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infinity-groupoids

simplicial sets and Kan complexes

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simplicial sets and Kan complexes

Following our general outline in motivation for sheaves, cohomology and higher stacks we want to look at sheaves with values not in Set but with values in nice topological spaces that can be modeled as infinity-groupoids.

A general way to handle infinity-groupoids is in terms of simplicial sets.

Recall the following definitions from our previous discussion:

the simplex category is the full subcategory of Cat on linear quivers, i.e. on categories freely generated on finite, non-empty linear graphs:

[n]:={012n}. [n] := \{0 \to 1 \to 2 \to \cdots \to n\} \,.

The collection of all functors between linear quivers

{01n}{01m} \{ 0 \to 1 \to \cdots \to n \} \to \{ 0 \to 1 \to \cdots \to m \}

is generated from those that map almost all generating morphisms kk+1k \to k+1 to another generating morphism, except at one position, where they

  • coface maps: δ i:=δ i n:[n1][n]\delta_i := \delta_i^n : [n-1] \hookrightarrow [n] is the injection missing i[n]i \in [n], i.e. the functor that maps a single generating morphism to the composite of two generating morphisms

    δ i n:[n1]n \delta^n_i : [n-1] \to n
    δ i n:((i1)i)((i1)i(i+1)) \delta^n_i : ((i-1) \to i) \mapsto ((i-1) \to i \to (i+1))
  • codegeneracy maps: σ i:=σ i n:[n+1][n]\sigma_i := \sigma_i^n : [n+1] \to [n] is the surjection such that σ i(i)=σ i(i+1)=i\sigma_i(i) = \sigma_i(i+1) = i, i.e. the functor that maps one generating morphism to an identity morphism

    σ i n:[n+1][n] \sigma^n_i : [n+1] \to [n]
    σ i n:(ii+1)Id i \sigma^n_i : (i \to i+1) \mapsto Id_i

These morphism generate Δ\Delta subject to the following relations, called the simplicial relations

δ j n+1δ i n=δ i n+1δ j1 n fori<j σ j nδ i n+1=σ i n1σ j+1 n forij \array{ \delta_j^{n+1} \circ \delta_i^n = \delta_i^{n+1}\circ \delta_{j-1}^n & for i \lt j \\ \sigma_j^n \circ \delta_i^{n+1} = \sigma_i^{n-1} \circ \sigma_{j+1}^n & for i \leq j }
σ j nδ i n+1={δ i nσ j1 n1 ifi<j Id n ifi=jori=j+1 δ i1 nσ j n1 ifi>j+1 \sigma_j^n \circ \delta_i^{n+1} = \left\lbrace \array{ \delta_i^n \circ \sigma_{j-1}^{n-1} & if i \lt j \\ Id_n & if i = j or i = j+1 \\ \delta^n_{i-1} \circ \sigma_{j}^{n-1} & if i \gt j +1 } \right.

A simplicial set is a presheaf on Δ\Delta.

The images of the morphisms δ\delta and σ\sigma under a given simplicial set give the face and boundary maps, that satisfy the simplicial identities

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

  • d is jd_i s_j can be written as

    • s j1d is_{j-1}d_i if i<ji \lt j,
    • idid if i=ji = j or j+1j+1,
    • s jd i1s_j d_{i-1} if i>j+1i \gt j+1,
  • s is j=s js i1s_i s_j = s_j s_{i-1} if i>ji \gt j.

We will get further intuition on the meaning and inner workins of simplicial sets from the following constructions and examples.

Recall also that, being a category of presheaves, there is canonically the cartesian closed monoidal structure on presheaves on SSet whose product is the objectwise product of sets:

(X×Y) n:=(X×Y)([n])=X(n)×Y(n). (X \times Y)_n := (X \times Y)([n]) = X(n) \times Y(n) \,.

nerve of a category

Every category gives rise to a simplicial set: its nerve.

Definition

Recall that the simplex category Δ\Delta is equivalent to the full subcategory

i:ΔCat i : \Delta \hookrightarrow Cat

of Cat on linear quivers, meaning that the object [n]Obj(Δ)[n] \in Obj(\Delta) can be identified with the category [n]={012n}[n] = \{0 \to 1 \to 2 \to \cdots \to n\}. The morphisms of Δ\Delta are all functors between these “linear quiver” categories.

For DD any locally small category, the nerve N(D)N(D) of DD is the simplicial set given by

N(D):Δ opCatCat(,D)Set, N(D) : \Delta^{op} \hookrightarrow Cat \stackrel{Cat(-,D)}{\to} Set \,,

where Cat is regarded as an ordinary 1-category with objects locally small categories, and morphisms being functors between these.

So the set N(D) nN(D)_n of nn-simplices of the nerve is the set of functors {01n}D\{0 \to 1 \to \cdots \to n\} \to D. This is clearly the same as the set of sequences of composable morphisms in DD of length nn:

N(D) n=Mor(D) t× sMor(D) t× s t× sMor(D) nfactors N(D)_n = \underbrace{ Mor(D) {}_t \times_s Mor(D) {}_t \times_s \cdots {}_t \times_s Mor(D)}_{n factors}

It follows that, for instance

  • for (d 0f 1d 1,d 1f 2d 2,d 2f 3d 3)N(D) 3(d_0 \stackrel{f_1}{\to} d_1, d_1 \stackrel{f_2}{\to} d_2, d_2 \stackrel{f_3}{\to} d_3) \in N(D)_3 the image under d 1:=N(D)(δ 1):N(D) 3N(D) 2d_1 := N(D)(\delta_1) : N(D)_3 \to N(D)_2 is obtained by composing the first two morphisms in this sequence: (d 0f 2f 1d 2,d 2f 3d 3)N(D) 2(d_0 \stackrel{f_2 \circ f_1}{\to} d_2, d_2 \stackrel{f_3}{\to} d_3) \in N(D)_2

