Simplicial homotopy

In algebraic topology, a simplicial homotopy is an analog of a homotopy between topological spaces for simplicial sets. Precisely,^{pg 23} if
are maps between simplicial sets, a simplicial homotopy from f to g is a map
such that the restriction of along is and the restriction along is ; see . In particular, and for all x in X.
Using the adjunction
the simplicial homotopy can also be thought of as a path in the simplicial set
A simplicial homotopy is in general not an equivalence relation. However, if is a Kan complex, then a homotopy from to is an equivalence relation. Indeed, a Kan complex is an ∞-groupoid; i.e., every morphism is invertible. Thus, if h is a homotopy from f to g, then the inverse of h is a homotopy from g to f, establishing that the relation is symmetric. The transitivity holds since a composition is possible.

Simplicial homotopy equivalence

If is a simplicial set and a Kan complex, then we form the quotient
where means are homotopic to each other. It is the set of the simplicial homotopy classes of maps from to. More generally, Quillen defines homotopy classes using the equivalence relation generated by the homotopy relation.
A map between Kan complexes is then called a simplicial homotopy equivalence if the homotopy class of it is bijective; i.e., there is some such that and.
An obvious pointed version of the above consideration also holds.

Simplicial homotopy group

Let be the pushout along the boundary and n-times. Then, as in usual algebraic topology, we define
for each pointed Kan complex X and an integer. It is the n-th simplicial homotopy group of X. For example, each class in amounts to a path-connected component of.
If is a pointed Kan complex, then the mapping space
from the base point to itself is also a Kan complex called the loop space of. It is also pointed with the base point the identity and so we can iterate:. It can be shown
as pointed Kan complexes. Thus,
Now, we have the identification for the homotopy [category of an ∞-category|homotopy category] of an ∞-category C and an endomorphism group is a group. So, is a group for. By the Eckmann-Hilton argument, is abelian for .
An analog of Whitehead's theorem holds: a map between Kan complexes is a homotopy equivalence if and only if for each choice of base points and each integer, is bijective.