# nLab 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

# Contents

## 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 $n$-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 $\Phi_1$ and $\Phi_2$, then it makes no sense to ask whether they are equal or not. Instead what makes sense to ask for is a gauge transformation $g$ that, if it exists, exhibits $\Phi_1$ as being gauge equivalent to $\Phi_2$ via $g$:

$\Phi_1 \underoverset{\simeq}{g}{\longrightarrow} \Phi_2 \,.$

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 $\Phi_3$ and a gauge transformation $g'$ from $\Phi_2$ to $\Phi_3$, then there is also the composite gauge transformation

$g' \circ g \Phi_1 \underoverset{\simeq}{g}{\longrightarrow} \Phi_2 \underoverset{\simeq}{g'}{\longrightarrow} \Phi_3$

and this composition is associative.

Moreover, these being equivalences means that they have inverses,

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

such that the compositions

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

and

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

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 $g^{-1}\circ g$ (transforming one way and then just transforming back) to be equal to the identity transformation $id$.

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

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

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 $n \in \mathbb{N}$ 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

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

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

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

###### Remark

The coordinate expression in def. 1 – also known as barycentric coordinates – is evidently just one of many possible ways to present topological $n$-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 $n = 0$ this is the point, $\Delta^0 = *$.

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

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

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

###### Definition

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

$\delta_k : \Delta^{n-1} \hookrightarrow \Delta^n$

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

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

which “omits” the $k$th canonical coordinate:

$(x_0, \cdots , x_{n-1}) \mapsto (x_0, \cdots, x_{k-1} , 0 , x_{k}, \cdots, x_{n-1}) \,.$
###### Example

The inclusion

$\delta_0 : \Delta^0 \to \Delta^1$

is the inclusion

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

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

$\delta_1 : \Delta^0 \to \Delta^1$

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

###### Definition

For $n \in \mathbb{N}$ and $0 \leq k \lt n$ the $k$th degenerate $(n)$-simplex (projection) is the surjective map

$\sigma_k : \Delta^{n} \to \Delta^{n-1}$

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

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

which sends

$(x_0, \cdots, x_n) \mapsto (x_0, \cdots, x_{k} + x_{k+1}, \cdots, x_n) \,.$
###### Definition

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

$\sigma : \Delta^n \to X$

from the topological $n$-simplex, def. 1, to $X$.

Write

$(Sing X)_n \coloneqq Hom_{Top}(\Delta^n , X)$

for the set of singular $n$-simplices of $X$.

So to a topological space $X$ is associated a sequence of sets

$(Sing X)_n \coloneqq Hom_{Top}(\Delta^n, X)$

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

$\delta_k : \Delta^{n-1} \to \Delta^{n}$

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

$\sigma_k : \Delta^{n+1} \to \Delta^n$

we dually have functions

$d_k \coloneqq Hom_{Top}(\delta_k, X) : (Sing X)_n \to (Sing X)_{n-1}$

that send each singular $n$-simplex to its $k$-face and functions

$s_k \coloneqq Hom_{Top}(\sigma_k,X) : (Sing X)_{n} \to (Sing X)_{n+1}$

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

###### Definition

A simplicial set $S \in sSet$ is

• for each $n \in \mathbb{N}$ a set $S_n \in Set$ – the set of $n$-simplices;

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

a function $d_i : S_{n} \to S_{n-1}$ – the $i$th face map on $n$-simplices;

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

a function $\sigma_i : S_{n} \to S_{n+1}$ – the $i$th degeneracy map on $n$-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. $d_i \circ d_j = d_{j-1} \circ d_i$ if $i \lt j$,

2. $s_i \circ s_j = s_j \circ s_{i-1}$ if $i \gt j$.

3. $d_i \circ s_j = \left\{ \array{ s_{j-1} \circ d_i & if \; i \lt j \\ id & if \; i = j \; or \; i = j+1 \\ s_j \circ d_{i-1} & if i \gt j+1 } \right.$

It is straightforward to check by explicit inspection that the evident injection and restriction maps between the sets of singular simplices make $(Sing X)_\bullet$ 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 $\Delta$ is the full subcategory of Cat on the free categories of the form

\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 $[n]$ as being the “spine” of the $n$-simplex. For instance for $n = 2$ we think of $0 \to 1 \to 2$ as the “spine” of the triangle. This becomes clear if we don’t just draw the morphisms that generate the category $[n]$, but draw also all their composites. For instance for $n = 2$ we have_

$[2] = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&\to&& 2 } \right\} \,.$
###### Proposition
$S : \Delta^{op} \to Set$

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 $\Delta^{op}([n],[n+1])$ and $\Delta^{op}([n],[n-1])$.

This makes the following evident:

###### Example

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

$\Delta^\bullet : \Delta \to Top \,.$

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

###### Definition

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

$(Sing X)_\bullet : \Delta^{op} \to Set$

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

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

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

##### Singular chain complex

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

###### Definition

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

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

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

Identifying an element $s \in S$ with the function $S \to \mathbb{Z}$, which sends $s$ to $1 \in \mathbb{Z}$ and all other elements to 0, this is written as

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

In this expression one calls $a_s \in \mathbb{Z}$ the coefficient of $s$ in the formal linear combination.

###### Remark

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

$(\sum_{s \in S} a_s \cdot s) + (\sum_{s \in S} b_s \cdot s) = \sum_{s \in S} (a_s + b_s) \cdot s \,.$

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

###### Definition

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

###### Definition

For $X$ a topological space, an $n$-chain on the singular simplicial complex $Sing X$ is called a singular $n$-chain on $X$.

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 $S$ a simplicial set, its alternating face map differential in degree $n$ is the linear map

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

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

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

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

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

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

\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 $d$;

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 $j$ by $j +1$;

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

###### Example

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

$\sigma^1 \in C_1(X) \,.$

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

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

or graphically (using notation as for orientals)

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

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

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

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

Hence the boundary of the boundary is:

\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 $S$ a simplicial set, we call the collection

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

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

(for all $n \in \mathbb{N}$) the alternating face map chain complex of $S$:

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

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

This motivates the general definition:

###### Definition

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

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

###### Definition

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

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

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

(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

$B_n \coloneqq im(\partial_n) \subset C_n(S)$

for the group of $n$-boundaries, and

$Z_n \coloneqq ker(\partial_n) \subset C_(S)$

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

###### Definition

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

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

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

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

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

###### Remark

So $H_0(C_\bullet(S)) = C_0(S)/im(\partial_0)$.

###### Example

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

$H_0(X) \simeq \mathbb{Z}[\pi_0(X)]$

is the free abelian group on the set of connected components of $X$.

###### Example

For $X$ a connected, orientable manifold of dimension $n$ we have

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

The precise choice of this isomorphism is a choice of orientation on $X$. With a choice of orientation, the element $1 \in \mathbb{Z}$ under this identification is called the fundamental class

$[X] \in H_n(X)$

of the manifold $X$.

###### Definition

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

$f_* \sigma : \Delta^n \stackrel{\sigma}{\to} X \stackrel{f}{\to} Y$

in $Y$. This is called the push-forward of $\sigma$ along $f$. Accordingly there is a push-forward map on groups of singular chains

$(f_*)_n : C_n(X) \to C_n(Y) \,.$
###### Proposition

These push-forward maps make all diagrams of the form

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

commute.

###### Proof

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

$f_* : Sing X \to Sing Y \,.$

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

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

###### Definition

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

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

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

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

Accordingly we have:

###### Proposition

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

$C_\bullet(-) : Top \to Ch_\bullet(Ab)$

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

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

$H_n(-) : Top \to Ab \,.$

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

$H_n(f) : H_n(X) \to H_n(Y)$

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

$f_n : H_n(C) \stackrel{\simeq}{\to} H_n(D) \,.$

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 $C_\bullet$ to $D_\bullet$.” 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

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

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 $\mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}$.

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

$C_\bullet \leftarrow \to \leftarrow \cdots \to D_\bullet$

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 $C_\bullet(-)$ 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 $(X,x)$ a pointed topological space, the Hurewicz homomorphism is the function

$\Phi : \pi_k(X,x) \to H_k(X)$

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

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

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

###### Proposition

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

$\mathbb{Z}[\pi_0(X)] \simeq H_0(X) \,.$

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 $X$ is connected.

###### Proposition

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

$\Phi : \pi_1(X,x) \to H_1(X)$

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

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

For higher connected $X$ we have the

###### Theorem

If $X$ is (n-1)-connected for $n \geq 2$ then

$\Phi : \pi_n(X,x) \to H_n(X)$

is an isomorphism.

This is known as the Hurewicz theorem.

This gives plenty of motivation for studying

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

### 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 $\mathcal{G}_\bullet$ is

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

• equipped with functions

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

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

such that

• $i$ takes values in endomorphisms;

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

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

• every morphism has an inverse under this composition.

###### Remark

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

$\mathcal{G}_1 = \left\{ \phi_0 \stackrel{k}{\to} \phi_1 \right\}$

and the set of pairs of composable morphisms is

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

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

###### Example

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

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

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

• $(\mathbf{B}G)_0 = \ast$;

• $(\mathbf{B}G)_1 = G$.