  • for (d 0f 1d 1)N(D) 1(d_0 \stackrel{f_1}{\to} d_1) \in N(D)_1 the image under s 1:=N(D)(σ 1):N(D) 1N(D) 2s_1 := N(D)(\sigma_1) : N(D)_1 \to N(D)_2 is obtained by inserting an identity morphism: (d 0f 1d 1,d 1Id d 1d 1)N(D) 2(d_0 \stackrel{f_1}{\to} d_1, d_1 \stackrel{Id_{d_1}}{\to} d_1) \in N(D)_2.

In this way, generally the face and degeneracy maps of the nerve of a category come from composition of morphisms and from inserting identity morphisms.

In particular in light of their generalization to nerves of higher categories, discussed below, the cells in the nerve N(D)N(D) have the following interpretation:

  • S 0={d|dObj(D)}S_0 = \{d | d \in Obj(D)\} is the collection of objects of DD;

  • S 1=Mor(D)={dfd|fMor(D)}S_1 = Mor(D) = \{d \stackrel{f}{\to} d' | f \in Mor(D)\} is the collection of morphisms of DD;

  • S 2={ d 1 f 1 ! f 2 d 0 f 2f 1 d 2|(f 1,f 2)Mor(D) t× sMor(D)}S_2 = \left\{ \left. \array{ && d_1 \\ & {}^{f_1}\nearrow &\Downarrow^{\exists !}& \searrow^{f_2} \\ d_0 &&\stackrel{f_2 \circ f_1}{\to}&& d_2 } \right| (f_1, f_2) \in Mor(D) {}_t \times_s Mor(D) \right\} is the collection of composable morphisms in DD: the 2-cell itself is to be read as the composition operation, which is unique for an ordinary category (there is just one way to compose to morphisms);

  • S 3={d 1 f 2 d 2 f 1 f 2f 1 f 3 d 0 f 3(f 2f 1) d 3!d 1 f 2 d 2 f 1 f 3f 2 f 3 d 0 (f 3f 2)f 1 d 3|(f 3,f 2,f 1)Mor(D) t× sMor(D) t× sMor(D)}S_3 = \left\{ \left. \array{ d_1 &\stackrel{f_2}{\to}& d_2 \\ {}^{f_1}\uparrow & {}^{f_2 \circ f_1}\nearrow & \downarrow^{f_3} \\ d_0 &\stackrel{f_3\circ (f_2\circ f_1)}{\to}& d_3 } \;\;\;\;\;\stackrel{\exists !}{\Rightarrow} \;\;\;\;\; \array{ d_1 &\stackrel{f_2}{\to}& d_2 \\ {}^{f_1}\uparrow & \searrow^{f_3\circ f_2} & \downarrow^{f_3} \\ d_0 &\stackrel{(f_3\circ f_2) \circ f_1}{\to}& d_3 } \right| (f_3,f_2, f_1) \in Mor(D) {}_t \times_s Mor(D) {}_t \times_s Mor(D) \right\} is the collection of triples of composable morphisms, to be read as the unique associators that relate one way to compose three morphisms using the above 2-cells to the other way.

Proposition

A simplicial set SS is the nerve of a locally small category CC if and only if all the commuting squares

S n+m d 0d 0 S m d n+m1d n+m S n d 0d 0 S 0 \array{ S_{n+m} &\stackrel{\cdots \circ d_0 \circ d_0}{\to}& S_m \\ {}^{\cdots d_{n+m-1}\circ d_{n+m}}\downarrow && \downarrow \\ S_n &\stackrel{d_0 \circ \cdots d_0}{\to}& S_0 }

are pullback diagrams.

Unwrapping this definition inductively in (n+m)(n+m), this says that a simplicial set is the nerve of a category if and only if all its cells in degree 2\geq 2 are unique compositors, associators, pentagonators, etc of composition of 1-morphisms. No non-trivial such structure cells appear and no further higher cells appear.

This characterization of categories in terms of nerves directly leads to the model of (infinity,1)-category in terms of complete Segal spaces by replacing in the above discussion sets by topological spaces (or something similar, like Kan complexes) and pullbacks by homotopy pullbacks.

Proposition

The nerve functor

N:CatSSet N : Cat \to SSet

is a full and faithful functor.

So functors between locally small categories are in bijections with morphisms of simplicial sets between their nerves.

examples

  • bar construction Let AA be a monoid (for instance a group) and write BA\mathbf{B} A for the corresponding one-object category with Mor(BA)=AMor(\mathbf{B} A) = A. Then the nerve N(BA)N(\mathbf{B} A) of BA\mathbf{B}A is the simplicial set which is the usual bar construction of AA
    N(BA)=(A×A×AA×AA*) N(\mathbf{B}A) = \left( \cdots A \times A \times A \stackrel{\to}{\stackrel{\to}{\to}} A \times A \stackrel{\to}{\to} A \to {*} \right)

    In particular, when A=GA = G is a discrete group, then the geometric realization |N(BG)||N(\mathbf{B} G)| of the nerve of BG\mathbf{B}G is the classifying topological space BG \cdots \simeq B G for GG-principal bundles.

simplex boundaries and horns

We want to understand special properties of simplicial sets that arise as nerves of groupoids. In order to do so we need the concepts of boundaries and horns of simplices, and notion of Kan complex derived from that.

