cone(f) in prop. 38 and by the above both of these cocones are homotopy-colimiting.

But in checking the claimed inverse of the induced map on homology groups, we verify this also explicity:

We first determine those cycles (x n1,y n)cone(f) n(x_{n-1}, y_n) \in cone(f)_n which lift a cycle z nz_n. By lemma 38 a lift of chains is any pair of the form (x n1,z^ n)(x_{n-1}, \hat z_n) where z^ n\hat z_n is a lift of z nz_n through Y nX nY_n \to X_n. So x n1x_{n-1} has to be found such that this pair is a cycle. By prop. 43 the differential acts on it by

cone(f)(x n1,z^ n)=( Xx n1, Yz^ n+x n1) \partial^{cone(f)} (x_{n-1}, \hat z_n) = ( -\partial^X x_{n-1} , \partial^Y \hat z_n + x_{n-1} )

and so the condition is that

x n1 Yz^ nx_{n-1} \coloneqq -\partial^Y \hat z_n (which implies Xx n1= X Yz^ n= Y Yz^ n=0\partial^X x_{n-1} = -\partial^X \partial^Y \hat z_n = -\partial^Y \partial^Y \hat z_n = 0 due to the fact that f nf_n is assumed to be an inclusion, hence that X\partial^X is the restriction of Y\partial^Y to elements in X nX_n).

This condition clearly has a unique solution for every lift z^ n\hat z_n and a lift z^ n\hat z_n always exists since p n:Y nZ np_n : Y_n \to Z_n is surjective, by assumption that we have a short exact sequence of chain complexes. This shows that H n(h )H_n(h_\bullet) is surjective.

To see that it is also injective we need to show that if a cycle ( Yz^ n,z^ n)cone(f) n(-\partial^Y \hat z_n, \hat z_n) \in cone(f)_n maps to a cycle z n=p n(z^ n)z_n = p_n(\hat z_n) that is trivial in H n(Z)H_n(Z) in that there is c n+1c_{n+1} with Zc n+1=z n\partial^Z c_{n+1} = z_n, then also the original cycle was trivial in homology, in that there is (x n,y n+1)(x_n, y_{n+1}) with

cone(f)(x n,y n+1)( Xx n, Yy n+1+x n)=( Yz^ n,z^ n). \partial^{cone(f)}(x_n, y_{n+1}) \coloneqq (-\partial^X x_n, \partial^Y y_{n+1} + x_n) = (-\partial^Y \hat z_n, \hat z_n) \,.

For that let c^ n+1Y n+1\hat c_{n+1} \in Y_{n+1} be a lift of c n+1c_{n+1} through p np_n, which exists again by surjectivity of p n+1p_{n+1}. Observe that

p n(z^ n Yc^ n+1)=z n Z(p nc^ n+1)=z n Z(c n+1)=0 p_{n}( \hat z_n - \partial^Y \hat c_{n+1}) = z_n -\partial^Z ( p_n \hat c_{n+1} ) = z_n - \partial^Z ( c_{n+1} ) = 0

by assumption on z nz_n and c n+1c_{n+1}, and hence that z^ n Yc^ n+1\hat z_n - \partial^Y \hat c_{n+1} is in X nX_n by exactness.

Hence (z n Yc^ n+1,c^ n+1)cone(f) n(z_n - \partial^Y \hat c_{n+1}, \hat c_{n+1}) \in cone(f)_n trivializes the given cocycle:

cone(f)(z^ n Yc^ n+1,c^ n+1) =( X(z^ n Yc^ n+1), Yc^ n+1+(z^ n Yc^ n+1)) =( Y(z^ n Yc^ n+1),z^ n) =( Yz^ n,z^ n). \begin{aligned} \partial^{cone(f)}( \hat z_n - \partial^Y \hat c_{n+1} , \hat c_{n+1}) & = (-\partial^X(\hat z_n - \partial^Y \hat c_{n+1} ), \partial^Y \hat c_{n+1} + (\hat z_n - \partial^Y \hat c_{n+1} ) ) \\ & = (-\partial^Y(\hat z_n - \partial^Y \hat c_{n+1}), \hat z_n ) \\ & = ( -\partial^Y \hat z_n, \hat z_n ) \end{aligned} \,.
Theorem

Let

X f Y Z X_\bullet \stackrel{f_\bullet}{\to} Y_\bullet \to Z_\bullet

be a short exact sequence of chain complexes.

Then the chain homology functor

H n():Ch (𝒜)𝒜 H_n(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A}

sends the homotopy cofiber sequence of ff, cor. 3, to the long exact sequence in homology induced by the given short exact sequence, hence to

H n(X )H n(Y )H n(Z )δH n1(X )H n1(Y )H n1(Z )δH n2(X ), H_n(X_\bullet) \to H_n(Y_\bullet) \to H_n(Z_\bullet) \stackrel{\delta}{\to} H_{n-1}(X_\bullet) \to H_{n-1}(Y_\bullet) \to H_{n-1}(Z_\bullet) \stackrel{\delta}{\to} H_{n-2}(X_\bullet) \to \cdots \,,

where δ n\delta_n is the nnth connecting homomorphism.

Proof

By lemma 1 the homotopy cofiber sequence is equivalen to the zigzag

cone(f)[1] h[1] cone(f) X[1] f[1] Y[1] Z[1] h X f Y Z . \array{ && && && && && cone(f)[1]_\bullet &\to& \cdots \\ && && && && && \downarrow^{\mathrlap{h[1]_\bullet}}_{\mathrlap{\simeq}} \\ && && cone(f)_\bullet &\to& X[1]_\bullet &\stackrel{f[1]_\bullet}{\to}& Y[1]_\bullet &\to& Z[1]_\bullet \\ && && \downarrow^{\mathrlap{h_\bullet}}_{\mathrlap{\simeq}} \\ X_\bullet &\stackrel{f_\bullet}{\to}& Y_\bullet &\stackrel{}{\to}& Z_\bullet } \,.

Observe that

H n(X[k] )H nk(X ). H_n( X[k]_\bullet) \simeq H_{n-k}(X_\bullet) \,.

It is therefore sufficient to check that

H n(cone(f) X[1] Z ):H n(Z )H n(cone(f) )H n1(X ) H_n \left( \array{ cone(f)_\bullet &\to& X[1]_\bullet \\ \downarrow^{\mathrlap{\simeq}} \\ Z_\bullet } \right) \;\; : \;\; H_n(Z_\bullet) \to H_n(cone(f)_\bullet) \to H_{n-1}(X_\bullet)

equals the connecting homomorphism δ n\delta_n induced by the short exact sequence.

By prop. 1 the inverse of the vertical map is given by choosing lifts and forming the corresponding element given by the connecting homomorphism. By prop. 45 the horizontal map is just the projection, and hence the assignment is of the form

[z n][x n1,y n][x n1]. [z_n] \mapsto [x_{n-1}, y_n] \mapsto [x_{n-1}] \,.

So in total the image of the zig-zag under homology sends

[z n] Z[ Yz^ n] X. [z_n]_Z \mapsto -[\partial^Y \hat z_n]_X \,.

By the discussion there, this is indeed the action of the connecting homomorphism.

