Schreiber infinity-groupoids

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• We had defined categories of sheaves with values in Set as geometric embeddings into categories of presheaves and noticed that

  • these are the homotopy categories of presheaf categories with respect to hypercovers/local isomorphisms;

  • and that a morphism of sheaves is an isomorphism precisely if on each stalk it is an isomorphism of sets (i.e. a bijection of sets).

• Following our motivation for sheaves, cohomology and higher stacks we want to pass now to presheaves that assign to an object not just a set, but something that behaves like a topological space, namely an infinity-groupoid.

• Here now we describe infinity-groupoids in terms of their standard model in terms of simplicial sets that satisfy the Kan condition.

• In particular we discuss the homotopical structure on $\infty$-groupoids, identifying the morphisms between them that are weak equivalences in that they should be regarded as invertible in a homotopical sense. This involves the standard notions of

  • simplicial homotopy

  • simplicial homotopy groups

  that we discuss.

• With this in hand we can then in the next installment give a simple definition of the homotopy category of infinity-stacks:

  • this is the homotopy category of $\infty$-groupoid valued sheaves obtained by regarding a morphism between two such as a weak equivalence if it induces a weak equivalence of $\infty$-groupoids on each stalk.

• The construction and discussion of this homotopy category of $\infty$-stacks is facilitated by realizing that it comes equipped with a structure that refines that of the multiplicative systems that we have seen before, namely that of a Brown category of fibrant objects. This is a structure similar to but more lightweight than the full model category structure on simplicial sheaves which we will not need to discuss here.

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:

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

The collection of all functors between linear quivers

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

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

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

  $$\delta^n_i : [n-1] \to n$$
  $$\delta^n_i : ((i-1) \to i) \mapsto ((i-1) \to i \to (i+1))$$

• codegeneracy maps: $\sigma_i := \sigma_i^n : [n+1] \to [n]$ is the surjection such that $\sigma_i(i) = \sigma_i(i+1) = i$, i.e. the functor that maps one generating morphism to an identity morphism

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

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

$$\array{ \delta_j^{n+1} \circ \delta_i^n = \delta_i^{n+1}\circ \delta_{j-1}^n & for i \lt j \\ \sigma_j^n \circ \delta_i^{n+1} = \sigma_i^{n-1} \circ \sigma_{j+1}^n & for i \leq j }$$

$$\sigma_j^n \circ \delta_i^{n+1} = \left\lbrace \array{ \delta_i^n \circ \sigma_{j-1}^{n-1} & if i \lt j \\ Id_n & if i = j or i = j+1 \\ \delta^n_{i-1} \circ \sigma_{j}^{n-1} & if i \gt j +1 } \right.$$

A simplicial set is a presheaf on $\Delta$.

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

• $d_i d_j = d_{j-1}d_i$ if $i \lt j$,

• $d_i s_j$ can be written as

  • $s_{j-1}d_i$ if $i \lt j$,
  • $id$ if $i = j$ or $j+1$,
  • $s_j d_{i-1}$ if $i \gt j+1$,

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

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:

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

nerve of a category

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

Definition

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

$$i : \Delta \hookrightarrow Cat$$

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

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

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

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

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

$$N(D)_n = \underbrace{ Mor(D) {}_t \times_s Mor(D) {}_t \times_s \cdots {}_t \times_s Mor(D)}_{n factors}$$

It follows that, for instance

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

• for $(d_0 \stackrel{f_1}{\to} d_1) \in N(D)_1$ the image under $s_1 := N(D)(\sigma_1) : N(D)_1 \to N(D)_2$ is obtained by inserting an identity morphism: $(d_0 \stackrel{f_1}{\to} d_1, d_1 \stackrel{Id_{d_1}}{\to} d_1) \in N(D)_2$.