In as far as the simplicial nn-simplex Δ n\Delta^n (a simplicial set) is a combinatorial model for the nn-ball, its boundary Δ n\partial \Delta^n is a combinatorial model for the (n1)(n-1)-sphere.

Definition

The boundary Δ n\partial \Delta^n of the simplicial nn-simplex Δ n\Delta^n is the simplicial set generated from the simplicial set Δ n\Delta^n minus its unique non-degenerate cell in dimension nn.

Regarding Δ n\Delta^n as the presheaf on on the simplex category that is represented by [n]Obj(Δ)[n] \in Obj(\Delta), then this means that Δ n\partial \Delta^n is the simplicial set generated from Δ\Delta minus the identity morphism Id [n]Id_{[n]}.

There is a canonical monomorphism

i n:Δ nΔ n, i_n : \partial \Delta^n \hookrightarrow \Delta^n \,,

the boundary inclusion .

The geometric realization of this is the inclusion of the (n1)(n-1)-sphere as the boundary of the nn-disk.

Simplicial boundary inclusions are one part of the cofibrant generation of the classical model structure on simplicial sets.

For low nn the boundaries of nn-simplices look like (see also the illustrations at oriental)

  • Δ 0=\partial \Delta^0 = \emptyset;

  • Δ 1={01}={0,1}=Δ 0Δ 0\partial \Delta^1 = \partial\{0 \to 1\} = \{0, 1\} = \Delta^0 \sqcup \Delta^0;

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

The horn Λ k[n]=Λ k nΔ n\Lambda_k[n] = \Lambda^n_k \hookrightarrow \Delta^n is the simplicial set obtained from the boundary of the n-simplex Δ n\partial \Delta^n of the standard simplicial nn-simplex Δ n\Delta^n by discarding the kkth face.

Definition

Let

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

be the standard simplicial nn-simplex in SimpSet.

Then, for each ii, 0in0 \leq i \leq n, we can form, within Δ[n]\Delta[n] , a subsimplicial set, Λ i[n]\Lambda^i[n], called the (n,i)(n,i)-horn or (n,i)(n,i)-box, by discarding the top dimensional non-degenerate nn-simplex (given by the identity map on [n][n]) and its i thi^{th} face. We must also discard all the degeneracies of those simplices.

The horn Λ k[n]\Lambda^k[n] is an outer horn if k=0k = 0 or k=nk = n.

Examples

The inner horn of the 2-simplex

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

with boundary

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

looks like

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

The two outer horns look like

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

and

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

respectively.

Kan fibrations

A Kan fibration is a morphism π:YX\pi : Y \to X of simplicial sets with the lifting property for all horn inclusions.

This means that for

Λ k[n] Y π Δ n X \array{ \Lambda^k[n] &\to& Y \\ \downarrow && \downarrow^\pi \\ \Delta^n &\to& X }

a commuting square, there always exists a lift

Λ k[n] Y π Δ n X. \array{ \Lambda^k[n] &\to& Y \\ \downarrow &\nearrow& \downarrow^\pi \\ \Delta^n &\to& X } \,.

Kan fibrations are combinatorial analogs of Serre fibrations of topological spaces. In fact, under the Quillen equivalence of the standard model structure on topological spaces and the standard model structure on simplicial sets, Kan fibrations map to Serre fibrations.

Recall the shape of the horns in low dimension.

  • -n=1n=1- The horns Λ 0 1\Lambda^1_0 and Λ 1 1\Lambda^1_1 of the 1-simplex are just copies of the 0-simplex Δ 0\Delta^0 regarded as the left and right endpoint of Δ 1\Delta^1. For n=1n= 1 the above condition says that for π:YX\pi : Y \to X a Kan fibration we have

    Y y π X π(y) f xY y f^ x^ π X π(y) f=π(f^) x=π(x) \array{ Y &\ni & y \\ \downarrow^\pi \\ X &\ni& \pi(y) &\stackrel{\forall f}{\to}& x } \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; \array{ Y &\ni& y &\stackrel{\exists \hat f}{\to}& \exists \hat x \\ \downarrow^\pi \\ X &\ni& \pi(y) &\stackrel{f = \pi(\hat f)}{\to}& x = \pi(x) }

    corresponding to the lifting diagram

    Λ 1 1 y Y f^ π Δ 1 f X. \array{ \Lambda_1^1 &\stackrel{y}{\to}& Y \\ \downarrow &{}^{\hat f}\nearrow& \downarrow^\pi \\ \Delta^1 &\stackrel{f}{\to}& X } \,.
  • -n=2n=2- the horn Λ 1 2\Lambda^2_1 consists of the two top sides of a triangle. For this the Kan condition says that for any two composable 1-cells in YY that have a “composite up to a 2-cell” in XX, there exists a corresponding “composite up to a 2-cell” in YY that projects down to the one in XX:

    y 2 Y y 1 y 3 π X π(y 2) h π(y 1) π(y 2) y 2 h^ Y y 1 y 3 π X π(y 2) h=π(h^) π(y 1) π(y 2) \array{ &&&&& y_2 \\ &&&& \nearrow && \searrow \\ Y &\ni& & y_1 &&&& y_3 \\ \downarrow^\pi \\ X &\ni& &&& \pi(y_2) \\ &&& & \nearrow &\Downarrow^{\forall h}& \searrow \\ &&& \pi(y_1) &&\to&& \pi(y_2) } \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; \array{ &&&&& y_2 \\ &&&& \nearrow &\Downarrow^{\exists \hat h}& \searrow \\ Y &\ni& & y_1 &&\stackrel{\exists}{\to}&& y_3 \\ \downarrow^\pi \\ X &\ni& &&& \pi(y_2) \\ &&& & \nearrow &\Downarrow^{h = \pi(\hat h)}& \searrow \\ &&& \pi(y_1) &&\to&& \pi(y_2) }