For $G$ and $K$ two groups, group homomorphisms $f \colon G \to K$ are in natural bijection with groupoid homomorphisms

$(\mathbf{B}f)_\bullet \;\colon\; (\mathbf{B}G)_\bullet \to (\mathbf{B}K)_\bullet \,.$

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

$(\mathbf{B}c)_\bullet \;\colon\; (\mathbf{B}G)_\bullet \to (\mathbf{B}U(1))_\bullet \,.$

Here, for the time being, all groups are discrete groups. Since the circle group $U(1)$ 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

$\flat U(1) \in Grp$

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

###### Example

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

$X//_\rho G = \left( X \times G \stackrel{\overset{\rho}{\longrightarrow}}{\underset{p_1}{\longrightarrow}} X \right)$

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

$x_1 \stackrel{g}{\to} x_2 = \rho(x_1)(g)$

for $x_1, x_2 \in X$, $g \in G$.

As an important special case we have:

###### Example

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

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

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

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

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

$\array{ G//G & \simeq & \mathbf{E}G \\ \downarrow && \downarrow \\ \ast //G & \simeq & \mathbf{B}G } \,.$

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

###### Example

The interval $I$ is the groupoid with

• $I_0 = \{a,b\}$;
• $I_1 = \{\mathrm{id}_a, \mathrm{id}_b, a \to b \}$.
###### Example

For $\Sigma$ a topological space, its fundamental groupoid $\Pi_1(\Sigma)$ is

• $\Pi_1(\Sigma)_0 =$ points in $X$;
• $\Pi_1(\Sigma)_1 =$ continuous paths in $X$ modulo homotopy that leaves the endpoints fixed.
###### Example

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

• $\mathcal{G}^I_0 = \mathcal{G}_1 = \left\{ \array{ \phi_0 \\ \downarrow^{\mathrlap{k}} \\ \phi_1 } \right\}$;

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

This comes with two canonical homomorphisms

$\mathcal{G}^I_\bullet \stackrel{\overset{ev_1}{\longrightarrow}}{\underset{ev_0}{\longrightarrow}} \mathcal{G}_\bullet$

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

###### Definition

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

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

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

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

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

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

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 $X,Y$ two groupoids, the mapping groupoid $[X,Y]$ or $Y^X$ is

• $[X,Y]_0 =$ homomorphisms $X \to Y$;
• $[X,Y]_1 =$ homotopies between such.
###### Definition

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

###### Example

The map

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

which picks any point and sends $n \in \mathbb{Z}$ to the loop based at that point which winds around $n$ 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 $X \stackrel{f}{\rightarrow} B \stackrel{g}{\leftarrow} Y$ their homotopy fiber product

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

is the limit cone

$\array{ X_\bullet \underset{B_\bullet}{\times} B^I_\bullet \underset{B_\bullet}{\times} Y_\bullet &\to& &\to& X_\bullet \\ \downarrow && && \downarrow^{\mathrlap{f_\bullet}} \\ && B^I_\bullet &\underset{(ev_0)_\bullet}{\to}& B_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ Y_\bullet &\underset{g_\bullet}{\to}& B_\bullet } \,,$

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

###### Remark

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

$(X_\bullet \underset{B_\bullet}{\times}Y_\bullet)_i = X_i \underset{B_i}{\times} Y_i \,.$
###### Example

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

$P \simeq X \underset{\mathbf{B}G}{\times} \ast \,.$

Namely both squares in the following diagram are pullback squares

$\array{ P &\to& \mathbf{E}G &\to& \ast_\bullet \\ \downarrow && && \downarrow^{\mathrlap{}} \\ && (\mathbf{B}G)^I_\bullet &\underset{(ev_0)_\bullet}{\to}& (\mathbf{B}G)_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ X_\bullet &\underset{}{\to}& (\mathbf{B}G)_\bullet } \,.$

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

###### Example

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

$\Omega X \coloneqq \ast \underset{X}{\times} \ast$

the loop space groupoid of $X$.

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

$G \simeq \Omega \mathbf{B}G = \ast \underset{\mathbf{B}G}{\times} \ast \,.$

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

###### Proof

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

$\array{ &\to& &\to& \ast \\ \downarrow && && \downarrow^{\mathrlap{}} \\ && (\mathbf{B}G)^I_\bullet &\underset{(ev_0)_\bullet}{\to}& \mathbf{B}G_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ \ast &\underset{}{\to}& \mathbf{B}G_\bullet } \,,$

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

###### Proposition
$[\Pi(S^1), X] \simeq X \underset{[\Pi(S^0), X]}{\times}X$
###### Proof

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

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

• 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 $X_\bullet \underset{[\Pi(S^0), X_\bullet]_\bullet}{\times} [\Pi(S^0), X_\bullet]^I_\bullet \underset{[\Pi(S^0), X_\bullet]_\bullet}{\times} X$ is the groupoid whose

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

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

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

###### Remark

Even though all these models of the circle $\Pi_1(S^1)$ 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 $G$ a discrete group, the free loop space object of its delooping $\mathbf{B}G$ is $G//_{ad} G$, the action groupoid, def. 9, of the adjoint action of $G$ on itself:

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

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

$[\Pi(S^1), \mathbf{B}\flat U(1)] \simeq \flat U(1)//_{ad} \flat U(1) \simeq (\flat U(1)) \times (\mathbf{B}\flat U(1)) \,.$
###### Example

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

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

Under the free loop space object construction this becomes

$[\Pi(S^1), \mathbf{B}c] \;\colon\; [\Pi(S^1), \mathbf{B}G] \to [\Pi(S^1), \mathbf{B}\flat U(1)]$

hence

$[\Pi(S^1), \mathbf{B}c] \;\colon\; G//_{ad}G \to \flat U(1) \times \mathbf{B}U(1) \,.$

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:

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

#### 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 $A$, 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 $A$ such that for all elements $a_1, a_2 \in A$ we have that the group product of $a_1$ with $a_2$ is the same as that of $a_2$ with $a_1$, which we write $a_1 + a_2 \in A$ (and the neutral element is denoted by $0 \in A$);

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

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 $A$, $B$ and $C$ abelian groups and $A \times B$ the cartesian product group, a bilinear map

$f : A \times B \to C$

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 $f : A \times B \to C$ is a function of sets that satisfies for all elements $a_1, a_2 \in A$ and $b_1, b_2 \in B$ the two relations

$f(a_1 + a_2, b_1) = f(a_1,b_1) + f(a_2, b_1)$

and

$f(a_1, b_1 + b_2) = f(a_1, b_1) + f(a_1, b_2) \,.$

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

$(a_1, b_1) + (a_2, b_2) = (a_1 + a_2 \;,\; b_1 + b_2) \,.$
$\phi : A \times B \to C$

hence satisfies

$\phi( a_1+a_2, b_1 + b_2 ) = \phi(a_1,b_1) + \phi(a_2, b_2)$

and hence in particular

$\phi( a_1+a_2, b_1 ) = \phi(a_1,b_1) + \phi(a_2, 0)$
$\phi( a_1, b_1 + b_2 ) = \phi(a_1,b_1) + \phi(0, b_2)$

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

###### Definition

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

• $(a_1,b)+(a_2,b)\sim (a_1+a_2,b)$

• $(a,b_1)+(a,b_2)\sim (a,b_1+b_2)$

for all $a, a_1, a_2 \in A$ and $b, b_1, b_2 \in B$.

In words: it is the group whose elements are presented by pairs of elements in $A$ and $B$ 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

$A \times B \stackrel{\otimes}{\to} A \otimes B \,.$

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

###### Proposition

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

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

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

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

This means:

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

$A \otimes (-) : Ab \to Ab \,,$
2. there is an associativity natural isomorphism $(A \otimes B) \otimes C \stackrel{\simeq}{\to} A \otimes (B \otimes C)$ 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 $A \otimes \mathbb{Z} \stackrel{\simeq}{\to} A$ 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 $A$ the map

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

which sends for $n \in \mathbb{N} \subset \mathbb{Z}$

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

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

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

This shows that $A \otimes \mathbb{Z} \to A$ 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 $a,b \in \mathbb{N}$ and positive, we have

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

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

###### Definition

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

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

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

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

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

there is a unique homomorphism $\phi : \oplus_{i \in I} A_i \to K$ such that $f_i = \phi \circ \iota_i$ for all $i \in I$.

Explicitly in terms of elements we have:

###### Proposition

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

$a_1 + a_2 + \cdots + a_k$

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

###### Example

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

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

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

$A \otimes (\oplus_{i \in I} B_i) \simeq \oplus_{i \in I} A \otimes B_o \,.$
###### Example

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

$\oplus_{i \in I} A \simeq (\oplus_{i \in I} \mathbb{Z}) \otimes A \simeq (\mathbb{Z}[I]) \otimes A \,.$
###### 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 $R$ equipped with

1. an element $1 \in R$

2. a bilinear operation, hence a group homomorphism

$\cdot : R \otimes R \to R$

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 $r, r_1, r_2 \in R$ we have

$r \cdot (r_1 + r_2) = r \cdot r_1 + r \cdot r_2$

and

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

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

• 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 $R$ a ring, the polynomials

$r_0 + r_1 x + r_2 x^2 + \cdots + r_n x^n$

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

###### Example

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

###### Example

For $X$ a topological space, the set of continuous functions $C(X,\mathbb{R})$ or $C(X,\mathbb{C})$ with values in the real numbers or complex numbers is a ring under pointwise (points in $X$) 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 $C(X, \mathbb{C})$ in example 26 – called the structure of a C-star algebra, then this construction

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

is an equivalence of categories between that of topological spaces, and the opposite category of $C^\ast$-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 $R$ to be a commutative ring.