In summary, the above says that for every chain map f :X Y f_\bullet : X_\bullet \to Y_\bullet we obtain maps

X fY (0 id Y )cone(f) (id X[1] 0)X[1] X_\bullet \stackrel{f}{\to} Y_\bullet \stackrel{ \left( \array{ 0 \\ id_{Y_\bullet} } \right) }{\to} cone(f)_\bullet \stackrel{ \left( \array{ id_{X[1]_\bullet} & 0 } \right) }{\to} X[1]_\bullet

which form a homotopy fiber sequence and such that this sequence continues by forming suspensions, hence for all nn \in \mathbb{Z} we have

X[n] fY[n] (0 id Y[n] )cone(f)[n] (id X[n+11] 0)X[n+1] X[n]_\bullet \stackrel{f}{\to} Y[n]_\bullet \stackrel{ \left( \array{ 0 \\ id_{Y[n]_\bullet} } \right) }{\to} cone(f)[n]_\bullet \stackrel{ \left( \array{ id_{X[n+11]_\bullet} & 0 } \right) }{\to} X[n+1]_\bullet

To amplify this quasi-cyclic behaviour one sometimes depicts the situation as follows:

X f Y [1] cone(f) \array{ X_\bullet &&\stackrel{f}{\to}&& Y_\bullet \\ & {}_{\mathllap{[1]}}\nwarrow && \swarrow \\ && cone(f)_\bullet }

and hence speaks of a “triangle”, or distinguished triangle or mapping cone triangle of ff.

Due to these “triangles” one calls the homotopy category of chain complexes localized at the quasi-isomorphisms, hence the derived category which we discuss below in 8), a triangulated category.

Double complexes and the diagram chasing lemmas

We have seen in the discussion of the connecting homomorphism in the homology long exact sequence in 4) above that given an exact sequence of chain complexes – hence in particular a chain complex of chain complexes – there are interesting ways to relate elements on the far right to elements on the far left in lower degree. In 5) we had given the conceptual explanation of this phenomenon in terms of long homotopy fiber sequences. But often it is just computationally useful to be able to efficiently establish and compute these “long diagram chase”-relations, independently of a homotopy-theoretic interpretation. Such computational tools we discuss here.

A chain complex of chain complex is called a double complex and so we first introduce this elementary notion and the corresponding notion notion of total complex. (Total complexes are similarly elementary to define but will turn out to play a deeper role as models for homotopy colimits, this we indicate further below in chapter V)).

There is a host of classical diagram-chasing lemmas that relate far-away entries in double complexes that enjoy suitable exactness properties. These go by names such as the snake lemma or the 3x3 lemma. The underlying mechanism of all these lemmas is made most transparent in the salamander lemma. This is fairly trivial to establish, and the notions it induces allow quick transparent proofs of all the other diagram-chasing lemmas.

The discussion to go here is kept at salamander lemma. See there.

Kan complexes – General homotopy types

With groupoids and chain complexes we have seen two kinds of objects which support concepts of homotopy theory, such as a concept of homotopy equivalence between them (equivalence of groupoids on the one hand, and quasi-isomorphism) on the other. In some sense these two cases are opposite extremes in the more general context of homotopy theory:

  • chain complexes have homotopical structure (e.g. chain homology) in arbitrary high degree, i.e. they may be homotopy n-types for arbitrary nn, but they are fully abelian in that there is never any nonabelian group structure in a chain complex, not is there any non-trivial action of the homology groups of a chain complex on each other;

  • groupoids have more general non-abelian structure, for every (nonabelian) group there is a groupoid which has this as its fundamental group, but this fundamental group (in degree 1) is already the highest homotopical structure they carry, groupoids are necessarily homotopy 1-types.

On the other hand, both groupoids and chain complexes naturally have incarnations in the joint context of simplicial sets. We now discuss how their common joint generalization is given by those simplicial sets whose simplices have a sensible notion of composition and inverses, the Kan complexes.

topologicalspaces higherpathgroupoid groupoids Grothendiecknerve Kancomplexesgroupoids DoldKancorrespondence chaincomplexes includedin simplicialsets \array{ && {topological \atop spaces} \\ && \downarrow^{\mathrlap{{higher \atop path}\atop groupoid}} & \\ groupoids &\stackrel{{Grothendieck \atop nerve}}{\longrightarrow}& { {\mathbf{Kan}\;\mathbf{complexes}} \atop {\simeq \infty-groupoids} } &\stackrel{{Dold-Kan \atop correspondence}}{\longleftarrow}& {chain \atop complexes} \\ && \downarrow^{\mathrlap{included \atop in}} \\ && {simplicial \atop sets} }

Kan complexes serve as a standard powerful model on wich the complete formulation of homotopy theory (without geometry) may be formulated.

Intuitive idea – Composition of higher order (gauge-)symmetries

An ∞-groupoid is first of all supposed to be a structure that has k-morphisms for all kk \in \mathbb{N}, which for k1k \geq 1 go between (k1)(k-1)-morphisms. A useful tool for organizing such collections of morphisms is the notion of a simplicial set. This is a functor on the opposite category of the simplex category Δ\Delta, whose objects are the abstract cellular kk-simplices, denoted [k][k] or Δ[k]\Delta[k] for all kk \in \mathbb{N}, and whose morphisms Δ[k 1]Δ[k 2]\Delta[k_1] \to \Delta[k_2] are all ways of mapping these into each other. So we think of such a simplicial set given by a functor

K:Δ opSet K : \Delta^{op} \to Set

as specifying

and generally

as well as specifying

  • functions ([n][n+1])(K n+1K n)([n] \hookrightarrow [n+1]) \mapsto (K_{n+1} \to K_n) that send n+1n+1-morphisms to their boundary nn-morphisms;

  • functions ([n+1][n])(K nK n+1)([n+1] \to [n]) \mapsto (K_{n} \to K_{n+1}) that send nn-morphisms to identity (n+1)(n+1)-morphisms on them.

The fact that KK is supposed to be a functor enforces that these assignments of sets and functions satisfy conditions that make consistent our interpretation of them as sets of kk-morphisms and source and target maps between these. These are called the simplicial identities.

But apart from this source-target matching, a generic simplicial set does not yet encode a notion of composition of these morphisms.

For instance for Λ 1[2]\Lambda^1[2] the simplicial set consisting of two attached 1-cells

Λ 1[2]={ 1 0 2} \Lambda^1[2] = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&&& 2 } \right\}

and for (f,g):Λ 1[2]K(f,g) : \Lambda^1[2] \to K an image of this situation in KK, hence a pair x 0fx 1gx 2x_0 \stackrel{f}{\to} x_1 \stackrel{g}{\to} x_2 of two composable 1-morphisms in KK, we want to demand that there exists a third 1-morphisms in KK that may be thought of as the composition x 0hx 2x_0 \stackrel{h}{\to} x_2 of ff and gg. But since we are working in higher category theory (and not to be evil), we want to identify this composite only up to a 2-morphism equivalence

x 1 f g x 0 h x 2. \array{ && x_1 \\ & {}^{\mathllap{f}}\nearrow &\Downarrow^{\mathrlap{\simeq}}& \searrow^{\mathrlap{g}} \\ x_0 &&\stackrel{h}{\to}&& x_2 } \,.

From the picture it is clear that this is equivalent to demanding that for Λ 1[2]Δ[2]\Lambda^1[2] \hookrightarrow \Delta[2] the obvious inclusion of the two abstract composable 1-morphisms into the 2-simplex we have a diagram of morphisms of simplicial sets

Λ 1[2] (f,g) K h Δ[2]. \array{ \Lambda^1[2] &\stackrel{(f,g)}{\to}& K \\ \downarrow & \nearrow_{\mathrlap{\exists h}} \\ \Delta[2] } \,.

A simplicial set where for all such (f,g)(f,g) a corresponding such hh exists may be thought of as a collection of higher morphisms that is equipped with a notion of composition of adjacent 1-morphisms.

For the purpose of describing groupoidal composition, we now want that this composition operation has all inverses. For that purpose, notice that for

Λ 2[2]={ 1 0 2} \Lambda^2[2] = \left\{ \array{ && 1 \\ & && \searrow \\ 0 &&\to&& 2 } \right\}

the simplicial set consisting of two 1-morphisms that touch at their end, hence for

(g,h):Λ 2[2]K (g,h) : \Lambda^2[2] \to K

two such 1-morphisms in KK, then if gg had an inverse g 1g^{-1} we could use the above composition operation to compose that with hh and thereby find a morphism ff connecting the sources of hh and gg. This being the case is evidently equivalent to the existence of diagrams of morphisms of simplicial sets of the form

Λ 2[2] (g,h) K f Δ[2]. \array{ \Lambda^2[2] &\stackrel{(g,h)}{\to}& K \\ \downarrow & \nearrow_{\mathrlap{\exists f}} \\ \Delta[2] } \,.