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 $N(D)$ have the following interpretation:

• $S_0 = \{d | d \in Obj(D)\}$ is the collection of objects of $D$;

• $S_1 = Mor(D) = \{d \stackrel{f}{\to} d' | f \in Mor(D)\}$ is the collection of morphisms of $D$;

• $S_2 = \left\{ \left. \array{ && d_1 \\ & {}^{f_1}\nearrow &\Downarrow^{\exists !}& \searrow^{f_2} \\ d_0 &&\stackrel{f_2 \circ f_1}{\to}&& d_2 } \right| (f_1, f_2) \in Mor(D) {}_t \times_s Mor(D) \right\}$ is the collection of composable morphisms in $D$: 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);

• $S_3 = \left\{ \left. \array{ d_1 &\stackrel{f_2}{\to}& d_2 \\ {}^{f_1}\uparrow & {}^{f_2 \circ f_1}\nearrow & \downarrow^{f_3} \\ d_0 &\stackrel{f_3\circ (f_2\circ f_1)}{\to}& d_3 } \;\;\;\;\;\stackrel{\exists !}{\Rightarrow} \;\;\;\;\; \array{ d_1 &\stackrel{f_2}{\to}& d_2 \\ {}^{f_1}\uparrow & \searrow^{f_3\circ f_2} & \downarrow^{f_3} \\ d_0 &\stackrel{(f_3\circ f_2) \circ f_1}{\to}& d_3 } \right| (f_3,f_2, f_1) \in Mor(D) {}_t \times_s Mor(D) {}_t \times_s Mor(D) \right\}$ 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 $S$ is the nerve of a locally small category $C$ if and only if all the commuting squares

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

are pullback diagrams.

Unwrapping this definition inductively in $(n+m)$, this says that a simplicial set is the nerve of a category if and only if all its cells in degree $\geq 2$ 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

$$N : Cat \to SSet$$

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

  In particular, when $A = G$ is a discrete group, then the geometric realization $|N(\mathbf{B} G)|$ of the nerve of $\mathbf{B}G$ is the classifying topological space $\cdots \simeq B G$ for $G$-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 $n$-simplex $\Delta^n$ (a simplicial set) is a combinatorial model for the $n$-ball, its boundary $\partial \Delta^n$ is a combinatorial model for the $(n-1)$-sphere.

Definition

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

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

There is a canonical monomorphism

$$i_n : \partial \Delta^n \hookrightarrow \Delta^n \,,$$

the boundary inclusion .

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

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

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

• $\partial \Delta^0 = \emptyset$;

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

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

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

Definition

Let

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

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

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

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

Examples

The inner horn 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.

Kan fibrations

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

This means that for

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

a commuting square, there always exists a lift

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

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.

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

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

  corresponding to the lifting diagram

  $$\array{ \Lambda_1^1 &\stackrel{y}{\to}& Y \\ \downarrow &{}^{\hat f}\nearrow& \downarrow^\pi \\ \Delta^1 &\stackrel{f}{\to}& X } \,.$$

• -$n=2$- the horn $\Lambda^2_1$ consists of the two top sides of a triangle. For this the Kan condition says that for any two composable 1-cells in $Y$ that have a “composite up to a 2-cell” in $X$, there exists a corresponding “composite up to a 2-cell” in $Y$ that projects down to the one in $X$:

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

  This corresponds to the lifting diagram

  $$\array{ \Lambda_2^1 &\stackrel{y}{\to}& Y \\ \downarrow &{}^{\hat h}\nearrow& \downarrow^\pi \\ \Delta^2 &\stackrel{h}{\to}& X } \,.$$
  • Crucial is this condition for the outer horns $\Lambda^n_0$ and $\Lambda^n_n$, 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 $\Lambda^2_2$ the picture is $$\array{ &&&&& y_2 \\ &&&& && \searrow \\ Y &\ni& & y_1 &&\to&& y_3 \\ \downarrow^\pi \\ X &\ni& &&& \pi(y_2) \\ &&& & \nearrow &\Downarrow^{\forall h}& \searrow \\ &&& \pi(y_1) &&\to&& \pi(y_2) } \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; \array{ &&&&& y_2 \\ &&&& {}^\exists\nearrow &\Downarrow^{\exists \hat h} & \searrow \\ Y &\ni& & y_1 &&\stackrel{\exists}{\to}&& y_3 \\ \downarrow^\pi \\ X &\ni& &&& \pi(y_2) \\ &&& & \nearrow &\Downarrow^{h = \pi(\hat h)}& \searrow \\ &&& \pi(y_1) &&\to&& \pi(y_2) }$$This corresponds to the lifting diagram$$\array{ \Lambda_2^2 &\stackrel{y}{\to}& Y \\ \downarrow &{}^{\hat h}\nearrow& \downarrow^\pi \\ \Delta^2 &\stackrel{h}{\to}& X } \,.$$

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 $S$ that satisfies the Kan condition,

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

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

• which means equivalently that for all diagrams

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

  there exists a diagonal morphism

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

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

  $$[\Delta[n], S] \to\gt [\Lambda^i[n],S] \,.$$

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 $(n-1)$ composable $n$-cells (a horn $\Lambda^k[n]$) there exists an $n$-cell – their composite – and an $(n-1)$-cell connecting the original $(n-1)$ $n$-cells with their composite. Depending on $k$, 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 $N(C)$ of a locally small category is a Kan complex if and only if $C$ is a groupoid.