###### Definition

A module $N$ over a ring $R$ is

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

2. equipped with a morphism

$\alpha : R \otimes N \to N$

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

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

and which is a bilinear function in that it satisfies

$(r, n_1 + n_2) \mapsto r n_1 + r n_2$

and

$(r_1 + r_2, n) \mapsto r_1 n + r_2 n$

for all $r, r_1, r_2 \in R$ and $n,n_1, n_2 \in N$;

3. such that the diagram

$\array{ R \otimes R \otimes N &\stackrel{\cdot_R \otimes Id_N}{\to}& R \otimes N \\ {}^{\mathllap{Id_R \otimes \alpha}}\downarrow && \downarrow^{\mathrlap{\alpha}} \\ R \otimes N &\to& N }$

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

$(r_1 \cdot r_2) n = r_1 (r_2 n) \,.$
4. such that the diagram

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

commutes, which means that on elements as above

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

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

###### Example

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

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

The module action is componentwise:

$r \cdot (r_1, r_2, \cdots, r_n) = (r \cdot r_1, r\cdot r_2, \cdot r \cdot r_n) \,.$
###### Example

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

This is the free module (over $R$) on the set $S$.

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

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

###### Example

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

###### Example

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

Every finitely-generated free $k$-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 $N$ a module and $\{n_i\}_{i \in I}$ a set of elements, the linear span

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

(hence the completion of this set under addition in $N$ and multiplication by $R$) is a submodule of $N$.

###### Example

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

###### Definition

Write $R$Mod for the category or $R$-modules and $R$-linear maps between them.

###### Example

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

###### Example

Let $X$ be a topological space and let

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

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

Given a complex vector bundle $E \to X$ on $X$, write $\Gamma(E)$ for its set of continuous sections. Since for each point $x \in X$ the fiber $E_x$ of $E$ over $x$ is a $\mathbb{C}$-module (by example 31), $\Gamma(X)$ is a $C(X,\mathbb{C})$-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 $X$ is Hausdorff and compact with ring of functions $C(X,\mathbb{C})$ – as in remark 15 above – then $\Gamma(X)$ is a projective $C(X,\mathbb{C})$-module and indeed there is an equivalence of categories between projective $C(X,\mathbb{C})$-modules and complex vector bundles over $X$. (We introduce the notion of projective modules below in Derived categories and derived functors.)

We now discuss a bunch of properties of the category $R$Mod which together will show that there is a reasonable concept of chain complexes of $R$-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 $R$ be a commutative ring. Then $R Mod$ 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 $0 \in \mathcal{C}$ is a zero object precisely if for every other object $A$ there is a unique morphism $A \to 0$ to the zero object as well as a unique morphism $0 \to A$ from the zero object.

###### Proposition

The trivial group is a zero object in Ab.

The trivial module is a zero object in $R$Mod.

###### Proof

Clearly the 0-module $0$ is a terminal object, since every morphism $N \to 0$ has to send all elements of $N$ to the unique element of $0$, and every such morphism is a homomorphism. Also, 0 is an initial object because a morphism $0 \to N$ 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 $N$.

###### Definition

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

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

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

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

###### Example

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

###### Example

More generally, for $R$ any ring, this is true in $R$Mod: 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 $f : A \to B$ is the pushout $coker(f)$ in

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

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

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

###### Example

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

###### Proposition

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

$R Mod$ has all cokernels. The cokernel of a homomorphism $f : N_1 \to N_2$ is the quotient abelian group

$coker f = \frac{N_2}{im(f)}$

of $N_2$ by the image of $f$.

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

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

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

###### Proof

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

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

###### Definition

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

###### Proposition

With def. 37 $R Mod$ composition of morphisms

$\circ : Hom(N_1, N_2) \times Hom(N_2, N_3) \to Hom(N_1,N_3)$

is a bilinear map, hence is equivalently a morphism

$Hom(N_1, N_2) \otimes Hom(N_2,N_3) \to Hom(N_1, N_3)$

out of the tensor product of abelian groups.

This makes $R Mod$ 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 $R Mod$ 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

$R Mod$ is an pre-additive category.

###### Proposition

$R Mod$ 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 $R$-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 $\prod_{i \in I} N_i$ consists of arbitrary tuples because it needs to have a projection map

$p_j : \prod_{i \in I} N_i \to N_j$

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

$\iota_j : N_j \to \oplus_{i \in I} N_i$

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 $N_j$, hence of finite formal sums of these.

Together cor. 2 and prop. 21 say that:

###### Corollary

$R Mod$ is an additive category.

###### Proposition

In $R Mod$

###### Proof

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

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

Now we finally have all the ingredients to talk about chain complexes of $R$-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 $R$Mod is

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

• and of morphisms $\partial_n : C_n \to C_{n-1}$

$\cdots \overset{\partial_3}{\to} C_2 \overset{\partial_2}{\to} C_1 \overset{\partial_1}{\to} C_0 \overset{\partial_0}{\to} C_{-1} \overset{\partial_{-1}}{\to} \cdots$

such that

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

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

###### Definition

For $C_\bullet$ a chain complex and $n \in \mathbb{N}$

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

• the elements of $C_n$ are called the $n$-chains;

• for $n \geq 1$ the elements in the kernel

$Z_n \coloneqq ker(\partial_{n-1})$

of $\partial_{n-1} : C_n \to C_{n-1}$ are called the $n$-cycles

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

$Z_0 \coloneqq C_0$

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

• the elements in the image

$B_n \coloneqq im(\partial_n)$

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

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

$0 \hookrightarrow B_n \hookrightarrow Z_n \hookrightarrow C_n \,.$
• the cokernel

$H_n \coloneqq Z_n/B_n$

is called the degree-$n$ chain homology of $C_\bullet$.

$0 \to B_n \to Z_n \to H_n \to 0 \,.$
###### Definition

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

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

commute, hence such that all the equations

$f_n \circ d^V_n = d^W_{n+1} \circ f_{n+1}$

hold.

###### Proposition

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

$B_n(f) : B_n(C_\bullet) \to B_n(D_\bullet)$

and

$Z_n(f) : Z_n(C_\bullet) \to Z_n(D_\bullet) \,.$

In particular it also respects chain homology

$H_n(f) : H_n(C_\bullet) \to H_n(D_\bullet) \,.$
###### Corollary

Conversely this means that taking chain homology is a functor

$H_n(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A}$

from the category of chain complexes in $\mathcal{A}$ to $\mathcal{A}$ 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 $X, Y \in Ch_\bullet(\mathcal{A})$ write $X \otimes Y \in Ch_\bullet(\mathcal{A})$ for the chain complex whose component in degree $n$ is given by the direct sum

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

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

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

(square as tensor product of interval with itself)

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

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

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

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

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

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

$I_1 = R \simeq R[\{(0 \to 1)\}]$

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

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

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

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

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

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

$S_\bullet \coloneqq I_\bullet \otimes I_\bullet \,.$

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

\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 $(a,b) \in R\otimes R$ of the previous basis elements. Notice that by the definition of tensor product of modules we have relations like

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

etc.

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

\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

\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 $(I \otimes I)_\bullet$ vanish (are the 0-module) because all higher terms in $I_\bullet$ 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

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

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

\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 $I_\bullet$ is 0, and hence so is any tensor product with it.

Similarly the oriented boundary of the square itself is computed to

\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 $(Ch_\bullet(R Mod), \otimes)$. The unit object is the chain complex concentrated in degree 0 on the tensor unit $R$ of $R Mod$.

###### Definition

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

###### Definition

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

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

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

$d f := d_Y \circ f - (-1)^{n} f \circ d_X \,.$

This defines a functor

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

This functor

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

is the internal hom of the category of chain complexes.

###### Proposition

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

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

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

###### Proof

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

$f_{k+1} \circ d_X = d_Y \circ f_k \,.$

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

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

$\lambda_{k+1} \circ d_X + d_Y \circ \lambda_k$

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

###### Proposition

The monoidal category $(Ch_\bullet, \otimes)$ 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 $R$ throughout and consider the category of modules over $R$, which we will abbreviate

$\mathcal{A} \coloneqq R Mod \,.$
###### Definition

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

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

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

$\cdots \to 0 \to 0 \to A \to B \to C \to 0 \to 0 \to \cdots \,.$

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

###### 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 $A \to B \to C$ being exact (at $B$) and $A \to B \to C$ being a “short exact sequence” in that $0 \to A \to B \to C \to 0$ is exact at $A$, $B$ and $C$. This is illustrated by the following proposition.

###### Proposition

Explicitly, a sequence of morphisms

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

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

1. $i$ is a monomorphism,

2. $p$ is an epimorphism,

3. and the image of $i$ equals the kernel of $p$ (equivalently, the coimage of $p$ equals the cokernel of $i$).

###### Proof

The third condition is the definition of exactness at $B$. So we need to show that the first two conditions are equivalent to exactness at $A$ and at $C$.

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

The sequence being exact at

$0 \to A \to B$

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

Dually, the sequence being exact at

$B \to C \to 0$

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

###### Example

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

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

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

$\cdots \to 0 \to C_{n+1} \stackrel{\partial_n}{\to} C_n \to 0 \to \cdots \,,$

then $\partial_n$ is an isomorphism and hence

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

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

###### Definition

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

1. There exists a section of $p$, hence a homomorphism $s \colon B\to C$ such that $p \circ s = id_C$.

2. There exists a retract of $i$, hence a homomorphism $r \colon B\to A$ such that $r \circ i = id_A$.

3. There exists an isomorphism of sequences with the sequence

$0\to A\to A\oplus C\to C\to 0$

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 $r \colon B \to A$ of $i \colon A \to B$. Write $P \colon B \stackrel{r}{\to} A \stackrel{i}{\to} B$ for the composite. Notice that by $r\circ i = id$ this is an idempotent: $P \circ P = P$, hence a projector.