Demanding that all such diagrams exist is therefore demanding that we have on 1-morphisms a composition operation with inverses in KK.

In order for this to qualify as an \infty-groupoid, this composition operation needs to satisfy an associativity law up to coherent 2-morphisms, which means that we can find the relevant tetrahedras in KK. These in turn need to be connected by pentagonators and ever so on. It is a nontrivial but true and powerful fact, that all these coherence conditions are captured by generalizing the above conditions to all dimensions in the evident way:

let Λ i[n]Δ[n]\Lambda^i[n] \hookrightarrow \Delta[n] be the simplicial set – called the iith nn-horn – that consists of all cells of the nn-simplex Δ[n]\Delta[n] except the interior nn-morphism and the iith (n1)(n-1)-morphism.

Then a simplicial set is called a Kan complex, if for all images f:Λ i[n]Kf : \Lambda^i[n] \to K of such horns in KK, the missing two cells can be found in KK- in that we can always find a horn filler σ\sigma in the diagram

Λ i[n] f K σ Δ[n]. \array{ \Lambda^i[n] &\stackrel{f}{\to}& K \\ \downarrow & \nearrow_{\mathrlap{\sigma}} \\ \Delta[n] } \,.

The basic example is the nerve N(C)sSetN(C) \in sSet of an ordinary groupoid CC, which is the simplicial set with N(C) kN(C)_k being the set of sequences of kk composable morphisms in CC. The nerve operation is a full and faithful functor from 1-groupoids into Kan complexes and hence may be thought of as embedding 1-groupoids in the context of general ∞-groupoids.

Definition

Recall the definition of simplicial sets from above. Let

Δ[n]=Δ(,[n])SimpSet \Delta[n] = \mathbf{\Delta}( -, [n]) \in Simp Set

be the standard simplicial nn-simplex in SimpSet.

Definition

For each ii, 0in0 \leq i \leq n, the (n,i)(n,i)-horn or (n,i)(n,i)-box is the subsimplicial set

Λ i[n]Δ[n] \Lambda^i[n] \hookrightarrow \Delta[n]

which is the union of all faces except the i thi^{th} one.

This is called an outer horn if k=0k = 0 or k=nk = n. Otherwise it is an inner horn.

Remark

Since sSet is a presheaf topos, unions of subobjects make sense and they are calculated objectwise, thus in this case dimensionwise. This way it becomes clear what the structure of a horn as a functor Λ k[n]:Δ opSet\Lambda^k[n]: \Delta^{op} \to Set must therefore be: it takes [m][m] to the collection of ordinal maps f:[m][n]f: [m] \to [n] which do not have the element kk in the image.

Example

The inner horn, def. 61 of the 2-simplex

Δ 2={ 1 0 2} \Delta^2 = \left\{ \array{ && 1 \\ & \nearrow &\Downarrow& \searrow \\ 0 &&\to&& 2 } \right\}

with boundary

Δ 2={ 1 0 2}\partial \Delta^2 = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&\to&& 2 } \right\}

looks like

Λ 1 2={ 1 0 2}. \Lambda^2_1 = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&&& 2 } \right\} \,.

The two outer horns look like

Λ 0 2={ 1 0 2}\Lambda^2_0 = \left\{ \array{ && 1 \\ & \nearrow && \\ 0 &&\to&& 2 } \right\}

and

Λ 2 2={ 1 0 2}\Lambda^2_2 = \left\{ \array{ && 1 \\ & && \searrow \\ 0 &&\to&& 2 } \right\}

respectively.

Definition

A Kan complex is a simplicial set SS that satisfies the Kan condition,

  • which says that all horns of the simplicial set have fillers/extend to simplices;

  • which means equivalently that the unique homomorphism SptS \to pt from SS to the point (the terminal simplicial set) is a Kan fibration;

  • which means equivalently that for all diagrams in sSet of the form

    Λ i[n] S Δ[n] ptΛ i[n] S Δ[n] \array{ \Lambda^i[n] &\to& S \\ \downarrow && \downarrow \\ \Delta[n] &\to& pt } \;\;\; \leftrightarrow \;\;\; \array{ \Lambda^i[n] &\to& S \\ \downarrow && \\ \Delta[n] }

    there exists a diagonal morphism

    Λ i[n] S Δ[n] ptΛ i[n] S Δ[n] \array{ \Lambda^i[n] &\to& S \\ \downarrow &\nearrow& \downarrow \\ \Delta[n] &\to& pt } \;\;\; \leftrightarrow \;\;\; \array{ \Lambda^i[n] &\to& S \\ \downarrow &\nearrow& \\ \Delta[n] }

    completing this to a commuting diagram;

  • which in turn means equivalently that the map from nn-simplices to (n,i)(n,i)-horns is an epimorphism

    [Δ[n],S][Λ i[n],S]. [\Delta[n], S]\, \twoheadrightarrow \,[\Lambda^i[n],S] \,.
Groupoids as Kan complexes – Grothendieck simplicial nerve
Definition

For 𝒢 \mathcal{G}_\bullet a groupoid, def. 21, its simplicial nerve N(𝒢 ) N(\mathcal{G}_\bullet)_\bullet is the simplicial set with

N(𝒢 ) n𝒢 1 × 𝒢 0 n N(\mathcal{G}_\bullet)_n \coloneqq \mathcal{G}_1^{\times_{\mathcal{G}_0}^n}

the set of sequences of composable morphisms of length nn, for nn \in \mathbb{N};

with face maps

d k:N(𝒢 ) n+1N(𝒢 ) n d_k \colon N(\mathcal{G}_\bullet)_{n+1} \to N(\mathcal{G}_\bullet)_{n}

being,

  • for n=0n = 0 the functions that remembers the kkth object;

  • for n1n \geq 1

    • the two outer face maps d 0d_0 and d nd_n are given by forgetting the first and the last morphism in such a sequence, respectively;

    • the n1n-1 inner face maps d 0<k<nd_{0 \lt k \lt n} are given by composing the kkth morphism with the k+1k+1st in the sequence.

The degeneracy maps

s k:N(𝒢 )nN(𝒢 ) n+1. s_k \colon N(\mathcal{G}_\bullet)n \to N(\mathcal{G}_\bullet)_{n+1} \,.

are given by inserting an identity morphism on x kx_k.

Remark

Spelling this out in more detail: write

𝒢 n={x 0f 0,1x 1f 1,2x 2f 2,3f n1,nx n} \mathcal{G}_n = \left\{ x_0 \stackrel{f_{0,1}}{\to} x_1 \stackrel{f_{1,2}}{\to} x_2 \stackrel{f_{2,3}}{\to} \cdots \stackrel{f_{n-1,n}}{\to} x_n \right\}

for the set of sequences of nn composable morphisms. Given any element of this set and 0<k<n0 \lt k \lt n , write

f i1,i+1f i,i+1f i1,i f_{i-1,i+1} \coloneqq f_{i,i+1} \circ f_{i-1,i}

for the comosition of the two morphism that share the iith vertex.

