Simplicial sets/categories

Example: singular chain complex

From a topological space , one defines a chain complex:

A simplicial set is a sequence of sets, together with a family of connecting maps

that satisfy

Consider the standard simplex

One has the maps

and

These are precisely opposite to the and in the definition definition above, and one gets the above and by composing with one of the or .

The simplex category has:

A simplicial set is a functor .

Every element can be decomposed into

A simplicial map is a sequence of maps that commute with and . Alternatively, it is a natural transformation

is a simplicial set. is a functor where gives .
is the “combinatorial -simplex”. . We have Every element in can be obtained from using face and degenerating maps: (c.f. the Yoneda lemma).
: and
, and
If and are two simplicial sets, then

We have a functor

given by (where has the discrete topology)

It is true that .

A homotopy between two simplicial maps is a map such that the following diagram commutes:

Warning: Being homotopic is not an equivalence relation on .

A simplicial set is Kan if and only if

that is to say, for all with , there exists such that .