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

$$\array{ && d_0 \\ & && \searrow^{f} \\ d_1 &&\stackrel{Id_{d_1}}{\to} && d_1 } \;\;\; \;\;\; and \;\;\; \;\;\; \array{ && d_1 \\ & {}^f\nearrow && \\ d_0 &&\stackrel{Id_{d_0}}{\to} && d_0 }$$

have fillers

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

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

$$\Pi(X)_n := Top(\Delta^n_{Top}, X)$$

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

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

$$\Delta_{Top} : \Delta \to Top$$

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

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

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

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 $I := \Delta^1$.

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

$$\array{ X \simeq X\times \Delta^0 &\stackrel{Id \times \delta^1}{\to}& X \times \Delta^1 & \stackrel{Id \times \delta^0}{\leftarrow}& X \times \Delta^0 \simeq X \\ & {}_f\searrow &\downarrow^\eta& \swarrow_{g} \\ && Y }$$

commutes.

Lemma

Precisely if $Y$ is a Kan complex is the relation

$$(f \sim g) \Leftrightarrow (\exists simplicial homotopy f \Rightarrow g : X \to Y )$$

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 $x,y : \Delta^0 \to Y$ is an equivalence relation when $Y$ is a Kan complex.

We identify in the following $x$ and $y$ with vertices in the image of these maps.

• -reflexivity- For every vertex $x \in Y_0$, the degenerate 1-simplex $s_0 x \in S_1$ has, by the simplicial identities, 0-faces $d_0 s_0 x = x$ and $d_1 s_0 x = x$.

  $$(d_1 s_0 x) \stackrel{s_0 x}{\to} (d_0 s_0 x)$$

  Therefore the morphism $\eta : \Delta^0 \times \Delta^1 \to Y$ that takes the unique non-degenerate 1-simplex in $\Delta^1$ to $s_0 x$ constitutes a homotopy from $x$ to itself.

• -transitivity- let $v_2 : x \to y$ and $v_0 : y \to z$ in $Y_1$ be 1-cells. Together they determine a map from the horn $\Lambda^2_1$ to $Y$,

  $$(v_2, v_2) : \Lambda^2_1 \to Y \,.$$

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

  $$\array{ \Lambda^2_1 &\stackrel{(v_0,v_2)}{\to}& Y \\ \downarrow & \nearrow_{\theta} \\ \Delta^2 } \,.$$

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

  $$\array{ && (d_0 d_2 \theta) = (d_1 d_0 \theta) \\ & {}^{d_2 \theta }\nearrow & \Downarrow \theta & \searrow^{d_0 \theta} \\ (d_1 d_2 \theta) = (d_1 d_1 \theta) && \stackrel{d_1 \theta}{\to} && (d_0 d_1 \theta) = (d_0 d_1 \theta) }$$

  that the 1-cell boundary bit $d_1 \theta$ in turn has 0-cell boundaries

  $$d_0 d_1 \theta = d_0 d_0 \theta = z$$

  and

  $$d_1 d_1 \theta = d_1 d_2 \theta = x \,.$$

  This means that $d_1 \theta$ is a homotopy $x \to z$.

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

  $$\array{ \Lambda^2_0 &\stackrel{v_1, v_2}{\to}& Y \\ \downarrow & \nearrow_{\theta'} \\ \Delta^2 }$$

  on the full 2-simplex.

  Now the 1-cell boundary $d_0 \theta'$ has, using the simplicial identities, 0-cell boundaries

  $$d_0 d_0 \theta' = d_0 d_1 \theta' = x$$

  and

  $$d_1 d_0 \theta' = d_0 d_2 \theta' = y$$

  and hence yields a homotopy $y \to x$. 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 $f : X \to Y$ is the Hom-adjunct of a morphism $\bar f : \Delta^0 \to [X,Y]$, and a homotopy $\eta : X \times \Delta^1 \to Y$ is the adjunct of a morphism $\bar \eta : \Delta^1 \to [X,Y]$. Therefore homotopies $\eta : f \Rightarrow g$ are in bijection with homotopies $\bar \eta : \bar f \to \bar g$.