With this, face map d kd_k acts simply by “removing the index kk”:

d 0:(x 0f 0,1x 1f 1,2x 2f n1,nx n)(x 1f 1,2x 2f n1,nx n) d_0 \colon (x_0 \stackrel{f_{0,1}}{\to} x_1 \stackrel{f_{1,2}}{\to} x_{2} \cdots \stackrel{f_{n-1,n}}{\to} x_n ) \mapsto (x_1 \stackrel{f_{1,2}}{\to} x_{2} \cdots \stackrel{f_{n-1,n}}{\to} x_n )
d 0<k<n:(x 0x k1f k1,kx kf k,k+1x k+1x n)(x 0x k1f k1,k+1x k+1x n) d_{0\lt k \lt n} \colon ( x_0 \cdots \stackrel{}{\to} x_{k-1} \stackrel{f_{k-1,k}}{\to} x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \stackrel{}{\to} \cdots x_n ) \mapsto ( x_0 \cdots \stackrel{}{\to} x_{k-1} \stackrel{f_{k-1,k+1}}{\to} x_{k+1} \stackrel{}{\to} \cdots x_n )
d n:(x 0f 0,1f n2,n1x n1f n1,nx n)(x 0f 0,1f n2,n1x n1). d_n \colon ( x_0 \stackrel{f_{0,1}}{\to} \cdots \stackrel{f_{n-2,n-1}}{\to} x_{n-1} \stackrel{f_{n-1,n}}{\to} x_n ) \mapsto ( x_0 \stackrel{f_{0,1}}{\to} \cdots \stackrel{f_{n-2,n-1}}{\to} x_{n-1} ) \,.

Similarly, writing

f k,kid x k f_{k,k} \coloneqq id_{x_k}

for the identity morphism on the object x kx_k, then the degenarcy map acts by “repeating the kkth index”

s k:(x 0x kf k,k+1x k+1)(x 0x kf k,kx kf k,k+1x k+1). s_k \colon ( x_0 \stackrel{}{\to} \cdots \to x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \to \cdots ) \mapsto ( x_0 \stackrel{}{\to} \cdots \to x_k \stackrel{f_{k,k}}{\to} x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \to \cdots ) \,.

This makes it manifest that these functions organise into a simplicial set.

Proposition

These collections of maps in def. 63 satisfy the simplicial identities, hence make the nerve 𝒢 \mathcal{G}_\bullet into a simplicial set. Moreover, this simplicial set is a Kan complex, where each horn has a unique filler (extension to a simplex).

(A 2-coskeletal Kan complex.)

Proposition

The nerve operation constitutes a full and faithful functor

N:GrpdKanCplxsSet. N \colon Grpd \to KanCplx \hookrightarrow sSet \,.
Chain complexes as Kan complexes – Dold-Kan-Moore correspondence

In the familiar construction of singular homology recalled above one constructs the alternating face map chain complex of the simplicial abelian group of singular simplices, def. 13. This construction is natural and straightforward, but the result chain complex tends to be very “large” even if its chain homology groups end up being very “small”. But in the context of homotopy theory one is to consider all objects notup to isomorphism, but of to weak equivalence, which for chain complexes means up to quasi-isomorphisms. Hence one should look for the natural construction of “smaller” chain complexes that are still quasi-isomorphic to these alternating face map complexes. This is accomplished by the normalized chain complex construction:

Definition

For AA a simplicial abelian group its alternating face map complex (CA) (C A)_\bullet of AA is the chain complex which

  • in degree nn is given by the group A nA_n itself

    (CA) n:=A n (C A)_n := A_n
  • with differential given by the alternating sum of face maps (using the abelian group structure on AA)

    n i=0 n(1) id i:(CA) n(CA) n1. \partial_n \coloneqq \sum_{i = 0}^n (-1)^i d_i \;\colon\; (C A)_n \to (C A)_{n-1} \,.
Lemma

The differential in def. 64 is well-defined in that it indeed squares to 0.

Proof

Using the simplicial identity, prop. 1, d id j=d j1d id_i \circ d_j = d_{j-1} \circ d_i for i<ji \lt j one finds:

n n+1 = i,j(1) i+jd id j = ij(1) i+jd id j+ i<j(1) i+jd id j = ij(1) i+jd id j+ i<j(1) i+jd j1d i = ij(1) i+jd id j ik(1) i+kd kd i =0. \begin{aligned} \partial_n \partial_{n+1} & = \sum_{i, j} (-1)^{i+j} d_i \circ d_{j} \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j + \sum_{i \lt j} (-1)^{i+j} d_i \circ d_j \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j + \sum_{i \lt j} (-1)^{i+j} d_{j-1} \circ d_i \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j - \sum_{i \leq k} (-1)^{i+k} d_{k} \circ d_i \\ &= 0 \end{aligned} \,.
Definition

Given a simplicial abelian group AA, its normalized chain complex or Moore complex is the \mathbb{N}-graded chain complex ((NA) ,)((N A)_\bullet,\partial ) which

  • is in degree nn the joint kernel

    (NA) n= i=1 nkerd i n (N A)_n=\bigcap_{i=1}^{n}ker\,d_i^n

    of all face maps except the 0-face;

  • with differential given by the remaining 0-face map

    n:=d 0 n| (NA) n:(NA) n(NA) n1. \partial_n := d_0^n|_{(N A)_n} : (N A)_n \rightarrow (N A)_{n-1} \,.
Remark

We may think of the elements of the complex NAN A, def. 65, in degree kk as being kk-dimensional disks in AA all whose boundary is captured by a single face:

  • an element gNG 1g \in N G_1 in degree 1 is a 1-disk

    1gg, 1 \stackrel{g}{\to} \partial g \,,
  • an element hNG 2h \in N G_2 is a 2-disk

    1 1 h h 1 1 1, \array{ && 1 \\ & {}^1\nearrow &\Downarrow^h& \searrow^{\partial h} \\ 1 &&\stackrel{1}{\to}&& 1 } \,,
  • a degree 2 element in the kernel of the boundary map is such a 2-disk that is closed to a 2-sphere

    1 1 h h=1 1 1 1, \array{ && 1 \\ & {}^1\nearrow &\Downarrow^h& \searrow^{\partial h = 1} \\ 1 &&\stackrel{1}{\to}&& 1 } \,,

etc.

Definition

Given a simplicial group AA (or in fact any simplicial set), then an element aA n+1a \in A_{n+1} is called degenerate (or thin) if it is in the image of one of the simplicial degeneracy maps s i:A nA n+1s_i \colon A_n \to A_{n+1}. All elements of A 0A_0 are regarded a non-degenerate. Write

D(A n+1) is i(A n)A n+1 D (A_{n+1}) \coloneqq \langle \cup_i s_i(A_{n}) \rangle \hookrightarrow A_{n+1}

for the subgroup of A n+1A_{n+1} which is generated by the degenerate elements (i.e. the smallest subgroup containing all the degenerate elements).

Definition

For AA a simplicial abelian group its alternating face maps chain complex modulo degeneracies, (CA)/(DA)(C A)/(D A) is the chain complex

  • which in degree 0 equals is just ((CA)/D(A)) 0A 0((C A)/D(A))_0 \coloneqq A_0;

  • which in degree n+1n+1 is the quotient group obtained by dividing out the group of degenerate elements, def. 66:

    ((CA)/D(A)) n+1:=A n+1/D(A n+1) ((C A)/D(A))_{n+1} := A_{n+1} / D(A_{n+1})
  • whose differential is the induced action of the alternating sum of faces on the quotient (which is well-defined by lemma 3).

Lemma

Def. 67 is indeed well defined in that the alternating face map differential respects the degenerate subcomplex.

Proof

Using the mixed simplicial identities we find that for s j(a)A ns_j(a) \in A_n a degenerate element, its boundary is

i(1) id is j(a) = i<j(1) is j1d i(a)+ i=j,j+1(1) ia+ i>j+1(1) is jd i1(a) = i<j(1) is j1d i(a)+ i>j+1(1) is jd i1(a) \begin{aligned} \sum_i (-1)^i d_i s_j(a) &= \sum_{i \lt j} (-1)^i s_{j-1} d_i(a) + \sum_{i = j, j+1} (-1)^i a + \sum_{i \gt j+1} (-1)^i s_j d_{i-1}(a) \\ &= \sum_{i \lt j} (-1)^i s_{j-1} d_i(a) + \sum_{i \gt j+1} (-1)^i s_j d_{i-1}(a) \end{aligned}

which is again a combination of elements in the image of the degeneracy maps.

