Paths and cylinders
A simplicial object in a category is an simplicial set internal to : a collection of objects in that behave as if were an object of -dimensional simplices internal to equipped with maps between these space that assign faces and degenerate simplices.
For instance, and there is a longer list further down this page, a simplicial object in is a collection of groups, together with face and degeneracy homomorphisms between them. This is just a simplicial group. We equally well have other important instances of the same idea, when we replace by other categories, or higher categories.
A simplicial object in a category is a functor , where is the simplicial indexing category.
More generally, a simplicial object in an (∞,1)-category is an (∞,1)-functor .
A cosimplicial object in is similarly a functor out of the opposite category, .
Accordingly, simplicial and cosimplicial objects in themselves form a category in an obvious way, namely the functor category and , respectively.
A simplicial object in is often specified by the objects, , which are the images under , of the objects of , together with a description of the face and degeneracy morphisms, and , which must satisfy the simplicial identities.
Category of simplicial objects
For a category, we write for the functor category from to : its category of simplicial objects.
Let be a category with all limits and colimits. This implies that it is tensored over Set
This induces a functor
which we shall also write just “”.
and for let
be given in degree by
We may regard the category of cosimplicial objects as an -enriched category using the above enrichment by identifying
- Peter May, Simplicial objects in algebraic topology , University of Chicago Press, 1967, (djvu)