simplicial homotopy group



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).


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


For XX 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(d 1,d 0)X 0×X 0X_1 \stackrel{(d_1,d_0)}{\longrightarrow} X_0 \times X_0

π 0(X):X 0/X 1. \pi_0(X) \colon X_0/X_1 \,.

For xX 0x \in X_0 a vertex and for nn \in \mathbb{N}, n1n \geq 1, then the underlying set of the nnth homotopy group of XX at xx – denoted π n(X,x)\pi_n(X,x) – is, the set of equivalence classes [α][\alpha] of morphisms

α:Δ nX \alpha \colon \Delta^n \to X

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

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

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}{\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

Δ[n]×Δ[1] Δ[0] x Δ[n]×Δ[1] η X. \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.


For XX a Kan complex, for xX 0x\in X_0, for n1n \geq 1 and for f,g:Δ[n]Xf,g \colon \Delta[n] \to X two representatives of π n(X,x)\pi_n(X,x) as in def. 1, consider 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 \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 Λ n+1[n]X\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+1[n] X θ Δ[n+1] \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 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 a product operation on π n(X,x)\pi_n(X,x) by

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

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


All the degenerate nn-simplices v 0in2v_{0 \leq i \leq n-2} in def. 2 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.


The product on homotopy group elements in def. 2 is well defined, in that it is independent of the choice of representatives ff, gg and of the extension θ\theta.

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


The product operation in def. 2 yields a group structure on π n(X,x)\pi_n(X,x), which is abelian for n2n \geq 2.

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



The simplicial homotopy groups of any simplicial set, not necessarily Kan, are those of any of its Kan fibrant replacements according to def. 1.


The first homotopy group, π 1(X,x)\pi_1(X,x), is also called the fundamental group of XX.


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, XX, is via its Dwyer-Kan loop groupoid and the Moore complex. This gives a simplicially enriched groupoid G(X)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 XX is reduced, that is to say, X 0X_0 is a singleton, and thus that G(X)G(X) is a simplicial group.)

The construction of G(X)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.)


There is an isomorphism between π n(X)\pi_n(X) as defined above and H n1(NG(X))H_{n-1}(N G(X)), the (n1)(n-1)th homology group of the Moore complex of the simplicial group, G(X)G(X).

Long exact sequences of a Kan fibration

For fcolonXYf colon X \longrightarrow Y a Kan fibration, for xX 0x\in X_0 any vertex, for yf(x)Yy \coloneqq f(x) \in Y its image and F xf 1(y)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

π n+1(Y,y)π n(F,x)π n(X,x)π n(Y,y)π n1(F,x) \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
π 1(F,x)π 0(F)π 0(X)π 0(Y) \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)



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


  • π 0𝒩(C,c)\pi_0 \mathcal{N}(C,c) is the set of isomorphism classes of CC with the class of cc as base point

  • π 1𝒩(C,c)\pi_1 \mathcal{N}(C,c) is the automorphism group Aut C(c)Aut_C(c) of cc

  • π n2𝒩(C,c)\pi_{n \geq 2} \mathcal{N}(C,c) is trivial

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 𝒩(f):𝒩(C)𝒩(D)\mathcal{N}(f) : \mathcal{N}(C) \to \mathcal{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.


Textbook accounts include

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)

Revised on March 6, 2016 06:31:41 by Anonymous Coward (