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

$0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0$

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

$0 \to A_n \to B_n \to C_n \to 0$

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

###### Definition

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

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

called the $n$th connecting homomorphism of the short exact sequence, by sending

$\delta_n : [c] \mapsto [\partial^B \hat c]_A \,,$

where

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

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

3. $[\partial^B \hat c]_A$ is the $A$-homology class of $\partial^B \hat c$ which is indeed in $A_{n-1} \hookrightarrow B_{n-1}$ by exactness (since $p(\partial^B \hat c) = \partial^C p(\hat c) = \partial^C c = 0$) and indeed in $Z_{n-1}(A) \hookrightarrow A_{n-1}$ since $\partial^A \partial^B \hat c = \partial^B \partial^B \hat c = 0$.

###### Proposition

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

###### Proof

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

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

###### Proposition

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

$\cdots \to H_n(A) \to H_n(B) \to H_n(C) \stackrel{\delta_n}{\to} H_{n-1}(A) \to H_{n-1}(B) \to H_{n-1}(C) \to \cdots \,.$
###### Proof

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

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

It remains to see that

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

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

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

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

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

###### Definition

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

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

in $\mathcal{A}$ such that

$f_n - g_n = \partial^D \circ \psi_n + \psi_{n-1} \partial^C \,.$
###### Remark

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

$\array{ \vdots && \vdots \\ \downarrow && \downarrow \\ C_{n+1} &\stackrel{f_{n+1} - g_{n+1}}{\to}& D_{n+1} \\ \downarrow^{\mathrlap{\partial^C_{n}}} &\nearrow_{\mathrlap{\psi_{n}}}& \downarrow^{\mathrlap{\partial^D_{n}}} \\ C_n &\stackrel{f_n - g_n}{\to}& D_n \\ \downarrow^{\mathrlap{\partial^C_{n-1}}} &\nearrow_{\mathrlap{\psi_{n-1}}}& \downarrow^{\mathrlap{\partial^D_{n-1}}} \\ C_{n-1} &\stackrel{f_{n-1} - g_{n-1}}{\to}& D_{n-1} \\ \downarrow && \downarrow \\ \vdots && \vdots } \,.$

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

$I_\bullet \coloneqq N_\bullet(C(\Delta[1]))$

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

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

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

###### Proposition

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

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

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

###### Proof

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

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

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

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

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

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

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

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

$f_n - g_n - \psi_{n-1} d_C = d_D \psi_{n} \,.$

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

###### Definition

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

$(f \sim g) \Leftrightarrow \exists (\psi : f \Rightarrow g) \,.$
###### Proposition

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

###### Definition

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

###### Proposition

This quotient is compatible with composition of chain maps.

Accordingly the following category exists:

###### Definition

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

$Hom_{\mathcal{K}_\bullet(\mathcal{A})}(C_\bullet, D_\bullet) \coloneqq Hom_{Ch_\bullet(\mathcal{A})}(C_\bullet, D_\bullet)_\sim \,.$

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

###### 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 $\mathcal{A}$. In the derived category one also quotients out chain homotopy, but one allows that first the domain of the two chain maps $f$ and $g$ is refined along a quasi-isomorphism.

###### Definition

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

$H_n(f) \colon H_n(C) \to H_n(D)$

is an isomorphism.

###### Remark

Quasi-isomorphisms are also called, more descriptively, homology isomorphisms or $H_\bullet$-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 $A_\bullet \to B_\bullet$ is the degreewise kernel of a chain map $B_\bullet \to C_\bullet$, then if $\hat A_\bullet \stackrel{\simeq}{\to} A_\bullet$ is a quasi-isomorphism (for instance a projective resolution of $A_\bullet$) then of course the composite chain map $\hat A_\bullet \to B_\bullet$ is in general far from being the degreewise kernel of $C_\bullet$. 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 $\mathcal{A}$ (for instance: not in the derived category of $\mathcal{A}$).

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 $A_\bullet \to B_\bullet \to C_\bullet$ is a short exact sequence of chain complexes, then the commuting square

$\array{ A_\bullet &\to& 0 \\ \downarrow && \downarrow \\ B_\bullet &\to& C_\bullet }$

is not only a pullback square in $Ch_\bullet(\mathcal{A})$, exhibiting $A_\bullet$ as the fiber of $B_\bullet \to C_\bullet$ over $0 \in C_\bullet$, 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 $\phi$:

$\array{ Q_\bullet &\to& 0 \\ \downarrow &\swArrow_{\phi}& \downarrow \\ B_\bullet &\to& C_\bullet }$

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

Equivalently, we have the formally dual result

###### Proposition

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

$\array{ A_\bullet &\to& 0 \\ \downarrow && \downarrow \\ B_\bullet &\to& C_\bullet }$

is not only a pushout square in $Ch_\bullet(\mathcal{A})$, exhibiting $C_\bullet$ as the cofiber of $A_\bullet \to B_\bullet$ over $0 \in C_\bullet$, 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 $\Omega(-)$ or suspension $\Sigma(-)$ 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 $f : A_\bullet \to B_\bullet$ looks like this:

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

here

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

• $A[1]$ etc. is the suspension of a chain complex of $A$, 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

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

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 $A_\bullet \to B_\bullet \to C_\bullet$.

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 $[0,1] \times X$ over a topological space, or the chain complex cylinder $I_\bullet \otimes X_\bullet$ 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 $f : X \to Y$ a morphism in a category with cylinder objects $cyl(-)$, the mapping cone or homotopy cofiber of $f$ is the colimit in the following diagram

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

in $C$ using any cylinder object $cyl(X)$ for $X$.

###### Remark

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

1. forming the cylinder over $X$;

2. gluing to one end of that the object $Y$ as specified by the map $f$.

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

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

Another interpretation of the mapping cone is just as important:

###### Remark

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

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

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

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

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 $C$, and in two ways by the following pasting diagram:

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

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

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

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

The pushout

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

defines the mapping cylinder $cyl(f)$ of $f$, the result of identifying one end of the cylinder over $X$ with $Y$, using $f$ as the gluing map.

The pushout

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

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

###### Remark

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

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

2. $cyl(f)$ is obtained from the cylinder over $X$ by gluing $Y$ to one end of the cylinder, as specified by the map $f$;

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

###### Definition

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

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

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

###### Definition

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

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

Denote by

$i_0 : *_\bullet \to I_\bullet$

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

$i_1 : *_\bullet \to I_\bullet$

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:

$I_\bullet = C_\bullet(\Delta[1]) \,.$

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

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

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

###### Proposition

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

$(I \otimes X)_\bullet \in Ch_\bullet$

is a cylinder object of $X_\bullet$ for the structure of a category of cofibrant objects on $Ch_\bullet$ 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 $(I \otimes X)_\bullet$ has components

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

and the differential is given by

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

hence in matrix calculus by

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

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

$(I \otimes X )_n = (R_{(0)} \otimes X_n ) \oplus (R_{(1)} \otimes X_n ) \oplus (R_{(0 \to 1)} \otimes X_{(n-1)} )$

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

\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 $X_\bullet$ into the cylinder are given in terms of def. 57 by

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

and

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

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

$X_n \hookrightarrow X_n \oplus X_n \oplus X_{n-1} \,.$

One part of definition 55 now reads:

###### Definition

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

$\array{ cyl(f)_\bullet &\leftarrow& Y_\bullet \\ \uparrow && \uparrow^{\mathrlap{f}} \\ I_\bullet \otimes X_\bullet &\stackrel{i_0}{\leftarrow}& X_\bullet } \,.$
###### Proposition

The components of $cyl(f)$ are

$cyl(f)_n = X_n \oplus Y_n \oplus X_{n-1}$

and the differential is given by

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

hence in matrix calculus by

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

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

the pushout identifies one direct summand $X_n$ with $Y_n$ along $f_n$ and so where previously a $id_{X_n}$ appeared on the diagonl, there is now $f_n$.

The last part of definition 55 now reads:

###### Definition

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

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

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

$cone(f)_n = Y_n \oplus X_{n-1}$

with differential given by

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

and hence in matrix calculus by

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

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

###### Proposition

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

$i_n : Y_n \to cone(f)_n = Y_n \oplus X_{n-1}$

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

###### Definition

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

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

for the chain map which in components

$p_n : cone(f)_n \to X[1]_n$

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

$p_n : Y_\n \oplus X_{n-1} \to X_{n-1} \,.$

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

###### Proof

By prop. 44 and def. 60 the sequence

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

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

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

Now, in the injective model structure on chain complexes all chain complexes are cofibrant objects and an inclusion such as $i : Y_\bullet \hookrightarrow cone(f)_\bullet$ 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 $p$ as the homotopy cofiber of $i$.

