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.
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.
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 adjunction between the product with a simplicial set and the internal-hom, which makes into a cartesian closed category.
Let be a Grothendieck topos equipped with an “interval” , i.e. a totally ordered object in the internal logic equipped with distinct top and bottom elements. Then we have the functor sending to the subobject which gives rise to a geometric morphism . Therefore, is the classifying topos of such “intervals”.
There are important model category structures on .
It is a two-valued topos, i.e. the only subobjects of are and . (This is not really a property of the internal logic, but we include it to contrast with the next point.)
It is not Boolean. In general, the complement of a simplicial subset is the full simplicial subset on the vertices of not contained in (“full” meaning it contains a simplex of as soon as it contains all its vertices). Thus, only if is a connected component of , i.e. any simplex with at least one vertex in lies entirely in .
Like any presheaf topos, it satisfies the dependent choice (assuming it holds in the metatheory); see Fourman and Scedrov. Moreover, natural numbers object is simply the discrete simplicial set of ordinary natural numbers.
Similarly, it satisfies Markov's principle.