Proposition

Given a simplicial abelian group AA, the evident composite of natural morphisms

NAiAp(CA)/(DA) N A \stackrel{i}{\hookrightarrow} A \stackrel{p}{\to} (C A)/(D A)

from the normalized chain complex, def. 65, into the alternating face map complex modulo degeneracies, def. 67, (inclusion followed by projection to the quotient) is a natural isomorphism of chain complexes.

e.g. (Goerss-Jardine, theorem III 2.1).

Corollary

For AA a simplicial abelian group, there is a splitting

C (A)N (A)D (A) C_\bullet(A) \simeq N_\bullet(A) \oplus D_\bullet(A)

of the alternating face map complex, def. 64 as a direct sum, where the first direct summand is naturally isomorphic to the normalized chain complex of def. 65 and the second is the degenerate cells from def. 67.

Proof

By prop. 48 there is an inverse to the diagonal composite in

CA p (CA)/(DA) i NA. \array{ C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) \\ {}^{\mathllap{i}}\uparrow & \nearrow \\ N A } \,.

This hence exhibits a splitting of the short exact sequence given by the quotient by DAD A.

0 DA CA p (CA)/(DA) 0 i iso NA. \array{ 0 &\to& D A &\hookrightarrow & C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) &\to & 0 \\ && && {}^{\mathllap{i}}\uparrow & \swarrow_{\mathrlap{\simeq}_{iso}} \\ && && N A } \,.
Theorem (Eilenberg-MacLane)

Given a simplicial abelian group AA, then the inclusion

NACA N A \hookrightarrow C A

of the normalized chain complex, def. 65 into the full alternating face map complex, def. 64, is a natural quasi-isomorphism and in fact a natural chain homotopy equivalence, i.e. the complex D (X)D_\bullet(X) is null-homotopic.

(Goerss-Jardine, theorem III 2.4)

Corollary

Given a simplicial abelian group AA, then the projection chain map

(CA)(CA)/(DA) (C A) \longrightarrow (C A)/(D A)

from its alternating face maps complex, def. 64, to the alternating face map complex modulo degeneracies, def. 67, is a quasi-isomorphism.

Proof

Consider the pre-composition of the map with the inclusion of the normalized chain complex, def. 65.

CA p (CA)/(DA) i NA \array{ C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) \\ {}^{\mathllap{i}}\uparrow & \nearrow \\ N A }

By theorem 4 the vertical map is a quasi-isomorphism and by prop. 48 the composite diagonal map is an isomorphism, hence in particular also a quasi-isomorphism. Since quasi-isomorphisms satisfy the two-out-of-three property, it follows that also the map in question is a quasi-isomorphism.

Example

Consider the 1-simplex Δ[1]\Delta[1] regarded as a simplicial set, and write [Δ[1]]\mathbb{Z}[\Delta[1]] for the simplicial abelian group which in each degree is the free abelian group on the simplices in Δ[1]\Delta[1].

This simplicial abelian group starts out as

[Δ[1]]=( 4 3 1 0 2) \mathbb{Z}[\Delta[1]] = \left( \cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}} \mathbb{Z}^4 \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} \mathbb{Z}^3 \stackrel{\overset{\partial_0}{\longrightarrow}}{\underset{\partial_1}{\longrightarrow}} \mathbb{Z}^2 \right)

(where we are indicating only the face maps for notational simplicity).

Here the first 2=\mathbb{Z}^2 = \mathbb{Z}\oplus \mathbb{Z}, the direct sum of two copies of the integers, is the group of 0-chains generated from the two endpoints (0)(0) and (1)(1) of Δ[1]\Delta[1], i.e. the abelian group of formal linear combinations of the form

2{a(0)+b(1)|a,b}. \mathbb{Z}^2 \simeq \left\{ a \cdot (0) + b \cdot (1) | a,b \in \mathbb{Z}\right\} \,.

The second 3\mathbb{Z}^3 \simeq \mathbb{Z}\oplus \mathbb{Z}\oplus \mathbb{Z} is the abelian group generated from the three (!) 1-simplicies in Δ[1]\Delta[1], namely the non-degenerate edge (01)(0\to 1) and the two degenerate cells (00)(0 \to 0) and (11)(1 \to 1), hence the abelian group of formal linear combinations of the form

3{a(00)+b(01)+c(11)|a,b,c}. \mathbb{Z}^3 \simeq \left\{ a \cdot (0\to 0) + b \cdot (0 \to 1) + c \cdot (1 \to 1) | a,b,c \in \mathbb{Z}\right\} \,.

The two face maps act on the basis 1-cells as

1:(ij)(i) \partial_1 \colon (i \to j) \mapsto (i)
0:(ij)(j). \partial_0 \colon (i \to j) \mapsto (j) \,.

Now of course most of the (infinitely!) many simplices inside Δ[1]\Delta[1] are degenerate. In fact the only non-degenerate simplices are the two 0-cells (0)(0) and (1)(1) and the 1-cell (01)(0 \to 1). Hence the alternating face maps complex modulo degeneracies, def. 67, of [Δ[1]]\mathbb{Z}[\Delta[1]] is simply this:

(C([Δ[1]]))/D([Δ[1]]))=(00(11) 2). (C (\mathbb{Z}[\Delta[1]])) / D (\mathbb{Z}[\Delta[1]])) = \left( \cdots \to 0 \to 0 \to \mathbb{Z} \stackrel{\left(1 \atop -1\right)}{\longrightarrow} \mathbb{Z}^2 \right) \,.

Notice that alternatively we could consider the topological 1-simplex Δ 1=[0,1]\Delta^1 = [0,1] and its singular simplicial complex Sing(Δ 1)Sing(\Delta^1) in place of the smaller Δ[1]\Delta[1], then the free simplicial abelian group (Sing(Δ 1))\mathbb{Z}(Sing(\Delta^1)) of that. The corresponding alternating face map chain complex C((Sing(Δ 1)))C(\mathbb{Z}(Sing(\Delta^1))) is “huge” in that in each positive degree it has a free abelian group on uncountably many generators. Quotienting out the degenerate cells still leaves uncountably many generators in each positive degree (while every singular nn-simplex in [0,1][0,1] is “thin”, only those whose parameterization is as induced by a degeneracy map are actually regarded as degenerate cells here). Hence even after normalization the singular simplicial chain complex is “huge”. Nevertheless it is quasi-isomorphic to the tiny chain complex found above.

The statement of the Dold-Kan correspondence now is the following.

Theorem

For AA an abelian category there is an equivalence of categories

N:A Δ opCh +(A):Γ N \;\colon\; A^{\Delta^{op}} \stackrel{\leftarrow}{\to} Ch_\bullet^+(A) \;\colon\; \Gamma

between

where

(Dold 58, Kan 58, Dold-Puppe 61).

Theorem (Kan)

For the case that AA is the category Ab of abelian groups, the functors NN and Γ\Gamma are nerve and realization with respect to the cosimplicial chain complex

[]:ΔCh +(Ab) \mathbb{Z}[-]: \Delta \to Ch_+(Ab)

that sends the standard nn-simplex to the normalized Moore complex of the free simplicial abelian group F (Δ n)F_{\mathbb{Z}}(\Delta^n) on the simplicial set Δ n\Delta^n, i.e.

Γ(V):[k]Hom Ch +(Ab)(N((Δ[k])),V). \Gamma(V) : [k] \mapsto Hom_{Ch_\bullet^+(Ab)}(N(\mathbb{Z}(\Delta[k])), V) \,.

This is due to (Kan 58).