    This corresponds to the lifting diagram

    Λ 2 1 y Y h^ π Δ 2 h X. \array{ \Lambda_2^1 &\stackrel{y}{\to}& Y \\ \downarrow &{}^{\hat h}\nearrow& \downarrow^\pi \\ \Delta^2 &\stackrel{h}{\to}& X } \,.
    • Crucial is this condition for the outer horns Λ 0 n\Lambda^n_0 and Λ n n\Lambda^n_n, where it says that the above works not only when edges are composable, but also when they mit with their sources or their targets. For

instance for the horn Λ 2 2\Lambda^2_2 the picture is $ y 2 Y y 1 y 3 π X π(y 2) h π(y 1) π(y 2) y 2 h^ Y y 1 y 3 π X π(y 2) h=π(h^) π(y 1) π(y 2) \array{ &&&&& y_2 \\ &&&& && \searrow \\ Y &\ni& & y_1 &&\to&& y_3 \\ \downarrow^\pi \\ X &\ni& &&& \pi(y_2) \\ &&& & \nearrow &\Downarrow^{\forall h}& \searrow \\ &&& \pi(y_1) &&\to&& \pi(y_2) } \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; \array{ &&&&& y_2 \\ &&&& {}^\exists\nearrow &\Downarrow^{\exists \hat h} & \searrow \\ Y &\ni& & y_1 &&\stackrel{\exists}{\to}&& y_3 \\ \downarrow^\pi \\ X &\ni& &&& \pi(y_2) \\ &&& & \nearrow &\Downarrow^{h = \pi(\hat h)}& \searrow \\ &&& \pi(y_1) &&\to&& \pi(y_2) } Thiscorrespondstotheliftingdiagram This corresponds to the lifting diagram Λ 2 2 y Y h^ π Δ 2 h X. \array{ \Lambda_2^2 &\stackrel{y}{\to}& Y \\ \downarrow &{}^{\hat h}\nearrow& \downarrow^\pi \\ \Delta^2 &\stackrel{h}{\to}& X } \,. $

Kan complexes

A Kan complex is a geometric model of an \infty-groupoid based on the shape modeled by the simplex category.

Definition

A Kan complex is a simplicial set SS that satisfies the Kan condition,

  • which says that all horns of the simplicial set have fillers,

  • which means equivalently that the unique morphism SptS \to pt from SS to the point is a Kan fibration,

  • which means equivalently that for all diagrams

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

    there exists a diagonal morphism

    Λ i[n] S Δ[n] ptΛ i[n] S Δ[n]. \array{ \Lambda^i[n] &\to& S \\ \downarrow &\nearrow& \downarrow \\ \Delta[n] &\to& pt } \;\;\; \leftrightarrow \;\;\; \array{ \Lambda^i[n] &\to& S \\ \downarrow &\nearrow& \\ \Delta[n] } \,.
  • This in turn means equivalently that the map from nn-simplices to (n,i)(n,i)-horns is an epimorphism

    [Δ[n],S]>[Λ i[n],S]. [\Delta[n], S] \to\gt [\Lambda^i[n],S] \,.

Remarks

  • Kan complexes are among the most convenient and popular models for infinity-groupoids. The horn filling condition from this point of view is read as guaranteeing that

    • for all collection of (n1)(n-1) composable nn-cells (a horn Λ k[n]\Lambda^k[n]) there exists an nn-cell – their composite – and an (n1)(n-1)-cell connecting the original (n1)(n-1) nn-cells with their composite. Depending on kk, this interpretation in terms of composition implies that one thinks of all cells as being reversible. Therefore this models an infinity-groupoid.
  • Whatever other definition of infinity-groupoid one considers, it is expected to map to a Kan complex under the nerve.

  • A slight weakening of the Kan condition, the weak Kan condition leads to the definition of quasi-category.

Examples of Kan complexes

Proposition

The nerve N(C)N(C) of a locally small category is a Kan complex if and only if CC is a groupoid.

The existence of inverse morphisms in DD corresponds to the fact that in the Kan complex N(D)N(D) the “outer” horns

d 0 f d 1 Id d 1 d 1and d 1 f d 0 Id d 0 d 0 \array{ && d_0 \\ & && \searrow^{f} \\ d_1 &&\stackrel{Id_{d_1}}{\to} && d_1 } \;\;\; \;\;\; and \;\;\; \;\;\; \array{ && d_1 \\ & {}^f\nearrow && \\ d_0 &&\stackrel{Id_{d_0}}{\to} && d_0 }

have fillers

d 0 f 1 f d 1 Id d 1 d 1and d 1 f f 1 d 0 Id d 0 d 0 \array{ && d_0 \\ & {}^{f^{-1}}\nearrow&& \searrow^{f} \\ d_1 &&\stackrel{Id_{d_1}}{\to} && d_1 } \;\;\; \;\;\; and \;\;\; \;\;\; \array{ && d_1 \\ & {}^f\nearrow && \searrow^{f^{-1}} \\ d_0 &&\stackrel{Id_{d_0}}{\to} && d_0 }

(even unique fillers, due to the above).