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

Definition

For $X$ a Kan complex

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

  • for each vertex, $x \in X_0$, the pointed set $\pi(X,x)$ to be the set $(X_0/X_1)$ with point the connected component $[x]$ of $x \in X_0$; this is actually a special case of the following definition, until the group structure.

• for every $n \geq 1$ and $x \in X_0$ the $n$th simplicial homotopy group of $X$ at $x$ to be the set

  • of equivalence classes of morphisms

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

    from the simplicial $n$-simplex $\Delta^n$ to $X$,

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

      meaning that all of the boundary of $\Delta^n$ maps to the single point $x$;

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

    $$\array{ \Delta^n \\ \downarrow^{i_0} & \searrow^{\alpha} \\ \Delta^n \times \Delta^1 &\stackrel{\eta}{\to}& X \\ \uparrow^{i_1} & \nearrow_{\alpha'} \\ \Delta^n }$$

  • that fixes the boundary

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

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

Definition (group structure on $\pi_n(X,x)$)

Let $n \geq 1$. For $f,g : \Delta^n \to X$ two representatives of $\pi_n(X,x)$, define the following $n$-simplices in $X_n$:

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

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

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

From the simplicial identities one finds that the boundary of the $n$-simplex arising as the $n$th boundary piece $d_n \theta$ of $\theta$ is constant on $x$

$$d_i d_{n} \theta = d_{n-1} d_i \theta = x$$

So $d_n \theta$ represents an element in $\pi_n(X,x)$ and we define the product operation by

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

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

Lemma

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

Lemma

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

weak homotopy equivalences of Kan simplicial sets

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

$$f_* : \pi_0(X) \to \pi_0(Y)$$

and

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

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

Lemma

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

Then

• $\pi_0 N(C) = set of isomorphism classes of C$

• $\pi_1 N(C,c) = automorphism group Aut_C(c) of c$

• $\pi_{n \geq 2} N(C,c) = 0$

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

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

Theorem

The morphisms $f : X \to Y$ 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 $\partial \Delta^n \hookrightarrow \Delta^n$:

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

Proof

A proof is in chapter I of

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

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

Theorem

A functor $F : C \to D$ is

• k-surjective for all $k$ and hence a surjective equivalence of categories precisely if under the nerve $N(F) : N(C) \to N(D)$ it induces an acyclic fibration of Kan complexes;

Proof

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

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

We check successively what this means for increasing $n$:

• $n= 0$. 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 $N(D)$ lifts through $N(F)$.

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

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

• $n=1$. 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 $a,b \in Obj(C)$ all elements in $Hom(F(a),F(b))$ are hit. Hence that $F$ is a full functor.

• $n=2$. 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 $\partial \Delta[2] \to N(C)$ 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 $a,b \in Obj(C)$ the map $F_{a,b} : Hom(a,b) \to Hom(F(a),F(b))$ is injective. Hence that $F$ is a faithful functor.

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

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

• Brown category of fibrant objects.

Definition

A category of fibrant objects $C$ 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:

• $C$ has finite products;

• $C$ has a terminal object;

• fibrations are preserved under pullback;

• acyclic fibrations are preserved under pullback;

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

  $$(X \stackrel{Id \times Id}{\to} X \times X) = (X \stackrel{\sigma}{\to} X^I \stackrel{(d_1 \times d_1)}{\to} X\times X)$$

  with $\sigma$ a weak equivalence and $(d_0,d_1)$ a fibration;

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

Proposition

Kan complexes form a Brown category of fibrant objects.

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

$$X^I = [\Delta^1, X] \,.$$

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 $R$ morphisms satisfying a right lifting property with respect to some class $L$ of morphisms is preserved under pullback.

Proof

Let $p : X \to Y$ be in $R$ and and let

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

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

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

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

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

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

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

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

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

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

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

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

• one is given by the morphisms

  1. $A \to Z \times_f X \to X$
  2. $A \stackrel{i}{\to} B \stackrel{g}{\to} Z$

  with universal morphism into the pullback being

  • $A \to Z \times_f X$

• the other by

  1. $A \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X \to X$
  2. $A \stackrel{i}{\to} B \stackrel{g}{\to} Z$.

  with universal morphism into the pullback being

  • $A \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X$.

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.