More explicitly we have the following

Proposition
  • For VCh +V \in Ch_\bullet^+ the simplicial abelian group Γ(V)\Gamma(V) is in degree nn given by

    Γ(V) n= [n]surj[k]V k \Gamma(V)_n = \bigoplus_{[n] \underset{surj}{\to} [k]} V_k

    and for θ:[m][n]\theta : [m] \to [n] a morphism in Δ\Delta the corresponding map Γ(V) nΓ(V) m\Gamma(V)_n \to \Gamma(V)_m

    θ *: [n]surj[k]V k [m]surj[r]V r \theta^* : \bigoplus_{[n] \underset{surj}{\to} [k]} V_k \to \bigoplus_{[m] \underset{surj}{\to} [r]} V_r

    is given on the summand indexed by some σ:[n][k]\sigma : [n] \to [k] by the composite

    V kd *V s [m]surj[r]V r V_k \stackrel{d^*}{\to} V_s \hookrightarrow \bigoplus_{[m] \underset{surj}{\to} [r]} V_r

    where

    [m]t[s]d[k] [m] \stackrel{t}{\to} [s] \stackrel{d}{\to} [k]

    is the epi-mono factorization of the composite [m]θ[n]σ[k][m] \stackrel{\theta}{\to} [n] \stackrel{\sigma}{\to} [k].

  • The natural isomorphism ΓNId\Gamma N \to Id is given on AsAb Δ opA \in sAb^{\Delta^{op}} by the map

    [n]surj[k](NA) kA n \bigoplus_{[n] \underset{surj}{\to} [k]} (N A)_k \to A_n

    which on the direct summand indexed by σ:[n][k]\sigma : [n] \to [k] is the composite

    NA kA kσ *A n. N A_k \hookrightarrow A_k \stackrel{\sigma^*}{\to} A_n \,.
  • The natural isomorphism IdNΓId \to N \Gamma is on a chain complex VV given by the composite of the projection

    VC(Γ(V))C(Γ(C))/D(Γ(V)) V \to C(\Gamma(V)) \to C(\Gamma(C))/D(\Gamma(V))

    with the inverse

    C(Γ(V))/D(Γ(V))NΓ(V) C(\Gamma(V))/D(\Gamma(V)) \to N \Gamma(V)

    of

    NΓ(V)C(Γ(V))C(Γ(V))/D(Γ(V)) N \Gamma(V) \hookrightarrow C(\Gamma(V)) \to C(\Gamma(V))/D(\Gamma(V))

    (which is indeed an isomorphism, as discussed at Moore complex).

This is spelled out in (Goerss-Jardine, prop. 2.2 in section III.2).

Proposition

With the explicit choice for ΓNId\Gamma N \stackrel{\simeq}{\to} Id as above we have that Γ\Gamma and NN form an adjoint equivalence (ΓN)(\Gamma \dashv N)

This is for instance (Weibel, exercise 8.4.2).

Remark

It follows that with the inverse structure maps, we also have an adjunction the other way round: (NΓ)(N \dashv \Gamma).

Hence in concclusion the Dold-Kan correspondence allows us to regard chain complexes (in non-negative degree) as, in particular, special simplicial sets. In fact as simplicial sets they are Kan complexes and hence infinity-groupoids:

Theorem (J. C. Moore)

The simplicial set underlying any simplicial group (by forgetting the group structure) is a Kan complex.

This is due to (Moore, 1954)

In fact, not only are the horn fillers guaranteed to exist, but there is an algorithm that provides explicit fillers. This implies that constructions on a simplicial group that use fillers of horns can often be adjusted to be functorial by using the algorithmically defined fillers. An argument that just uses ‘existence’ of fillers can be refined to give something more ‘coherent’.

Proof

Let GG be a simplicial group.

Here is the explicit algorithm that computes the horn fillers:

Let (y 0,,y k1,,y k+1,,y n)(y_0,\ldots, y_{k-1}, -,y_{k+1}, \ldots, y_n) give a horn in G n1G_{n-1}, so the y iy_is are (n1)(n-1) simplices that fit together as if they were all but one, the k thk^{th} one, of the faces of an nn-simplex. There are three cases:

  1. if k=0k = 0:

    • Let w n=s n1y nw_n = s_{n-1}y_n and then w i=w i+1(s i1d iw i+1) 1s i1y iw_i = w_{i+1}(s_{i-1}d_i w_{i+1})^{-1}s_{i-1}y_i for i=n,,1i = n, \ldots, 1, then w 1w_1 satisfies d iw 1=y id_i w_1 = y_i, i0i\neq 0;
  2. if 0<k<n0\lt k \lt n:

    • Let w 0=s 0y 0w_0 = s_0 y_0 and w i=w i1(s id iw i1) 1s iy iw_i = w_{i-1}(s_i d_i w_{i-1})^{-1}s_i y_i for i=0,,k1i = 0, \ldots, k-1, then take w n=w k1(s n1d nwk1) 1s n1y nw_n = w_{k-1}(s_{n-1}d_nw_{k-1})^{-1}s_{n-1}y_n, and finally a downwards induction given by w i=w i+1(s i1d iw i+1) 1s i1y iw_i = w_{i+1}(s_{i-1}d_{i}w_{i+1})^{-1}s_{i-1}y_i, for i=n,,k+1i = n, \ldots, k+1, then w k+1w_{k+1} gives d iw k+1=y id_{i}w_{k+1} = y_i for iki \neq k;
  3. if k=nk=n:

    • use w 0=s 0y 0w_0 = s_0 y_0 and w i=w i1(s id iw i1) 1s iy iw_i = w_{i-1}(s_i d_i w_{i-1})^{-1}s_i y_i for i=0,,n1i = 0, \ldots, n-1, then w n1w_{n-1} satisfies d iw n1=y id_i w_{n-1} = y_i, ini\neq n.

Abelian homotopy theory

We have seen in Chain complexes – Abelian homotopy types that the most interesting properties of the category of chain complexes is all secretly controled by the phenomenon of chain homotopy and quasi-isomorphism. Strictly speaking these two phenomena point beyond plain category theory to the richer context of general abstract homotopy theory. Here we discuss properties of the category of chain complexes from this genuine homotopy-theoretic point of view. The result of passing the category of chain complexes to genuine homotopy theory is called the derived category (of the underlying abelian category 𝒜\mathcal{A}, say of modules) and we start in 7) with a motivation of the phenomenon of this “homotopy derivation” and the discussion of the necessary resolutions of chain complexes. This naturally gives rise to the general notion of derived functors which we discuss in 8). Examples of these are ubiquituous in homological algebra, but as in ordinary enriched category theory two stand out as being of more fundamental importance, the derived functor “Ext” of the hom-functor and the derived functor “Tor” of the tensor product functor. Their properties and uses we discuss in 9).

Chain homotopy and resolutions

We now come back to the category 𝒦(𝒜)\mathcal{K}(\mathcal{A}) of def. 53, the “homotopy category of chain complexes” in which chain-homotopic chain maps are identified. This would seem to be the right context to study the homotopy theory of chain complexes, but one finds that there are still chain maps which ought to be identified in homotopy theory, but which are still not identified in 𝒦(𝒜)\mathcal{K}(\mathcal{A}). This is our motivating example 45 below.

We discuss then how this problem is fixed by allowing to first “resolve” chain complexes quasi-isomorphically by “good representatives” called projective resolutions or injective resolutions. Many of the computations in the following sections – and in homological algebra in general – come down to operating on such resolutions. We end this section by prop. 58 below, which shows that the above problem indeed goes away when allowing chain complexes to be resolved.

In the next section, 8), we discuss how this process of forming resolutions functorially extends to the whole category of modules.

So we start here with this simple example that shows the problem with bare chain homotopies and indicates how these have to be resolved:

Example

In Ch (𝒜)Ch_\bullet(\mathcal{A}) for 𝒜=\mathcal{A} = Ab consider the chain map

0 0 0 2 id 0 2 mod2 2. \array{ \cdots &\to& 0 &\to& 0 &\to& 0 &\to& \mathbb{Z}_2 \\ && \downarrow && \downarrow && \downarrow && \downarrow^{\mathrlap{id}} \\ \cdots &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z} &\stackrel{mod\,2}{\to}& \mathbb{Z}_2 } \,.