It is in this sense that a simplicial set that is a Kan complex but which does not necessarily have the above pullback property that makes it a nerve of an ordinary groupoid models an infinity-groupoid.

fundamental \infty-groupoid of a topological space

For XX a topological space, its singular simplicial complex is the simplicial set Π(X)\Pi(X) (often denoted S(X)S(X)) whose set of nn-simplices is the hom-set

Π(X) n:=Top(Δ Top n,X) \Pi(X)_n := Top(\Delta^n_{Top}, X)

in Top of continuous maps from the standard topological nn-simplex Δ Top n\Delta^n_{Top} into XX.

Using the fact that the Δ Top n\Delta^n_{Top} arrange themselves into a cosimplicial space

Δ Top:ΔTop \Delta_{Top} : \Delta \to Top

in the obvious way, the (Π(X) n)(\Pi(X)_n) become a simplicial set in the corresponding obvious way. For instance the face maps are induced by restricting maps to XX along the face inclusions δ i:Δ n1Δ n\delta^i : \Delta^{n-1} \hookrightarrow \Delta^n.

That Π(X)\Pi(X) is indeed a Kan complex is intuitively clear. Technically it follows from the fact that the inclusions Λ n Top kΔ Top n{{\Lambda^n}_{Top}}_k \hookrightarrow \Delta^n_{Top} of topological horns into tolological simplices are retracts.

The infinity-groupoid represented by the Kan complex Π(X)\Pi(X) is the fundamental infinity-groupoid of XX.

simplicial homotopies and homotopy groups

A simplicial homotopy is a homotopy in the classical model structure on simplicial sets.

Definition

SSet has a cylinder functor given by cartesian product with the standard 1-simplex I:=Δ 1I := \Delta^1.

Therefore for f,g:XYf,g : X \to Y two morphisms of simplicial sets, a homotopy η:fg\eta : f \Rightarrow g is a morphism η:X×Δ 1Y\eta : X \times \Delta^1 \to Y such that the diagram

XX×Δ 0 Id×δ 1 X×Δ 1 Id×δ 0 X×Δ 0X f η g Y \array{ X \simeq X\times \Delta^0 &\stackrel{Id \times \delta^1}{\to}& X \times \Delta^1 & \stackrel{Id \times \delta^0}{\leftarrow}& X \times \Delta^0 \simeq X \\ & {}_f\searrow &\downarrow^\eta& \swarrow_{g} \\ && Y }

commutes.

Lemma

Precisely if YY is a Kan complex is the relation

(fg)(simplicialhomotopyfg:XY) (f \sim g) \Leftrightarrow (\exists simplicial homotopy f \Rightarrow g : X \to Y )

an equivalence relation.

Proof

Since Kan complexes are precisely the fibrant objects with respect to the standard model structure on simplicial sets this follows from general statements about homotopy in model categories.

The following is a direct proof.

We first show that the homotopy between points x,y:Δ 0Yx,y : \Delta^0 \to Y is an equivalence relation when YY is a Kan complex.

We identify in the following xx and yy with vertices in the image of these maps.

  • -reflexivity- For every vertex xY 0x \in Y_0, the degenerate 1-simplex s 0xS 1s_0 x \in S_1 has, by the simplicial identities, 0-faces d 0s 0x=xd_0 s_0 x = x and d 1s 0x=xd_1 s_0 x = x.

    (d 1s 0x)s 0x(d 0s 0x) (d_1 s_0 x) \stackrel{s_0 x}{\to} (d_0 s_0 x)

    Therefore the morphism η:Δ 0×Δ 1Y\eta : \Delta^0 \times \Delta^1 \to Y that takes the unique non-degenerate 1-simplex in Δ 1\Delta^1 to s 0xs_0 x constitutes a homotopy from xx to itself.

  • -transitivity- let v 2:xyv_2 : x \to y and v 0:yzv_0 : y \to z in Y 1Y_1 be 1-cells. Together they determine a map from the horn Λ 1 2\Lambda^2_1 to YY,

    (v 2,v 2):Λ 1 2Y. (v_2, v_2) : \Lambda^2_1 \to Y \,.

    By the Kan complex property there is an extension θ\theta of this morphism through the 2-simplex Delta 2Delta^2

    Λ 1 2 (v 0,v 2) Y θ Δ 2. \array{ \Lambda^2_1 &\stackrel{(v_0,v_2)}{\to}& Y \\ \downarrow & \nearrow_{\theta} \\ \Delta^2 } \,.

    If we again identify θ\theta with its image (the image of its unique non-degenerate 2-cell) in Y 2Y_2, then using the simplicial identities we find

    (d 0d 2θ)=(d 1d 0θ) d 2θ θ d 0θ (d 1d 2θ)=(d 1d 1θ) d 1θ (d 0d 1θ)=(d 0d 1θ) \array{ && (d_0 d_2 \theta) = (d_1 d_0 \theta) \\ & {}^{d_2 \theta }\nearrow & \Downarrow \theta & \searrow^{d_0 \theta} \\ (d_1 d_2 \theta) = (d_1 d_1 \theta) && \stackrel{d_1 \theta}{\to} && (d_0 d_1 \theta) = (d_0 d_1 \theta) }

    that the 1-cell boundary bit d 1θd_1 \theta in turn has 0-cell boundaries

    d 0d 1θ=d 0d 0θ=z d_0 d_1 \theta = d_0 d_0 \theta = z

    and

    d 1d 1θ=d 1d 2θ=x. d_1 d_1 \theta = d_1 d_2 \theta = x \,.

    This means that d 1θd_1 \theta is a homotopy xzx \to z.