Accordingly one says:

###### Corollary

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

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

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

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

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

$0 \to X_\bullet \stackrel{f_\bullet}{\to} Y_\bullet \stackrel{p_\bullet}{\to} Z_\bullet \to 0$

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

###### Lemma

Let

$X_\bullet \stackrel{f_\bullet}{\to} Y_\bullet \stackrel{p_\bullet}{\to} Z_\bullet$

The collection of linear maps

$h_n : Y_n \oplus X_{n-1} \to Y_n \stackrel{}{\to} Z_n$

constitutes a chain map

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

This is a quasi-isomorphism. The inverse of $H_n(h_\bullet)$ is given by sending a representing cycle $z \in Z_n$ to

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

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

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

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

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

$\partial^{cone(f)} (x_{n-1}, \hat z_n) = ( -\partial^X x_{n-1} , \partial^Y \hat z_n + x_{n-1} )$

and use that $x_{n-1}$ is in the kernel of $p_n$ by exactness, hence

\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

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

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

Abstractly, this already implies that $cone(f)_\bullet \to Z_\bullet$ is a quasi-isomorphism, for this diagram gives a morphism of cocones under the diagram defining $cone(f)$ 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 $(x_{n-1}, y_n) \in cone(f)_n$ which lift a cycle $z_n$. By lemma 38 a lift of chains is any pair of the form $(x_{n-1}, \hat z_n)$ where $\hat z_n$ is a lift of $z_n$ through $Y_n \to X_n$. So $x_{n-1}$ has to be found such that this pair is a cycle. By prop. 43 the differential acts on it by

$\partial^{cone(f)} (x_{n-1}, \hat z_n) = ( -\partial^X x_{n-1} , \partial^Y \hat z_n + x_{n-1} )$

and so the condition is that

$x_{n-1} \coloneqq -\partial^Y \hat z_n$ (which implies $\partial^X x_{n-1} = -\partial^X \partial^Y \hat z_n = -\partial^Y \partial^Y \hat z_n = 0$ due to the fact that $f_n$ is assumed to be an inclusion, hence that $\partial^X$ is the restriction of $\partial^Y$ to elements in $X_n$).

This condition clearly has a unique solution for every lift $\hat z_n$ and a lift $\hat z_n$ always exists since $p_n : Y_n \to Z_n$ is surjective, by assumption that we have a short exact sequence of chain complexes. This shows that $H_n(h_\bullet)$ is surjective.

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

$\partial^{cone(f)}(x_n, y_{n+1}) \coloneqq (-\partial^X x_n, \partial^Y y_{n+1} + x_n) = (-\partial^Y \hat z_n, \hat z_n) \,.$

For that let $\hat c_{n+1} \in Y_{n+1}$ be a lift of $c_{n+1}$ through $p_n$, which exists again by surjectivity of $p_{n+1}$. Observe that

$p_{n}( \hat z_n - \partial^Y \hat c_{n+1}) = z_n -\partial^Z ( p_n \hat c_{n+1} ) = z_n - \partial^Z ( c_{n+1} ) = 0$

by assumption on $z_n$ and $c_{n+1}$, and hence that $\hat z_n - \partial^Y \hat c_{n+1}$ is in $X_n$ by exactness.

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

\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

$X_\bullet \stackrel{f_\bullet}{\to} Y_\bullet \to Z_\bullet$

Then the chain homology functor

$H_n(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A}$

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

$H_n(X_\bullet) \to H_n(Y_\bullet) \to H_n(Z_\bullet) \stackrel{\delta}{\to} H_{n-1}(X_\bullet) \to H_{n-1}(Y_\bullet) \to H_{n-1}(Z_\bullet) \stackrel{\delta}{\to} H_{n-2}(X_\bullet) \to \cdots \,,$

where $\delta_n$ is the $n$th connecting homomorphism.

###### Proof

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

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

Observe that

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

It is therefore sufficient to check that

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

equals the connecting homomorphism $\delta_n$ 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

$[z_n] \mapsto [x_{n-1}, y_n] \mapsto [x_{n-1}] \,.$

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

$[z_n]_Z \mapsto -[\partial^Y \hat z_n]_X \,.$

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

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

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

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

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

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

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

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

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 $n$, 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.

$\array{ && {topological \atop spaces} \\ && \downarrow^{\mathrlap{{higher \atop path}\atop groupoid}} & \\ groupoids &\stackrel{{Grothendieck \atop nerve}}{\longrightarrow}& { {\mathbf{Kan}\;\mathbf{complexes}} \atop {\simeq \infty-groupoids} } &\stackrel{{Dold-Kan \atop correspondence}}{\longleftarrow}& {chain \atop complexes} \\ && \downarrow^{\mathrlap{included \atop in}} \\ && {simplicial \atop sets} }$

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 $k \in \mathbb{N}$, which for $k \geq 1$ go between $(k-1)$-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 $\Delta$, whose objects are the abstract cellular $k$-simplices, denoted $[k]$ or $\Delta[k]$ for all $k \in \mathbb{N}$, and whose morphisms $\Delta[k_1] \to \Delta[k_2]$ are all ways of mapping these into each other. So we think of such a simplicial set given by a functor

$K : \Delta^{op} \to Set$

as specifying

• a set $[0] \mapsto K_0$ of objects;

• a set $[1] \mapsto K_1$ of morphisms;

• a set $[2] \mapsto K_2$ of 2-morphisms;

• a set $[3] \mapsto K_3$ of 3-morphisms;

and generally

• a set $[k] \mapsto K_k$ of k-morphisms

as well as specifying

• functions $([n] \hookrightarrow [n+1]) \mapsto (K_{n+1} \to K_n)$ that send $n+1$-morphisms to their boundary $n$-morphisms;

• functions $([n+1] \to [n]) \mapsto (K_{n} \to K_{n+1})$ that send $n$-morphisms to identity $(n+1)$-morphisms on them.

The fact that $K$ 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 $k$-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 $\Lambda^1[2]$ the simplicial set consisting of two attached 1-cells

$\Lambda^1[2] = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&&& 2 } \right\}$

and for $(f,g) : \Lambda^1[2] \to K$ an image of this situation in $K$, hence a pair $x_0 \stackrel{f}{\to} x_1 \stackrel{g}{\to} x_2$ of two composable 1-morphisms in $K$, we want to demand that there exists a third 1-morphisms in $K$ that may be thought of as the composition $x_0 \stackrel{h}{\to} x_2$ of $f$ and $g$. 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

$\array{ && x_1 \\ & {}^{\mathllap{f}}\nearrow &\Downarrow^{\mathrlap{\simeq}}& \searrow^{\mathrlap{g}} \\ x_0 &&\stackrel{h}{\to}&& x_2 } \,.$

From the picture it is clear that this is equivalent to demanding that for $\Lambda^1[2] \hookrightarrow \Delta[2]$ the obvious inclusion of the two abstract composable 1-morphisms into the 2-simplex we have a diagram of morphisms of simplicial sets

$\array{ \Lambda^1[2] &\stackrel{(f,g)}{\to}& K \\ \downarrow & \nearrow_{\mathrlap{\exists h}} \\ \Delta[2] } \,.$

A simplicial set where for all such $(f,g)$ a corresponding such $h$ 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

$\Lambda^2[2] = \left\{ \array{ && 1 \\ & && \searrow \\ 0 &&\to&& 2 } \right\}$

the simplicial set consisting of two 1-morphisms that touch at their end, hence for

$(g,h) : \Lambda^2[2] \to K$

two such 1-morphisms in $K$, then if $g$ had an inverse $g^{-1}$ we could use the above composition operation to compose that with $h$ and thereby find a morphism $f$ connecting the sources of $h$ and $g$. This being the case is evidently equivalent to the existence of diagrams of morphisms of simplicial sets of the form

$\array{ \Lambda^2[2] &\stackrel{(g,h)}{\to}& K \\ \downarrow & \nearrow_{\mathrlap{\exists f}} \\ \Delta[2] } \,.$

Demanding that all such diagrams exist is therefore demanding that we have on 1-morphisms a composition operation with inverses in $K$.

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 $K$. 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 $\Lambda^i[n] \hookrightarrow \Delta[n]$ be the simplicial set – called the $i$th $n$-horn – that consists of all cells of the $n$-simplex $\Delta[n]$ except the interior $n$-morphism and the $i$th $(n-1)$-morphism.

Then a simplicial set is called a Kan complex, if for all images $f : \Lambda^i[n] \to K$ of such horns in $K$, the missing two cells can be found in $K$- in that we can always find a horn filler $\sigma$ in the diagram

$\array{ \Lambda^i[n] &\stackrel{f}{\to}& K \\ \downarrow & \nearrow_{\mathrlap{\sigma}} \\ \Delta[n] } \,.$

The basic example is the nerve $N(C) \in sSet$ of an ordinary groupoid $C$, which is the simplicial set with $N(C)_k$ being the set of sequences of $k$ composable morphisms in $C$. 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

$\Delta[n] = \mathbf{\Delta}( -, [n]) \in Simp Set$

be the standard simplicial $n$-simplex in SimpSet.

###### Definition

For each $i$, $0 \leq i \leq n$, the $(n,i)$-horn or $(n,i)$-box is the subsimplicial set

$\Lambda^i[n] \hookrightarrow \Delta[n]$

which is the union of all faces except the $i^{th}$ one.

This is called an outer horn if $k = 0$ or $k = n$. 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 $\Lambda^k[n]: \Delta^{op} \to Set$ must therefore be: it takes $[m]$ to the collection of ordinal maps $f: [m] \to [n]$ which do not have the element $k$ in the image.

