Paths and cylinders
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 be a fibrant simplicial set, i.e. a Kan complex.
For 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
For a vertex and for , , then the underlying set of the th homotopy group of at – denoted – is, the set of equivalence classes of morphisms
from the simplicial -simplex to , such that these take the boundary of the simplex to , i.e. such that they fit into a commuting diagram in sSet of the form
where two such maps are taken to be equivalent is they are related by a simplicial homotopy
that fixes the boundary in that it fits into a commuting diagram in sSet of the form
These sets are taken to be equipped with the following group structure.
For a Kan complex, for , for and for two representatives of as in def. 1, consider the following -simplices in :
This corresponds to a morphism from a horn of the -simplex into . By the Kan complex property of this morphism has an extension through the -simplex
From the simplicial identities one finds that the boundary of the -simplex arising as the th boundary piece of is constant on
So represents an element in and we define a product operation on by
(e.g. Goerss-Jardine 96, p. 26)
The product on homotopy group elements in def. 2 is well defined, in that it is independent of the choice of representatives , and of the extension .
e.g. (Goerss-Jardine 96, lemma 7.1)
The product operation in def. 2 yields a group structure on , which is abelian for .
e.g. (Goerss-Jardine 96, theorem 7.2)
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, , is via its Dwyer-Kan loop groupoid and the Moore complex. This gives a simplicially enriched groupoid , 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 is reduced, that is to say, is a singleton, and thus that is a simplicial group.)
The construction of 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 as defined above and , the th homology group of the Moore complex of the simplicial group, .
Long exact sequences of a Kan fibration
For a Kan fibration, for any vertex, for its image and the fiber at that point, then the induced homomorphism of simplicial homotopy groups form a long exact sequence of homotopy groups
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 be a groupoid and its nerve.
is the set of isomorphism classes of with the class of as base point
is the automorphism group of
In particular a functor of groupoids is a equivalence of categories if under the nerve it induces a weak equivalence of Kan complexes:
- that is an isomorphism implies that is an essentially surjective functor and is implied by 's being a full functor;
- that is an isomorphism is equivalent to '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)