  • -symmetry- In a similar manner, suppose that v 2:xyv_2 : x \to y is a 1-cell in Y 1Y_1 that constitutes a homotopy from xx to yy. Let v 1:=s 0xv_1 := s_0 x be the degenerate 1-cell on xx. Since d 1v 1=d 1v 2d_1 v_1 = d_1 v_2 together they define a map Λ 0 2v 1,v 2Y\Lambda^2_0 \stackrel{v_1, v_2}{\to} Y which by the Kan property of YY we may extend to a map θ\theta'

    Λ 0 2 v 1,v 2 Y θ Δ 2 \array{ \Lambda^2_0 &\stackrel{v_1, v_2}{\to}& Y \\ \downarrow & \nearrow_{\theta'} \\ \Delta^2 }

    on the full 2-simplex.

    Now the 1-cell boundary d 0θd_0 \theta' has, using the simplicial identities, 0-cell boundaries

    d 0d 0θ=d 0d 1θ=x d_0 d_0 \theta' = d_0 d_1 \theta' = x

    and

    d 1d 0θ=d 0d 2θ=y d_1 d_0 \theta' = d_0 d_2 \theta' = y

    and hence yields a homotopy yxy \to x. So being homotopic is a symmetric relation on vertices in a Kan complex.

Finally we use the fact that SSet is a cartesian closed category to deduce from this statements about vertices the corresponding statement for all map:

a morphism f:XYf : X \to Y is the Hom-adjunct of a morphism f¯:Δ 0[X,Y]\bar f : \Delta^0 \to [X,Y], and a homotopy η:X×Δ 1Y\eta : X \times \Delta^1 \to Y is the adjunct of a morphism η¯:Δ 1[X,Y]\bar \eta : \Delta^1 \to [X,Y]. Therefore homotopies η:fg\eta : f \Rightarrow g are in bijection with homotopies η¯:f¯g¯\bar \eta : \bar f \to \bar g.

simplicial homotopy groups

Recall that a Kan complex is a special simplicial set that behaves like a combinatorial model for a sufficiently nice topological space.

The simplicial homotopy groups of a Kan complex are accordingly the combinatorial analog of the homotopy groups of topological spaces: instead of being maps from topological spheres modulo maps from topological disks, they are maps from the boundary of a simplex modulo those from the simplex itself.

Accordingly, the definition of the discussion of simplicial homotopy groups is essentially literally the same as that of ordinary homotopy groups. One technical difference is for instance that the definition of the group structure is slightly more non-immediate for simplicial homotopy groups than for topological homotopy groups (see below).

As for ordinary homotopy groups, an nnth simplicial homotopy ‘group’ is really an nn-tuply groupal 00-groupoid. That is, for n=0n = 0, it is not a group at all but rather a pointed set; for n=1n = 1, it is a group; and for n2n \geq 2, it is an abelian group. On the other hand, we could drop the base vertex and move to the nnth simplicial homotopy ‘groupoid’, which is really an nn-groupoid.

Definition

For XX a Kan complex

  • the 00th simplicial homotopy groupoid Π 0(X)\Pi_0(X) is the discrete groupoid on the set (X 0/X 1)(X_0/X_1) of connected components of XX, i.e. on the set of equivalence classes of 0-cells under simplicial homotopy;

    • for each vertex, xX 0x \in X_0, the pointed set π(X,x)\pi(X,x) to be the set (X 0/X 1)(X_0/X_1) with point the connected component [x][x] of xX 0x \in X_0; this is actually a special case of the following definition, until the group structure.
  • for every n1n \geq 1 and xX 0x \in X_0 the nnth simplicial homotopy group of XX at xx to be the set

    • of equivalence classes of morphisms

      α:Δ nX \alpha : \Delta^n \to X

      from the simplicial nn-simplex Δ n\Delta^n to XX,

      • such that they fit into the diagram
        Δ n Δ 0 Δ n α X; \array{ \partial \Delta^n &\to& \Delta^0 \\ \downarrow && \downarrow \\ \Delta^n &\stackrel{\alpha}{\to}& X };

        meaning that all of the boundary of Δ n\Delta^n maps to the single point xx;

    • where two such maps α,α\alpha, \alpha' are taken to be equivalent is they are related by a simplicial homotopy η\eta

      Δ n i 0 α Δ n×Δ 1 η X i 1 α Δ n \array{ \Delta^n \\ \downarrow^{i_0} & \searrow^{\alpha} \\ \Delta^n \times \Delta^1 &\stackrel{\eta}{\to}& X \\ \uparrow^{i_1} & \nearrow_{\alpha'} \\ \Delta^n }
    • that fixes the boundary

      Δ n×Δ 1 Δ 0 Δ n×Δ 1 η X. \array{ \partial \Delta^n \times \Delta^1 &\to& \Delta^0 \\ \downarrow && \downarrow \\ \Delta^n \times \Delta^1 &\stackrel{\eta}{\to}& X } \,.

These pointed sets are taken to be equipped with the following group structure.

Definition (group structure on π n(X,x)\pi_n(X,x))

Let n1n \geq 1. For f,g:Δ nXf,g : \Delta^n \to X two representatives of π n(X,x)\pi_n(X,x), define the following nn-simplices in X nX_n:

v i={s 0s 0s 0(x) for0in2 f fori=n1 g fori=n+1 v_i = \left\{ \array{ s_0 \circ s_0 \circ \cdots \circ s_0 (x) & for 0 \leq i \leq n-2 \\ f & for i = n-1 \\ g & for i = n+1 } \right.