###### Example

The inner horn, def. 61 of the 2-simplex

$\Delta^2 = \left\{ \array{ && 1 \\ & \nearrow &\Downarrow& \searrow \\ 0 &&\to&& 2 } \right\}$

with boundary

$\partial \Delta^2 = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&\to&& 2 } \right\}$

looks like

$\Lambda^2_1 = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&&& 2 } \right\} \,.$

The two outer horns look like

$\Lambda^2_0 = \left\{ \array{ && 1 \\ & \nearrow && \\ 0 &&\to&& 2 } \right\}$

and

$\Lambda^2_2 = \left\{ \array{ && 1 \\ & && \searrow \\ 0 &&\to&& 2 } \right\}$

respectively.

###### Definition

A Kan complex is a simplicial set $S$ 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 $S \to pt$ from $S$ to the point (the terminal simplicial set) is a Kan fibration;

• which means equivalently that for all diagrams in sSet of the form

$\array{ \Lambda^i[n] &\to& S \\ \downarrow && \downarrow \\ \Delta[n] &\to& pt } \;\;\; \leftrightarrow \;\;\; \array{ \Lambda^i[n] &\to& S \\ \downarrow && \\ \Delta[n] }$

there exists a diagonal morphism

$\array{ \Lambda^i[n] &\to& S \\ \downarrow &\nearrow& \downarrow \\ \Delta[n] &\to& pt } \;\;\; \leftrightarrow \;\;\; \array{ \Lambda^i[n] &\to& S \\ \downarrow &\nearrow& \\ \Delta[n] }$

completing this to a commuting diagram;

• which in turn means equivalently that the map from $n$-simplices to $(n,i)$-horns is an epimorphism

$[\Delta[n], S]\, \twoheadrightarrow \,[\Lambda^i[n],S] \,.$
##### Groupoids as Kan complexes – Grothendieck simplicial nerve
###### Definition

For $\mathcal{G}_\bullet$ a groupoid, def. 21, its simplicial nerve $N(\mathcal{G}_\bullet)_\bullet$ is the simplicial set with

$N(\mathcal{G}_\bullet)_n \coloneqq \mathcal{G}_1^{\times_{\mathcal{G}_0}^n}$

the set of sequences of composable morphisms of length $n$, for $n \in \mathbb{N}$;

with face maps

$d_k \colon N(\mathcal{G}_\bullet)_{n+1} \to N(\mathcal{G}_\bullet)_{n}$

being,

• for $n = 0$ the functions that remembers the $k$th object;

• for $n \geq 1$

• the two outer face maps $d_0$ and $d_n$ are given by forgetting the first and the last morphism in such a sequence, respectively;

• the $n-1$ inner face maps $d_{0 \lt k \lt n}$ are given by composing the $k$th morphism with the $k+1$st in the sequence.

The degeneracy maps

$s_k \colon N(\mathcal{G}_\bullet)n \to N(\mathcal{G}_\bullet)_{n+1} \,.$

are given by inserting an identity morphism on $x_k$.

###### Remark

Spelling this out in more detail: write

$\mathcal{G}_n = \left\{ x_0 \stackrel{f_{0,1}}{\to} x_1 \stackrel{f_{1,2}}{\to} x_2 \stackrel{f_{2,3}}{\to} \cdots \stackrel{f_{n-1,n}}{\to} x_n \right\}$

for the set of sequences of $n$ composable morphisms. Given any element of this set and $0 \lt k \lt n$, write

$f_{i-1,i+1} \coloneqq f_{i,i+1} \circ f_{i-1,i}$

for the comosition of the two morphism that share the $i$th vertex.

With this, face map $d_k$ acts simply by “removing the index $k$”:

$d_0 \colon (x_0 \stackrel{f_{0,1}}{\to} x_1 \stackrel{f_{1,2}}{\to} x_{2} \cdots \stackrel{f_{n-1,n}}{\to} x_n ) \mapsto (x_1 \stackrel{f_{1,2}}{\to} x_{2} \cdots \stackrel{f_{n-1,n}}{\to} x_n )$
$d_{0\lt k \lt n} \colon ( x_0 \cdots \stackrel{}{\to} x_{k-1} \stackrel{f_{k-1,k}}{\to} x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \stackrel{}{\to} \cdots x_n ) \mapsto ( x_0 \cdots \stackrel{}{\to} x_{k-1} \stackrel{f_{k-1,k+1}}{\to} x_{k+1} \stackrel{}{\to} \cdots x_n )$
$d_n \colon ( x_0 \stackrel{f_{0,1}}{\to} \cdots \stackrel{f_{n-2,n-1}}{\to} x_{n-1} \stackrel{f_{n-1,n}}{\to} x_n ) \mapsto ( x_0 \stackrel{f_{0,1}}{\to} \cdots \stackrel{f_{n-2,n-1}}{\to} x_{n-1} ) \,.$

Similarly, writing

$f_{k,k} \coloneqq id_{x_k}$

for the identity morphism on the object $x_k$, then the degenarcy map acts by “repeating the $k$th index”

$s_k \colon ( x_0 \stackrel{}{\to} \cdots \to x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \to \cdots ) \mapsto ( x_0 \stackrel{}{\to} \cdots \to x_k \stackrel{f_{k,k}}{\to} x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \to \cdots ) \,.$

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 $\mathcal{G}_\bullet$ 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

$N \colon Grpd \to KanCplx \hookrightarrow sSet \,.$
##### 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 $A$ a simplicial abelian group its alternating face map complex $(C A)_\bullet$ of $A$ is the chain complex which

• in degree $n$ is given by the group $A_n$ itself

$(C A)_n := A_n$
• with differential given by the alternating sum of face maps (using the abelian group structure on $A$)

$\partial_n \coloneqq \sum_{i = 0}^n (-1)^i d_i \;\colon\; (C A)_n \to (C A)_{n-1} \,.$
###### Lemma

The differential in def. 64 is well-defined in that it indeed squares to 0.

###### Proof

Using the simplicial identity, prop. 1, $d_i \circ d_j = d_{j-1} \circ d_i$ for $i \lt j$ one finds:

\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 $A$, its normalized chain complex or Moore complex is the $\mathbb{N}$-graded chain complex $((N A)_\bullet,\partial )$ which

• is in degree $n$ the joint kernel

$(N A)_n=\bigcap_{i=1}^{n}ker\,d_i^n$

of all face maps except the 0-face;

• with differential given by the remaining 0-face map

$\partial_n := d_0^n|_{(N A)_n} : (N A)_n \rightarrow (N A)_{n-1} \,.$
###### Remark

We may think of the elements of the complex $N A$, def. 65, in degree $k$ as being $k$-dimensional disks in $A$ all whose boundary is captured by a single face:

• an element $g \in N G_1$ in degree 1 is a 1-disk

$1 \stackrel{g}{\to} \partial g \,,$
• an element $h \in N G_2$ is a 2-disk

$\array{ && 1 \\ & {}^1\nearrow &\Downarrow^h& \searrow^{\partial h} \\ 1 &&\stackrel{1}{\to}&& 1 } \,,$
• a degree 2 element in the kernel of the boundary map is such a 2-disk that is closed to a 2-sphere

$\array{ && 1 \\ & {}^1\nearrow &\Downarrow^h& \searrow^{\partial h = 1} \\ 1 &&\stackrel{1}{\to}&& 1 } \,,$

etc.

###### Definition

Given a simplicial group $A$ (or in fact any simplicial set), then an element $a \in A_{n+1}$ is called degenerate (or thin) if it is in the image of one of the simplicial degeneracy maps $s_i \colon A_n \to A_{n+1}$. All elements of $A_0$ are regarded a non-degenerate. Write

$D (A_{n+1}) \coloneqq \langle \cup_i s_i(A_{n}) \rangle \hookrightarrow A_{n+1}$

for the subgroup of $A_{n+1}$ which is generated by the degenerate elements (i.e. the smallest subgroup containing all the degenerate elements).

###### Definition

For $A$ a simplicial abelian group its alternating face maps chain complex modulo degeneracies, $(C A)/(D A)$ is the chain complex

• which in degree 0 equals is just $((C A)/D(A))_0 \coloneqq A_0$;

• which in degree $n+1$ is the quotient group obtained by dividing out the group of degenerate elements, def. 66:

$((C A)/D(A))_{n+1} := A_{n+1} / D(A_{n+1})$
• 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 $s_j(a) \in A_n$ a degenerate element, its boundary is

\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 $A$, the evident composite of natural morphisms

$N A \stackrel{i}{\hookrightarrow} A \stackrel{p}{\to} (C A)/(D A)$

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.

###### Corollary

For $A$ a simplicial abelian group, there is a splitting

$C_\bullet(A) \simeq N_\bullet(A) \oplus D_\bullet(A)$

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

$\array{ C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) \\ {}^{\mathllap{i}}\uparrow & \nearrow \\ N A } \,.$

This hence exhibits a splitting of the short exact sequence given by the quotient by $D A$.

$\array{ 0 &\to& D A &\hookrightarrow & C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) &\to & 0 \\ && && {}^{\mathllap{i}}\uparrow & \swarrow_{\mathrlap{\simeq}_{iso}} \\ && && N A } \,.$
###### Theorem (Eilenberg-MacLane)

Given a simplicial abelian group $A$, then the inclusion

$N A \hookrightarrow C A$

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 $D_\bullet(X)$ is null-homotopic.

###### Corollary

Given a simplicial abelian group $A$, then the projection chain map

$(C A) \longrightarrow (C A)/(D A)$

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.

