The category , or for short, is the category of simplicial sets.
This is the functor category from the opposite category of the simplex category to the category Set of sets:
Its objects are simplicial sets.
Like all categories of presheaves on a small category, the category SimpSet of simplicial sets is complete and cocomplete (with limits and colimits constructed levelwise) and cartesian closed. In fact, like all presheaf categories, it is a topos.
As described at closed monoidal structure on presheaves the cartesian tensor product of simplicial sets and is the simplicial set
where the product on the right is the cartesian product in Set.
One central reason why simplicial sets are useful and important is that this simple monoidal structure (“disturbingly simple minded” in the words of Friedman08, p. 24) actually does fully capture the standard monoidal structure on topological spaces under geometric realization
For and simplicial sets, we have
where on the right the cartesian product is in the nice category of compactly generated Hausdorff spaces.
See also products of simplices.
As described at closed monoidal structure on presheaves the internal hom of simplicial sets is the simplicial set
where is the standard simplicial -simplex, the image of under the Yoneda embedding. This formula is clearly representing a Kan extension.
The maps and described in the examples are actually functors, both of which have left adjoints. These adjoint pairs are examples of a very general sort of adjunction involving simplicial sets, of which there are many examples.
Let be any cocomplete category and let be a functor. We define the right adjoint as follows. Given an object the -simplices of are defined to be the set of morphisms in from to . Face and degeneracy maps are given by precomposition by the appropriate (dual) maps in the image of . is defined on morphisms by postcomposition.
The left adjoint is defined to be the left Kan extension of along the Yoneda embedding . Because the is full and faithful, we will have , i.e., . By specifying , we have already defined a functor to on the represented simplicial sets; is the unique cocontinuous extension of this functor to . It can be described explicitly on objects as a coend, or as a weighted colimit.
(Easy) abstract nonsense shows that and form an adjoint pair .
Here are some examples:
Let and be the functor (the inclusion of posets into categories). The right adjoint is the nerve functor described above. The left adjoint takes a simplicial set to its fundamental category.
Let and be the functor . The right adjoint is the total singular complex functor described above. The left adjoint is called geometric realization. As a consequence of the Kan extension construction, the geometric realization of the represented simplicial set is the standard -simplex .
(Barycentric) subdivision and extension .
The homotopy coherent nerve functor and its left adjoint where SimpCat? denotes the category of simplicially enriched categories, i.e., categories enriched in .
The adjunction between the product with a simplicial set and the internal-hom, which makes into a cartesian closed category.
There are important model category structures on .
The standard model structure on simplicial sets presents the (∞,1)-category ∞Grpd of ∞-groupoids.
The model structure for quasi-categories on presents the (∞,2)-category of (∞,1)-categories (∞,1)Cat.