related by the Dold-Kan correspondence
A simplicial homotopy is a homotopy in the classical model structure on simplicial sets. It can also be defined combinatorially; in that form one can define a homotopy 2-cell between morphisms of simplicial objects in any category .
SSet has a cylinder functor given by cartesian product with the standard 1-simplex, . (In fact, one can define simplicial cylinders, , more generally, for example for being a simplicial object in an cocomplete category ,(see below).)
Remark: Since in the standard model structure on simplicial sets every simplicial set is cofibrant, this indeed defines left homotopies.
Given morphisms of simplicial objects in any category , a simplicial homotopy is a family of morphisms, , of , such that , and
The above formulae give one of the, at least, two forms of the combinatorial specfication of a homotopy between and . (When trying to construct a specific homotopy using a combinatorial form, check which convention is being used!) The two forms correspond to different conventions such as saying that this is a homotopy from to , or reversing the labelling of the .
It is fairly easy to prove that the combinatorial definition of homotopy agrees with the one via the cylinder both for simplicial sets and for simplicial objects in any finitely cocomplete category, . This uses the fact that the category of simplicial objects in a cocomplete category, , has copowers with finite simplicial sets and hence in particular with . (As there are explicit formulae for the construction of copowers …)
Tim: With only my own resources available, I was unable to find them so was hoping someone kind would come up with them. They derive from the coend formulae/Kan extension formulae using some combinatorics to discuss the indexing sets. I think Quillen gave some form of them, but have not got a copy of HA. I needed them recently and could not find them in any of the usual sources, and did not manage to work them out using the Kan extension idea either (Help please anyone). We could do with those formulae or with a reference to them at least.
In the case of the category of (not necessarily abelian) groups, the combinatorial definition equals the one via cylinder only if the role of “cylinder” for a group is played by a simplicial object in the category of groups which in degree equals the free product of copies of , indexed by the set (noted by Swan and quoted in exercise 8.3.5 of Weibel: Homological algebra).
Tim Porter: Perhaps we need an explicit description of copowers in simplicial objects also. I pointed out in an edit above that the combinatorial description is much more general than just for simplicial objects in an abelian category.
Can specific references to Swan be given, anyone?
Zoran Škoda: I agree that one should talk about copowers etc.
The following is a direct proof.
We first show that the homotopy between points is an equivalence relation when is a Kan complex.
We identify in the following and with vertices in the image of these maps.
-reflexivity- For every vertex , the degenerate 1-simplex has, by the simplicial identities, 0-faces and .
Therefore the morphism that takes the unique non-degenerate 1-simplex in to constitutes a homotopy from to itself.
-transitivity- let and in be 1-cells. Together they determine a map from the horn to ,
If we again identify with its image (the image of its unique non-degenerate 2-cell) in , then using the simplicial identities we find
that the 1-cell boundary bit in turn has 0-cell boundaries
This means that is a homotopy .
-symmetry- In a similar manner, suppose that is a 1-cell in that constitutes a homotopy from to . Let be the degenerate 1-cell on . Since together they define a map which by the Kan property of we may extend to a map
on the full 2-simplex.
Now the 1-cell boundary has, using the simplicial identities, 0-cell boundaries
and hence yields a homotopy . So being homotopic is a symmetric relation on vertices in a Kan complex.
Let be an abelian category and homotopic morphisms of simplicial objects in . Then the induced maps of their normalized chain complexes are chain homotopic.
Let be an abelian category. The morphisms of simplicial objects (variant: of unbounded chain complexes) in , which are homotopic to zero, form an ideal. More precisely
being homotopic is an equivalence relation on the class of morphism,
and implies ,
if (resp. ) exists and if then (resp. ).
Cylinder based homotopy is also discussed extensively in