The codomain of this map is an exact sequence, hence is quasi-isomorphic to the 0-chain complex. Thereofore in homotopy theory it should behave entirely as the 0-complex itself. In particular, every chain map to it should be chain homotopic to the zero morphism (have a null homotopy).

But the above chain map is chain homotopic precisely only to itself. This is because the degree-0 component of any chain homotopy out of this has to be a homomorphism of abelian groups 2\mathbb{Z}_2 \to \mathbb{Z}, and this must be the 0-morphism, because \mathbb{Z} is a free group, but 2\mathbb{Z}_2 is not.

This points to the problem: the components of the domain chain complex are not free enough to admit sufficiently many maps out of it.

Consider therefore a free resolution of the above domain complex by the quasi-isomorphism

0 0 2 mod2 0 0 0 2, \array{ \cdots &\to& 0 &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z} \\ && \downarrow && \downarrow && \downarrow && \downarrow^{\mathrlap{mod\,2}} \\ \cdots &\to& 0 &\to& 0 &\to& 0 &\to& \mathbb{Z}_2 } \,,

where now the domain complex consists entirely of free groups. The composite of this with the original chain map is now

0 0 2 0 mod2 0 2 mod2 2. \array{ \cdots &\to& 0 &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z} \\ && \downarrow && \downarrow && \downarrow^{0} && \downarrow^{\mathrlap{mod\,2}} \\ \cdots &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z} &\stackrel{mod\,2}{\to}& \mathbb{Z}_2 } \,.

This is the corresponding resolution of the original chain map. And this indeed has a null homotopy:

0 0 2 id 0 id mod2 0 2 mod2 2. \array{ \cdots &\to& 0 &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z} \\ && \downarrow &\swarrow& \downarrow &\swarrow_{-id}& \downarrow^{0} &\swarrow_{\mathrlap{id}}& \downarrow^{\mathrlap{mod\,2}} \\ \cdots &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z} &\stackrel{mod\,2}{\to}& \mathbb{Z}_2 } \,.

So resolving the domain by a sufficiently free complex makes otherwise missing chain homotopies exist. Below in lemma 7 we discuss the general theory behind the kind of situation of this example. But to get there we first need some basic notions and facts.

Notably, in general it is awkward to insist on actual free resolutions. But it is easy to see, and this we discuss now, that essentially just as well is a resolution by modules which are direct summands of free modules.

Definition

An object PP of a category CC is a projective object if it has the left lifting property against epimorphisms.

This means that PP is projective if for any morphism f:PBf:P \to B and any epimorphism q:ABq:A \to B, ff factors through qq by some morphism PAP\to A.

A q P f B. \array{ && A \\ &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{q}} \\ P &\stackrel{f}{\to}& B } \,.

An equivalent way to say this is that:

Definition

An object PP is projective precisely if the hom-functor Hom(P,)Hom(P,-) preserves epimorphisms.

Remark

The point of this lifting property will become clear when we discuss the construction of projective resolutions a bit further below: they are built by applying this property degreewise to obtain suitable chain maps.

We will be interested in projective objects in the category RRMod: projective modules. Before we come to that, notice the following example (which the reader may on first sight feel is pedantic and irrelevant, but for the following it is actually good to make this explicit).

Example

In the category Set of sets the following are equivalent

Remark

We will assume here throughout the axiom of choice in Set, as usual. The point of the above example, however, is that one could just as well replace Set by another “base topos” which will behave essentially precisely like Set, but in general will not validate the axiom of choice. Homological algebra in such a more general context is the theory of complexes of abelian sheaves/sheaves of abelian groups and ultimately the theory of abelian sheaf cohomology.

This is a major aspect of homological algebra. While we will not discuss this further here in this introduction, the reader might enjoy keeping in mind that all of the following discussion of resolutions of RR-modules goes through in this wider context of sheaves of modules except for subtleties related to the (partial) failure of example 46 for the category of sheaves.

We now characterize projective modules.

Lemma

Assuming the axiom of choice, a free module NR (S)N \simeq R^{(S)} is projective.

Proof

Explicitly: if SSetS \in Set and F(S)=R (S)F(S) = R^{(S)} is the free module on SS, then a module homomorphism F(S)NF(S) \to N is specified equivalently by a function f:SU(N)f : S \to U(N) from SS to the underlying set of NN, which can be thought of as specifying the images of the unit elements in R (S) sSRR^{(S)} \simeq \oplus_{s \in S} R of the |S|{\vert S\vert} copies of RR.

Accordingly then for N˜N\tilde N \to N an epimorphism, the underlying function U(N˜)U(N)U(\tilde N) \to U(N) is an epimorphism, and the axiom of choice in Set says that we have all lifts f˜\tilde f in

U(N˜) f˜ S f U(N). \array{ && U(\tilde N) \\ & {}^{\tilde f} \nearrow & \downarrow \\ S &\stackrel{f}{\to}& U(N) } \,.

By adjunction these are equivalently lifts of module homomorphisms

N˜ R (S) N. \array{ && \tilde N \\ & \nearrow & \downarrow \\ R^{(S)} &\stackrel{}{\to}& N } \,.
Lemma

If NRModN \in R Mod is a direct summand of a free module, hence if there is NRModN' \in R Mod and SSetS \in Set such that

R (S)NN, R^{(S)} \simeq N \oplus N' \,,

then NN is a projective module.

Proof

Let K˜K\tilde K \to K be a surjective homomorphism of modules and f:NKf : N \to K a homomorphism. We need to show that there is a lift f˜\tilde f in

K˜ f˜ N f K. \array{ && \tilde K \\ & {}^{\mathllap{\tilde f}}\nearrow & \downarrow \\ N &\stackrel{f}{\to}& K } \,.

By definition of direct sum we can factor the identity on NN as

id N:NNNN. id_N : N \to N \oplus N' \to N \,.

Since NNN \oplus N' is free by assumption, and hence projective by lemma 4, there is a lift f^\hat f in

K˜ f^ N NN K. \array{ && && \tilde K \\ && & {}^{\mathllap{\hat f}}\nearrow & \downarrow \\ N &\to& N \oplus N' &\to& K } \,.

Hence f˜:NNNf^K˜\tilde f : N \to N \oplus N' \stackrel{\hat f}{\to} \tilde K is a lift of ff.

Proposition

An RR-module NN is projective precisely if it is the direct summand of a free module.

Proof

By lemma 5 if NN is a direct summand then it is projective. So we need to show the converse.

Let F(U(N))F(U(N)) be the free module on the set U(N)U(N) underlying NN, hence the direct sum

F(U(n))= nU(n)R. F(U(n)) = \oplus_{n \in U(n)} R \,.

There is a canonical module homomorphism

nU(N)RN \oplus_{n \in U(N)} R \to N

given by sending the unit 1R n1 \in R_n of the copy of RR in the direct sum labeled by nU(n)n \in U(n) to nNn \in N.

(Abstractly this is the counit ϵ:F(U(N))N\epsilon : F(U(N)) \to N of the free/forgetful-adjunction (FU)(F \dashv U).)

This is clearly an epimorphism. Thefore if NN is projective, there is a section ss of ϵ\epsilon. This exhibits NN as a direct summand of F(U(N))F(U(N)).

We discuss next how to build resolutions of chain complexes by projective modules. But before we come to that it is useful to also introduce the dual notion. So far we have concentrated on chain complexes with degrees in the natural numbers: non-negative degrees. For a discussion of resolutions we need a more degree-symmetric perspective, which of course is straightforward to obtain.