$\array{ C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) \\ {}^{\mathllap{i}}\uparrow & \nearrow \\ N A }$

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 $\Delta[1]$ regarded as a simplicial set, and write $\mathbb{Z}[\Delta[1]]$ for the simplicial abelian group which in each degree is the free abelian group on the simplices in $\Delta[1]$.

This simplicial abelian group starts out as

$\mathbb{Z}[\Delta[1]] = \left( \cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}} \mathbb{Z}^4 \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} \mathbb{Z}^3 \stackrel{\overset{\partial_0}{\longrightarrow}}{\underset{\partial_1}{\longrightarrow}} \mathbb{Z}^2 \right)$

(where we are indicating only the face maps for notational simplicity).

Here the first $\mathbb{Z}^2 = \mathbb{Z}\oplus \mathbb{Z}$, the direct sum of two copies of the integers, is the group of 0-chains generated from the two endpoints $(0)$ and $(1)$ of $\Delta[1]$, i.e. the abelian group of formal linear combinations of the form

$\mathbb{Z}^2 \simeq \left\{ a \cdot (0) + b \cdot (1) | a,b \in \mathbb{Z}\right\} \,.$

The second $\mathbb{Z}^3 \simeq \mathbb{Z}\oplus \mathbb{Z}\oplus \mathbb{Z}$ is the abelian group generated from the three (!) 1-simplicies in $\Delta[1]$, namely the non-degenerate edge $(0\to 1)$ and the two degenerate cells $(0 \to 0)$ and $(1 \to 1)$, hence the abelian group of formal linear combinations of the form

$\mathbb{Z}^3 \simeq \left\{ a \cdot (0\to 0) + b \cdot (0 \to 1) + c \cdot (1 \to 1) | a,b,c \in \mathbb{Z}\right\} \,.$

The two face maps act on the basis 1-cells as

$\partial_1 \colon (i \to j) \mapsto (i)$
$\partial_0 \colon (i \to j) \mapsto (j) \,.$

Now of course most of the (infinitely!) many simplices inside $\Delta[1]$ are degenerate. In fact the only non-degenerate simplices are the two 0-cells $(0)$ and $(1)$ and the 1-cell $(0 \to 1)$. Hence the alternating face maps complex modulo degeneracies, def. 67, of $\mathbb{Z}[\Delta[1]]$ is simply this:

$(C (\mathbb{Z}[\Delta[1]])) / D (\mathbb{Z}[\Delta[1]])) = \left( \cdots \to 0 \to 0 \to \mathbb{Z} \stackrel{\left(1 \atop -1\right)}{\longrightarrow} \mathbb{Z}^2 \right) \,.$

Notice that alternatively we could consider the topological 1-simplex $\Delta^1 = [0,1]$ and its singular simplicial complex $Sing(\Delta^1)$ in place of the smaller $\Delta[1]$, then the free simplicial abelian group $\mathbb{Z}(Sing(\Delta^1))$ of that. The corresponding alternating face map chain complex $C(\mathbb{Z}(Sing(\Delta^1)))$ 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 $n$-simplex in $[0,1]$ 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 $A$ an abelian category there is an equivalence of categories

$N \;\colon\; A^{\Delta^{op}} \stackrel{\leftarrow}{\to} Ch_\bullet^+(A) \;\colon\; \Gamma$

between

where

###### Theorem (Kan)

For the case that $A$ is the category Ab of abelian groups, the functors $N$ and $\Gamma$ are nerve and realization with respect to the cosimplicial chain complex

$\mathbb{Z}[-]: \Delta \to Ch_+(Ab)$

that sends the standard $n$-simplex to the normalized Moore complex of the free simplicial abelian group $F_{\mathbb{Z}}(\Delta^n)$ on the simplicial set $\Delta^n$, i.e.

$\Gamma(V) : [k] \mapsto Hom_{Ch_\bullet^+(Ab)}(N(\mathbb{Z}(\Delta[k])), V) \,.$

This is due to (Kan 58).

More explicitly we have the following

###### Proposition
• For $V \in Ch_\bullet^+$ the simplicial abelian group $\Gamma(V)$ is in degree $n$ given by

$\Gamma(V)_n = \bigoplus_{[n] \underset{surj}{\to} [k]} V_k$

and for $\theta : [m] \to [n]$ a morphism in $\Delta$ the corresponding map $\Gamma(V)_n \to \Gamma(V)_m$

$\theta^* : \bigoplus_{[n] \underset{surj}{\to} [k]} V_k \to \bigoplus_{[m] \underset{surj}{\to} [r]} V_r$

is given on the summand indexed by some $\sigma : [n] \to [k]$ by the composite

$V_k \stackrel{d^*}{\to} V_s \hookrightarrow \bigoplus_{[m] \underset{surj}{\to} [r]} V_r$

where

$[m] \stackrel{t}{\to} [s] \stackrel{d}{\to} [k]$

is the epi-mono factorization of the composite $[m] \stackrel{\theta}{\to} [n] \stackrel{\sigma}{\to} [k]$.

• The natural isomorphism $\Gamma N \to Id$ is given on $A \in sAb^{\Delta^{op}}$ by the map

$\bigoplus_{[n] \underset{surj}{\to} [k]} (N A)_k \to A_n$

which on the direct summand indexed by $\sigma : [n] \to [k]$ is the composite

$N A_k \hookrightarrow A_k \stackrel{\sigma^*}{\to} A_n \,.$
• The natural isomorphism $Id \to N \Gamma$ is on a chain complex $V$ given by the composite of the projection

$V \to C(\Gamma(V)) \to C(\Gamma(C))/D(\Gamma(V))$

with the inverse

$C(\Gamma(V))/D(\Gamma(V)) \to N \Gamma(V)$

of

$N \Gamma(V) \hookrightarrow C(\Gamma(V)) \to C(\Gamma(V))/D(\Gamma(V))$

(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 $\Gamma N \stackrel{\simeq}{\to} Id$ as above we have that $\Gamma$ and $N$ form an adjoint equivalence $(\Gamma \dashv N)$

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: $(N \dashv \Gamma)$.

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 $G$ be a simplicial group.

Here is the explicit algorithm that computes the horn fillers:

Let $(y_0,\ldots, y_{k-1}, -,y_{k+1}, \ldots, y_n)$ give a horn in $G_{n-1}$, so the $y_i$s are $(n-1)$ simplices that fit together as if they were all but one, the $k^{th}$ one, of the faces of an $n$-simplex. There are three cases:

1. if $k = 0$:

• Let $w_n = s_{n-1}y_n$ and then $w_i = w_{i+1}(s_{i-1}d_i w_{i+1})^{-1}s_{i-1}y_i$ for $i = n, \ldots, 1$, then $w_1$ satisfies $d_i w_1 = y_i$, $i\neq 0$;
2. if $0\lt k \lt n$:

• Let $w_0 = s_0 y_0$ and $w_i = w_{i-1}(s_i d_i w_{i-1})^{-1}s_i y_i$ for $i = 0, \ldots, k-1$, then take $w_n = w_{k-1}(s_{n-1}d_nw_{k-1})^{-1}s_{n-1}y_n$, and finally a downwards induction given by $w_i = w_{i+1}(s_{i-1}d_{i}w_{i+1})^{-1}s_{i-1}y_i$, for $i = n, \ldots, k+1$, then $w_{k+1}$ gives $d_{i}w_{k+1} = y_i$ for $i \neq k$;
3. if $k=n$:

• use $w_0 = s_0 y_0$ and $w_i = w_{i-1}(s_i d_i w_{i-1})^{-1}s_i y_i$ for $i = 0, \ldots, n-1$, then $w_{n-1}$ satisfies $d_i w_{n-1} = y_i$, $i\neq n$.

#### 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 $\mathcal{A}$, 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 $\mathcal{K}(\mathcal{A})$ 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 $\mathcal{K}(\mathcal{A})$. 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 $Ch_\bullet(\mathcal{A})$ for $\mathcal{A} =$ Ab consider the chain map

$\array{ \cdots &\to& 0 &\to& 0 &\to& 0 &\to& \mathbb{Z}_2 \\ && \downarrow && \downarrow && \downarrow && \downarrow^{\mathrlap{id}} \\ \cdots &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z} &\stackrel{mod\,2}{\to}& \mathbb{Z}_2 } \,.$

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 $\mathbb{Z}_2 \to \mathbb{Z}$, and this must be the 0-morphism, because $\mathbb{Z}$ is a free group, but $\mathbb{Z}_2$ 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

$\array{ \cdots &\to& 0 &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z} \\ && \downarrow && \downarrow && \downarrow && \downarrow^{\mathrlap{mod\,2}} \\ \cdots &\to& 0 &\to& 0 &\to& 0 &\to& \mathbb{Z}_2 } \,,$

where now the domain complex consists entirely of free groups. The composite of this with the original chain map is now

$\array{ \cdots &\to& 0 &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z} \\ && \downarrow && \downarrow && \downarrow^{0} && \downarrow^{\mathrlap{mod\,2}} \\ \cdots &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z} &\stackrel{mod\,2}{\to}& \mathbb{Z}_2 } \,.$

This is the corresponding resolution of the original chain map. And this indeed has a null homotopy:

$\array{ \cdots &\to& 0 &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z} \\ && \downarrow &\swarrow& \downarrow &\swarrow_{-id}& \downarrow^{0} &\swarrow_{\mathrlap{id}}& \downarrow^{\mathrlap{mod\,2}} \\ \cdots &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z} &\stackrel{mod\,2}{\to}& \mathbb{Z}_2 } \,.$

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 $P$ of a category $C$ is a projective object if it has the left lifting property against epimorphisms.

