nLab simplicial homotopy group

Idea

Simplicial homotopy groups are the basic invariants of simplicial sets/Kan complexes in simplicial homotopy theory.

Given that a Kan complex is a special simplicial set that behaves like a combinatorial model for a 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 homotopy groups of topological spaces. 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).

Definition

Recall the classical model structure on simplicial sets. Let $X$ be a fibrant simplicial set, i.e. a Kan complex.

Definition

For $X$ a Kan complex, then its 0th homotopy group (or set of connected components) is the set of equivalence classes of vertices modulo the equivalence relation $X_1 \stackrel{(d_1,d_0)}{\longrightarrow} X_0 \times X_0$

$$\pi_0(X) \colon X_0/X_1 \,.$$

For $x \in X_0$ a vertex and for $n \in \mathbb{N}$, $n \geq 1$, then the underlying set of the $n$th homotopy group of $X$ at $x$ – denoted $\pi_n(X,x)$ – is, the set of equivalence classes $[\alpha]$ of morphisms

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

from the simplicial $n$-simplex $\Delta^n$ to $X$, such that these take the boundary of the simplex to $x$, i.e. such that they fit into a commuting diagram in sSet of the form

$$\array{ \partial \Delta[n] & \longrightarrow & \Delta[0] \\ \downarrow && \downarrow^{\mathrlap{x}} \\ \Delta[n] &\stackrel{\alpha}{\longrightarrow}& X } \,,$$

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

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

that fixes the boundary in that it fits into a commuting diagram in sSet of the form

$$\array{ \partial \Delta[n] \times \Delta[1] & \longrightarrow & \Delta[0] \\ \downarrow && \downarrow^{\mathrlap{x}} \\ \Delta[n] \times \Delta[1] &\stackrel{\eta}{\longrightarrow}& X } \,.$$

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

Definition

For $X$ a Kan complex, for $x\in X_0$, for $n \geq 1$ and for $f,g \colon \Delta[n] \to X$ two representatives of $\pi_n(X,x)$ as in def. , consider the following $n$-simplices in $X_n$:

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

This corresponds to 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 $\Delta[n]$

$$\array{ \Lambda^{n+1}[n] & \longrightarrow & X \\ \downarrow & \nearrow_{\mathrlap{\theta}} \\ \Delta[n+1] }$$

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 a product operation on $\pi_n(X,x)$ by

$$[f]\cdot [g] \coloneqq [d_n \theta] \,.$$

(e.g. Goerss-Jardine 96, p. 26)

Remark

All the degenerate $n$-simplices $v_{0 \leq i \leq n-2}$ in def. 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 product on homotopy group elements in def. is well defined, in that it is independent of the choice of representatives $f$, $g$ and of the extension $\theta$.

e.g. (Goerss-Jardine 96, lemma 7.1)

Lemma

The product operation in def. yields a group structure on $\pi_n(X,x)$, which is abelian for $n \geq 2$.

e.g. (Goerss-Jardine 96, theorem 7.2)

Finally:

Definition

The simplicial homotopy groups of any simplicial set, not necessarily Kan, are those of any of its Kan fibrant replacements according to def. .

Remark

The first homotopy group, $\pi_1(X,x)$, is also called the fundamental group of $X$.

Properties

Relation to topological homotopy groups

The simplicial homotopy groups of a Kan complex coincide with the homotopy groups of its geometric realization, see e.g. (Goerss-Jardine 96, page 60).

Relation to homotopy equivalence

A morphism of simplicial sets which induces an isomorphism on all simplicial homotopy groups is called a weak homotopy equivalence. If it goes between Kan complexes then it is actually a homotopy equivalence.

Relation to chain homology groups of associated Moore complexes

Another way to get the group structure on the homotopy groups of a Kan complex, $X$, is via its Dwyer-Kan loop groupoid and the Moore complex. This gives a simplicially enriched groupoid $G(X)$, or if we restricted to the pointed case, and just look at the loops at the base vertex, a simplicial group. (We will assume for the sake of simplicity that $X$ is reduced, that is to say, $X_0$ is a singleton, and thus that $G(X)$ is a simplicial group.)

The construction of $G(X)$ is then given by the free group functor on the various levels, shifted by 1, and with a twist in the zeroth face map (see Dwyer-Kan loop groupoid and simplify to the reduced case.)

Proposition

There is an isomorphism between $\pi_n(X)$ as defined above and $H_{n-1}(N G(X))$, the $(n-1)$th homology group of the Moore complex of the simplicial group, $G(X)$.

Long exact sequences of a Kan fibration

For $f \colon X \longrightarrow Y$ a Kan fibration, for $x\in X_0$ any vertex, for $y \coloneqq f(x) \in Y$ its image and $F_x \coloneqq f^{-1}(y)$ the fiber at that point, then the induced homomorphism of simplicial homotopy groups form a long exact sequence of homotopy groups

$$\cdots \to \pi_{n+1}(Y,y) \stackrel{}{\longrightarrow} \pi_n(F,x) \stackrel{}{\longrightarrow} \pi_n(X,x) \stackrel{}{\longrightarrow} \pi_n(Y,y) \stackrel{}{\longrightarrow} \pi_{n-1}(F,x) \to \cdots$$

$$\cdots \to \pi_1(F,x) \longrightarrow \pi_0(F) \stackrel{}{\longrightarrow} \pi_0(X) \longrightarrow \pi_0(Y)$$

i.e. a long exact sequence of groups ending in a long exact sequence of pointed sets.

(e.g. Goerss-Jardine 96, lemma 7.3)

Examples

Lemma

Let $C$ be a groupoid and $\mathcal{N}(C)$ its nerve.

Then

• $\pi_0 \mathcal{N}(C,c)$ is the set of isomorphism classes of $C$ with the class of $c$ as base point

• $\pi_1 \mathcal{N}(C,c)$ is the automorphism group $Aut_C(c)$ of $c$

• $\pi_{n \geq 2} \mathcal{N}(C,c)$ is trivial

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

References

Textbook accounts include

• Paul Goerss, Rick Jardine, chapter I, section 7 of Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1996)

Originally homotopy groups of simplicial sets had been defined in terms of the ordinary homotopy groups of the topological spaces realizing them. Apparently the first or one of the first discussions of the purely combinatorial definition is

• Dan Kan, A combinatorial definition of homotopy groups, Annals of Mathematics Second Series, Vol. 67, No. 2 (Mar., 1958), pp. 282-312 (jstor)