Definition

A cochain complex C C^\bullet in 𝒜=RMod\mathcal{A} = R Mod is a sequence of morphism

C 0d 0C 1d 1C 2d 2 C^0 \stackrel{d^0}{\to} C^1 \stackrel{d^1}{\to} C^2 \stackrel{d^2}{\to} \cdots

in 𝒜\mathcal{A} such that dd=0d\circ d = 0. A homomorphism of cochain complexes f :C D f^\bullet : C^\bullet \to D^\bullet is a collection of morphisms {f n:C nD n}\{f^n : C^n \to D^n\} such that d D nf n=f nd C nd^n_D \circ f^n = f^n \circ d^n_C for all nn \in \mathbb{N}.

We write Ch (𝒜)Ch^\bullet(\mathcal{A}) for the category of cochain complexes.

Example

Let N𝒜N \in \mathcal{A} be a fixed module and C Ch (𝒜)C_\bullet \in Ch_\bullet(\mathcal{A}) a chain complex. Then applying degreewise the hom-functor out of the components of C C_\bullet into NN yields a cochain complex in Mod\mathbb{Z} Mod \simeq Ab:

Hom 𝒜(C ,N)=[Hom 𝒜(C 0,N)Hom 𝒜( 0,N)Hom 𝒜(C 1,N)Hom 𝒜( 1,N)Hom 𝒜(C 2,N)Hom 𝒜( 2,N)]. Hom_{\mathcal{A}}(C_\bullet, N) = \left[ Hom_{\mathcal{A}}(C_0, N) \stackrel{Hom_{\mathcal{A}}(\partial_0, N)}{\to} Hom_{\mathcal{A}}(C_1, N) \stackrel{Hom_{\mathcal{A}}(\partial_1, N)}{\to} Hom_{\mathcal{A}}(C_2, N) \stackrel{Hom_{\mathcal{A}}(\partial_2, N)}{\to} \cdots \right] \,.
Example

In example 47 let 𝒜=\mathcal{A} = \mathbb{Z}Mod == Ab, let N=N = \mathbb{Z} and let C =[Sing(X)]C_\bullet = \mathbb{Z}[Sing(X)] be the singular simplicial complex of a topological space XX. Write

C (X)Hom [SingX],. C^\bullet(X) \coloneqq Hom_{\mathbb{Z}[Sing X], \mathbb{Z}} \,.

Then H (C(X))H^\bullet(C(X)) is called the singular cohomology of XX.

Remark

Example 47 is just a special case of the internal hom of def. 43: we may regard cochain complexes in non-negative degree equivalently as chain complexes in positive degree.

Accordingly we say for C C^\bullet a cochain complex that

  • an element in C nC^n is an nn-cochain

  • an element in im(d n1)im(d^{n-1}) is an nn-coboundary

  • al element in ker(d n)ker(d^n) is an nn-cocycle.

But equivalently we may regard a cochain in degree nn as a chain in degree (n)(-n) and so forth. And this is the perspective used in all of the following.

The role of projective objects, def. 68, for chain complexes is played, dually, by injective objects for cochain complexes:

Definition

An object II a category is injective if all diagrams of the form

X I Z \array{ X &\to& I \\ {}^{}\downarrow \\ Z }

with XZX \to Z a monomorphism admit an extension

X I Z. \array{ X &\to& I \\ {}^{}\downarrow & \nearrow_{\mathrlap{\exists}} \\ Z } \,.

Since we are interested in refining modules by projective or injective modules, we have the following terminology.

Definition

A category

We have essentially already seen the following statement.

Proposition

Assuming the axiom of choice, the category RRMod has enough projectives.

Proof

Let F(U(N))F(U(N)) be the free module on the set U(N)U(N) underlying NN. By lemma 4 this is a projective module.

The canonical morphism

F(U(n))= nU(n)RN F(U(n)) = \oplus_{n \in U(n)} R \to N

is clearly a surjection, hence an epimorphism in RRMod.

We now show that similarly RModR Mod has enough injectives. This is a little bit more work and hence we proceed with a few preparatory statements.

The following basic statement of algebra we cite here without proof (but see at injective object for details).

Proposition

Assuming the axiom of choice, an abelian group AA is injective as a \mathbb{Z}-module precisely if it is a divisible group, in that for all integers nn \in \mathbb{N} we have nG=Gn G = G.

Example

By prop. 53 the following abelian groups are injective in Ab.

The group of rational numbers \mathbb{Q} is injective in Ab, as is the additive group of real numbers \mathbb{R} and generally that underlying any field. The additive group underlying any vector space is injective. The quotient of any injective group by any other group is injective.

Example

Not injective in Ab are the cyclic groups /n\mathbb{Z}/n\mathbb{Z}.

Proposition

Assuming the axiom of choice, the category \mathbb{Z}Mod \simeq Ab has enough injectives.

Proof

By prop. 53 an abelian group is an injective \mathbb{Z}-module precisely if it is a divisible group. So we need to show that every abelian group is a subgroup of a divisible group.

To start with, notice that the group \mathbb{Q} of rational numbers is divisible and hence the canonical embedding \mathbb{Z} \hookrightarrow \mathbb{Q} shows that the additive group of integers embeds into an injective \mathbb{Z}-module.

Now by the discussion at projective module every abelian group AA receives an epimorphism ( sS)A(\oplus_{s \in S} \mathbb{Z}) \to A from a free abelian group, hence is the quotient group of a direct sum of copies of \mathbb{Z}. Accordingly it embeds into a quotient A˜\tilde A of a direct sum of copies of \mathbb{Q}.

ker = ker ( sS) ( sS) A A˜ \array{ ker &\stackrel{=}{\to}& ker \\ \downarrow && \downarrow \\ (\oplus_{s \in S} \mathbb{Z}) &\hookrightarrow& (\oplus_{s \in S} \mathbb{Q}) \\ \downarrow && \downarrow \\ A &\hookrightarrow& \tilde A }

Here A˜\tilde A is divisible because the direct sum of divisible groups is again divisible, and also the quotient group of a divisible groups is again divisble. So this exhibits an embedding of any AA into a divisible abelian group, hence into an injective \mathbb{Z}-module.

Proposition

Assuming the axiom of choice, for RR a ring, the category RRMod has enough injectives.

The proof uses the following lemma.

Write U:RModAbU\colon R Mod \to Ab for the forgetful functor that forgets the RR-module structure on a module NN and just remembers the underlying abelian group U(N)U(N).

Lemma

The functor U:RModAbU\colon R Mod \to Ab has a right adjoint

R *:AbRMod R_* : Ab \to R Mod

given by sending an abelian group AA to the abelian group

U(R *(A))Ab(U(R),A) U(R_*(A)) \coloneqq Ab(U(R),A)

equipped with the RR-module struture by which for rRr \in R an element (U(R)fA)U(R *(A))(U(R) \stackrel{f}{\to} A) \in U(R_*(A)) is sent to the element rfr f given by

rf:rf(rr). r f : r' \mapsto f(r' \cdot r) \,.

This is called the coextension of scalars along the ring homomorphism R\mathbb{Z} \to R.

The unit of the (UR *)(U \dashv R_*) adjunction

ϵ N:NR *(U(N)) \epsilon_N : N \to R_*(U(N))

is the RR-module homomorphism

ϵ N:NHom Ab(U(R),U(N)) \epsilon_N : N \to Hom_{Ab}(U(R), U(N))

given on nNn \in N by

j(n):rrn. j(n) : r \mapsto r n \,.
Proof

of prop. 55

Let NRModN \in R Mod. We need to find a monomorphism NN˜N \to \tilde N such that N˜\tilde N is an injective RR-module.

By prop. 54 there exists a monomorphism

i:U(N)D i \colon U(N) \hookrightarrow D

of the underlying abelian group into an injective abelian group DD.

Now consider the (UR *)(U \dashv R_*)-adjunct

NR *(D) N \to R_*(D)

of ii, hence the composite