This means that $P$ is projective if for any morphism $f:P \to B$ and any epimorphism $q:A \to B$, $f$ factors through $q$ by some morphism $P\to A$.

$\array{ && A \\ &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{q}} \\ P &\stackrel{f}{\to}& B } \,.$

An equivalent way to say this is that:

###### Definition

An object $P$ is projective precisely if the hom-functor $Hom(P,-)$ 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 $R$Mod: 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 $R$-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 $N \simeq R^{(S)}$ is projective.

###### Proof

Explicitly: if $S \in Set$ and $F(S) = R^{(S)}$ is the free module on $S$, then a module homomorphism $F(S) \to N$ is specified equivalently by a function $f : S \to U(N)$ from $S$ to the underlying set of $N$, which can be thought of as specifying the images of the unit elements in $R^{(S)} \simeq \oplus_{s \in S} R$ of the ${\vert S\vert}$ copies of $R$.

Accordingly then for $\tilde N \to N$ an epimorphism, the underlying function $U(\tilde N) \to U(N)$ is an epimorphism, and the axiom of choice in Set says that we have all lifts $\tilde f$ in

$\array{ && U(\tilde N) \\ & {}^{\tilde f} \nearrow & \downarrow \\ S &\stackrel{f}{\to}& U(N) } \,.$

By adjunction these are equivalently lifts of module homomorphisms

$\array{ && \tilde N \\ & \nearrow & \downarrow \\ R^{(S)} &\stackrel{}{\to}& N } \,.$
###### Lemma

If $N \in R Mod$ is a direct summand of a free module, hence if there is $N' \in R Mod$ and $S \in Set$ such that

$R^{(S)} \simeq N \oplus N' \,,$

then $N$ is a projective module.

###### Proof

Let $\tilde K \to K$ be a surjective homomorphism of modules and $f : N \to K$ a homomorphism. We need to show that there is a lift $\tilde f$ in

$\array{ && \tilde K \\ & {}^{\mathllap{\tilde f}}\nearrow & \downarrow \\ N &\stackrel{f}{\to}& K } \,.$

By definition of direct sum we can factor the identity on $N$ as

$id_N : N \to N \oplus N' \to N \,.$

Since $N \oplus N'$ is free by assumption, and hence projective by lemma 4, there is a lift $\hat f$ in

$\array{ && && \tilde K \\ && & {}^{\mathllap{\hat f}}\nearrow & \downarrow \\ N &\to& N \oplus N' &\to& K } \,.$

Hence $\tilde f : N \to N \oplus N' \stackrel{\hat f}{\to} \tilde K$ is a lift of $f$.

###### Proposition

An $R$-module $N$ is projective precisely if it is the direct summand of a free module.

###### Proof

By lemma 5 if $N$ is a direct summand then it is projective. So we need to show the converse.

Let $F(U(N))$ be the free module on the set $U(N)$ underlying $N$, hence the direct sum

$F(U(n)) = \oplus_{n \in U(n)} R \,.$

There is a canonical module homomorphism

$\oplus_{n \in U(N)} R \to N$

given by sending the unit $1 \in R_n$ of the copy of $R$ in the direct sum labeled by $n \in U(n)$ to $n \in N$.

(Abstractly this is the counit $\epsilon : F(U(N)) \to N$ of the free/forgetful-adjunction $(F \dashv U)$.)

This is clearly an epimorphism. Thefore if $N$ is projective, there is a section $s$ of $\epsilon$. This exhibits $N$ as a direct summand of $F(U(N))$.

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 $C^\bullet$ in $\mathcal{A} = R Mod$ is a sequence of morphism

$C^0 \stackrel{d^0}{\to} C^1 \stackrel{d^1}{\to} C^2 \stackrel{d^2}{\to} \cdots$

in $\mathcal{A}$ such that $d\circ d = 0$. A homomorphism of cochain complexes $f^\bullet : C^\bullet \to D^\bullet$ is a collection of morphisms $\{f^n : C^n \to D^n\}$ such that $d^n_D \circ f^n = f^n \circ d^n_C$ for all $n \in \mathbb{N}$.

We write $Ch^\bullet(\mathcal{A})$ for the category of cochain complexes.

###### Example

Let $N \in \mathcal{A}$ be a fixed module and $C_\bullet \in Ch_\bullet(\mathcal{A})$ a chain complex. Then applying degreewise the hom-functor out of the components of $C_\bullet$ into $N$ yields a cochain complex in $\mathbb{Z} Mod \simeq$ Ab:

$Hom_{\mathcal{A}}(C_\bullet, N) = \left[ Hom_{\mathcal{A}}(C_0, N) \stackrel{Hom_{\mathcal{A}}(\partial_0, N)}{\to} Hom_{\mathcal{A}}(C_1, N) \stackrel{Hom_{\mathcal{A}}(\partial_1, N)}{\to} Hom_{\mathcal{A}}(C_2, N) \stackrel{Hom_{\mathcal{A}}(\partial_2, N)}{\to} \cdots \right] \,.$
###### Example

In example 47 let $\mathcal{A} = \mathbb{Z}$Mod $=$ Ab, let $N = \mathbb{Z}$ and let $C_\bullet = \mathbb{Z}[Sing(X)]$ be the singular simplicial complex of a topological space $X$. Write

$C^\bullet(X) \coloneqq Hom_{\mathbb{Z}[Sing X], \mathbb{Z}} \,.$

Then $H^\bullet(C(X))$ is called the singular cohomology of $X$.

###### 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 $C^\bullet$ a cochain complex that

• an element in $C^n$ is an $n$-cochain

• an element in $im(d^{n-1})$ is an $n$-coboundary

• al element in $ker(d^n)$ is an $n$-cocycle.

But equivalently we may regard a cochain in degree $n$ as a chain in degree $(-n)$ 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 $I$ a category is injective if all diagrams of the form

$\array{ X &\to& I \\ {}^{}\downarrow \\ Z }$

with $X \to Z$ a monomorphism admit an extension

$\array{ X &\to& I \\ {}^{}\downarrow & \nearrow_{\mathrlap{\exists}} \\ Z } \,.$

Since we are interested in refining modules by projective or injective modules, we have the following terminology.

###### Definition

A category

• has enough projectives if for every object $X$ there is a projective object $Q$ equipped with an epimorphism $Q \to X$;

• has enough injectives if for every object $X$ there is an injective object $P$ equipped with a monomorphism $X \to P$.

We have essentially already seen the following statement.

###### Proposition

Assuming the axiom of choice, the category $R$Mod has enough projectives.

###### Proof

Let $F(U(N))$ be the free module on the set $U(N)$ underlying $N$. By lemma 4 this is a projective module.

The canonical morphism

$F(U(n)) = \oplus_{n \in U(n)} R \to N$

is clearly a surjection, hence an epimorphism in $R$Mod.

We now show that similarly $R Mod$ 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 $A$ is injective as a $\mathbb{Z}$-module precisely if it is a divisible group, in that for all integers $n \in \mathbb{N}$ we have $n G = G$.

###### 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 $\mathbb{Z}/n\mathbb{Z}$.

###### 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 $A$ receives an epimorphism $(\oplus_{s \in S} \mathbb{Z}) \to A$ 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 $\tilde A$ of a direct sum of copies of $\mathbb{Q}$.

$\array{ ker &\stackrel{=}{\to}& ker \\ \downarrow && \downarrow \\ (\oplus_{s \in S} \mathbb{Z}) &\hookrightarrow& (\oplus_{s \in S} \mathbb{Q}) \\ \downarrow && \downarrow \\ A &\hookrightarrow& \tilde A }$

Here $\tilde A$ 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 $A$ into a divisible abelian group, hence into an injective $\mathbb{Z}$-module.

###### Proposition

Assuming the axiom of choice, for $R$ a ring, the category $R$Mod has enough injectives.

The proof uses the following lemma.

Write $U\colon R Mod \to Ab$ for the forgetful functor that forgets the $R$-module structure on a module $N$ and just remembers the underlying abelian group $U(N)$.

###### Lemma

The functor $U\colon R Mod \to Ab$ has a right adjoint

$R_* : Ab \to R Mod$

given by sending an abelian group $A$ to the abelian group

$U(R_*(A)) \coloneqq Ab(U(R),A)$

equipped with the $R$-module struture by which for $r \in R$ an element $(U(R) \stackrel{f}{\to} A) \in U(R_*(A))$ is sent to the element $r f$ given by

$r f : r' \mapsto f(r' \cdot r) \,.$

This is called the coextension of scalars along the ring homomorphism $\mathbb{Z} \to R$.

The unit of the $(U \dashv R_*)$ adjunction

$\epsilon_N : N \to R_*(U(N))$

is the $R$-module homomorphism

$\epsilon_N : N \to Hom_{Ab}(U(R), U(N))$

given on $n \in N$ by

$j(n) : r \mapsto r n \,.$
###### Proof

of prop. 55

Let $N \in R Mod$. We need to find a monomorphism $N \to \tilde N$ such that $\tilde N$ is an injective $R$-module.

By prop. 54 there exists a monomorphism

$i \colon U(N) \hookrightarrow D$

of the underlying abelian group into an injective abelian group $D$.

Now consider the $(U \dashv R_*)$-adjunct

$N \to R_*(D)$

of $i$, hence the composite