This is designed such that it yields a morphism Λ n n+1X\Lambda^{n+1}_n \to X from a horn of the (n+1)(n+1)-simplex into XX. By the Kan complex property of XX this morphism has an extension θ\theta through the (n+1)(n+1)-simplex Δ n\Delta^n

Λ n n+1 X θ Δ n+1 \array{ \Lambda^{n+1}_n &\to& X \\ \downarrow & \nearrow_{\theta} \\ \Delta^{n+1} }

From the simplicial identities one finds that the boundary of the nn-simplex arising as the nnth boundary piece d nθd_n \theta of θ\theta is constant on xx

d id nθ=d n1d iθ=x d_i d_{n} \theta = d_{n-1} d_i \theta = x

So d nθd_n \theta represents an element in π n(X,x)\pi_n(X,x) and we define the product operation by

[f][g]:=[d nθ]. [f]\cdot [g] := [d_n \theta] \,.

Remark: All the degenerate nn-simplices v 0in2v_{0 \leq i \leq n-2} above are just there so that the gluing of the two nn-cells ff and gg to each other can be regarded as forming the boundary of an (n+1)(n+1)-simplex except for one face. BY the Kan extension property that missing face exists, namely d nθd_n \theta. This is a choice of gluing composite of ff with gg.

Lemma

The above product on homotopy group elements is indeed well defined, in that it is independent of the choice of representatives ff, gg and of the extension θ\theta.

Lemma

For n2n \geq 2 all the groups π n(X,x)\pi_n(X,x) are abelian.

weak homotopy equivalences of Kan simplicial sets

For XX and YY fibrant simplicial sets, i.e. Kan complexes, a morphism f:XYf : X \to Y is a weak equivalence with respect to the classical model structure on simplicial sets if

f *:π 0(X)π 0(Y) f_* : \pi_0(X) \to \pi_0(Y)

and

f *:π n(X,x)π n(Y,f(x)) f_* : \pi_n(X,x) \to \pi_n(Y,f(x))

are isomorphisms for all choices of base vertex xX 0x \in X_0.

Lemma

Let CC be a groupoid and N(C)N(C) its nerve.

Then

  • π 0N(C)=setofisomorphismclassesofC\pi_0 N(C) = set of isomorphism classes of C

  • π 1N(C,c)=automorphismgroupAut C(c)ofc\pi_1 N(C,c) = automorphism group Aut_C(c) of c

  • π n2N(C,c)=0\pi_{n \geq 2} N(C,c) = 0

In particular a functor f:CDf : C \to D of groupoids is a equivalence of categories if under the nerve it induces a weak equivalence Nf:N(C)N(D)N f : N(C) \to N(D) of Kan complexes:

  • that π 0𝒩(f,c):π 0(C,c)π 0(D,f(c))\pi_0 \mathcal{N}(f,c) : \pi_0(C,c) \to \pi_0(D,f(c)) is an isomorphism implies that ff is an essentially surjective functor and is implied by ff's being a full functor;
  • that π 1𝒩(f,c):π 1(C,c)π 1(D,f(c))\pi_1 \mathcal{N}(f,c) : \pi_1(C,c) \to \pi_1(D,f(c)) is an isomorphism is equivalent to ff's being a full and faithful functor.
Theorem

The morphisms f:XYf : X \to Y of Kan complexes that are both Kan fibrations as well as weak equivalences in that they induce isomorphisms on all simplicial homotopy groups (i.e. the acyclic fibrations of Kan complexes) are precisely the morphisms that have the right lifting property with respect to all boundary inclusions Δ nΔ n\partial \Delta^n \hookrightarrow \Delta^n:

Δ[n] X f Δ[n] Y. \array{ \partial \Delta[n] &\to& X \\ \downarrow &{}^\exists\nearrow& \downarrow^f \\ \Delta[n] &\to& Y } \,.
Proof

A proof is in chapter I of

  • Goerss-Jardine, Simplicial homotopy theory. Explicitly, it is theorem 7.10 here.

Let C,DC, D be ordinary groupoids and N(C)N(C), N(D)N(D) their ordinary nerves. We’d like to show in detail that

Theorem

A functor F:CDF : C \to D is

Proof

We know that both N(C)N(C) and N(D)N(D) are Kan complexes. By the above theorem it suffices to show that N(f)N(f) being a surjective equivalence is the same as having all lifts

Δ[n] N(C) N(F) Δ[n] N(D). \array{ \partial \Delta[n] &\to& N(C) \\ \downarrow &{}^\exists\nearrow& \downarrow^{N(F)} \\ \Delta[n] &\to& N(D) } \,.

We check successively what this means for increasing nn:

  • n=0n= 0. In degree 0 the boundary inclusion is that of the empty set into the point *\emptyset \hookrightarrow {*}. The lifting property in this case amounts to saying that every point in N(D)N(D) lifts through N(F)N(F).

    N(C) N(F) * N(D) N(C) N(F) * N(D). \array{ \emptyset &\to& N(C) \\ \downarrow &{}^\exists\nearrow& \downarrow^{N(F)} \\ {*} &\to& N(D) } \Leftrightarrow \array{ && N(C) \\ &{}^\exists\nearrow& \downarrow^{N(F)} \\ {*} &\to& N(D) } \,.

    This precisely says that N(F)N(F) is surjective on 0-cells and hence that FF is surjective on objects.

  • n=1n=1. In degree 1 the boundary inclusion is that of a pair of points as the endpoints of the interval {,}{}\{\circ, \bullet\} \hookrightarrow \{\circ \to \bullet\}. The lifting property here evidently is equivalent to saying that for all objects a,bObj(C)a,b \in Obj(C) all elements in Hom(F(a),F(b))Hom(F(a),F(b)) are hit. Hence that FF is a full functor.

  • n=2n=2. In degree 2 the boundary inclusion is that of the triangle as the boundary of a filled triangle. It is sufficient to restrict attention to the case that the map Δ[2]N(C)\partial \Delta[2] \to N(C) sends the top left edge of the triangle to an identity. Then the lifting property here evidently is equivalent to saying that for all objects a,bObj(C)a,b \in Obj(C) the map F a,b:Hom(a,b)Hom(F(a),F(b))F_{a,b} : Hom(a,b) \to Hom(F(a),F(b)) is injective. Hence that FF is a faithful functor.

    ( a Id a f a g b)N(F)( a Id a = F(f) a F(g) b) \left( \array{ && a \\ & {}^{Id_a}\nearrow && \searrow^{f} \\ a &&\stackrel{g}{\to}&& b } \right) \stackrel{N(F)}{\mapsto} \left( \array{ && a \\ & {}^{Id_a}\nearrow &\Downarrow^=& \searrow^{F(f)} \\ a &&\stackrel{F(g)}{\to}&& b } \right)

Kan complexes form a Brown category of fibrant objects

The 1-category of Kan complexes equipped with the information of which morphisms are fibrations (namely Kan fibrations) and which are weak equivalences (namely those inducing isomorphisms on simplicial homotopy groups) forms a flavor of homotopical category which is a

Definition

A category of fibrant objects CC is a category with weak equivalences equipped with a further subcollection of its collection of morphisms called fibrations. (As usual, those morphisms which are both weak equivalences and fibrations are called acyclic fibrations.)

This data has to satisfy the following properties:

  • CC has finite products;

  • CC has a terminal object;

  • fibrations are preserved under pullback;

  • acyclic fibrations are preserved under pullback;

  • for every object XX there exists a path object X IX^I, namely a diagram

    (XId×IdX×X)=(XσX I(d 1×d 1)X×X) (X \stackrel{Id \times Id}{\to} X \times X) = (X \stackrel{\sigma}{\to} X^I \stackrel{(d_1 \times d_1)}{\to} X\times X)

    with σ\sigma a weak equivalence and (d 0,d 1)(d_0,d_1) a fibration;

  • all objects are fibrant, i.e. all morphisms to the terminal object are fibrations.

The path object of any XX can be chosen to be the internal hom

X I=[Δ 1,X]. X^I = [\Delta^1, X] \,.

The stability of fibrations and acyclic fibrations follows from the above fact that both are characterized by a right lifting property, by the following standard lemma

Lemma

A class RR morphisms satisfying a right lifting property with respect to some class LL of morphisms is preserved under pullback.

Proof

Let p:XYp : X \to Y be in RR and and let

Z× fX X f *p p Z f Y \array{ Z \times_f X &\to& X \\ \downarrow^{f^* p} && \downarrow^p \\ Z &\stackrel{f}{\to} & Y }

be a pullback diagram. We need to show that f *pf^* p has the right lifting property with respect to all i:ABi : A \to B in LL. So let

A Z× fX i f *p B g Z \array{ A &\to& Z \times_f X \\ \downarrow^i && \downarrow^{f^* p} \\ B &\stackrel{g}{\to}& Z }

be any commuting square. We need to construct a diagonal lift of that square. To that end, first compose with the pullback square from above to obtain the commuting diagram

A Z× fX X i f *p p B g Z f Y. \array{ A &\to& Z \times_f X &\to& X \\ \downarrow^i && \downarrow^{f^* p} && \downarrow^p \\ B &\stackrel{g}{\to}& Z &\stackrel{f}{\to}& Y } \,.

By the right lifting property of pp, there is a diagonal lift of the total outer diagram

A X i (fg)^ p B fg Y. \array{ A &\to& X \\ \downarrow^i &{}^{\hat {(f g)}}\nearrow& \downarrow^p \\ B &\stackrel{f g}{\to}& Y } \,.

By the pullback property this gives rise to the lift g^\hat g in

Z× fX X g^ f *p p B g Z f Y. \array{ && Z \times_f X &\to& X \\ &{}^{\hat g} \nearrow& \downarrow^{f^* p} && \downarrow^p \\ B &\stackrel{g}{\to}& Z &\stackrel{f}{\to}& Y } \,.

In order for g^\hat g to qualify as the intended lift of the total diagram, it remains to show that

A Z× fX i g^ B \array{ A &\to& Z \times_f X \\ \downarrow^i & {}^{\hat g}\nearrow \\ B }

commutes. To do so we notice that we obtain two pullback cones with tip AA:

  • one is given by the morphisms

    1. AZ× fXXA \to Z \times_f X \to X
    2. AiBgZA \stackrel{i}{\to} B \stackrel{g}{\to} Z

    with universal morphism into the pullback being

    • AZ× fXA \to Z \times_f X
  • the other by

    1. AiBg^Z× fXXA \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X \to X
    2. AiBgZA \stackrel{i}{\to} B \stackrel{g}{\to} Z.

    with universal morphism into the pullback being

    • AiBg^Z× fXA \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X.

The commutativity of the diagrams that we have established so far shows that the first and second morphisms here equal each other, respectively. By the fact that the universal morphism into a pullback diagram is unique this implies the required identity of morphisms.

Last revised on June 30, 2009 at 16:16:33. See the history of this page for a list of all contributions to it.