The category $SimpSet$, or $sSet$ for short, is the category of simplicial sets.
This is the functor category from the opposite category $\Delta^{op}$ of the simplex category $\Delta$ 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 $S \otimes T = S \times T$ of simplicial sets $S$ and $T$ 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 $|\cdot| : SSet \to Top$
For $S$ and $T$ 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 $[S,T]$ of simplicial sets is the simplicial set
where $\Delta[n] = Hom_{\mathbf{\Delta}}(-,[n])$ is the standard simplicial $n$-simplex, the image of $[n] \in \mathbf{\Delta}$ under the Yoneda embedding. This formula is clearly representing a Kan extension.
The maps $N: \Cat \rightarrow \Simp\Set$ and $S: \Top \rightarrow \Simp\Set$ 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 $E$ be any cocomplete category and let $F: \Delta \rightarrow E$ be a functor. We define the right adjoint $R : E \rightarrow \Simp\Set$ as follows. Given an object $e \in E$ the $n$-simplices of $Re$ are defined to be the set $E(F[n],e)$ of morphisms in $E$ from $F[n]$ to $e$. Face and degeneracy maps are given by precomposition by the appropriate (dual) maps in the image of $F$. $R$ is defined on morphisms by postcomposition.
The left adjoint $L$ is defined to be the left Kan extension of $F$ along the Yoneda embedding $y: \Delta \rightarrow \Simp\Set$. Because the $y$ is full and faithful, we will have $Ly = F$, i.e., $L (\Delta[n]) = F[n]$. By specifying $F$, we have already defined a functor to $E$ on the represented simplicial sets; $L$ is the unique cocontinuous extension of this functor to $\Simp\Set$. It can be described explicitly on objects as a coend, or as a weighted colimit.
(Easy) abstract nonsense shows that $L$ and $R$ form an adjoint pair $L \dashv R$.
Here are some examples:
Let $E = \Cat$ and $F$ be the functor $[n] \mapsto [n]$ (the inclusion of posets into categories). The right adjoint is the nerve functor $N$ described above. The left adjoint ${\tau}_1$ takes a simplicial set to its fundamental category.
Let $E = \Top$ and $F$ be the functor $[n] \mapsto {\Delta}_n$. The right adjoint is the total singular complex functor $S$ 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 $\Delta[n]$ is the standard $n$-simplex ${\Delta}^n$.
(Barycentric) subdivision and extension $\sd: \Simp\Set \leftrightarrow \Simp\Set :\ex$.
The homotopy coherent nerve functor and its left adjoint $\Simp\Set \leftrightarrow \Simp\Cat$ where SimpCat? denotes the category of simplicially enriched categories, i.e., categories enriched in $\Simp\Set$.
The adjunction $- \times X: \SimpSet \leftrightarrow \SimpSet :(-)^X$ between the product with a simplicial set $X$ and the internal-hom, which makes $\Simp\Set$ into a cartesian closed category.
Let $E$ be a Grothendieck topos equipped with an “interval” $I$, i.e. a totally ordered object in the internal logic equipped with distinct top and bottom elements. Then we have the functor $\Delta \to E$ sending $[n]$ to the subobject $\{ (x_1,x_2,\dots,x_n) \;|\; x_1 \le x_2 \le \dots \le x_n \} \hookrightarrow I^n$ which gives rise to a geometric morphism $E\to \SimpSet$. Therefore, $\SimpSet$ is the classifying topos of such “intervals”.
There are important model category structures on $sSet$.
The standard model structure on simplicial sets presents the (∞,1)-category ∞Grpd of ∞-groupoids.
The model structure for quasi-categories on $sSet$ presents the (∞,2)-category of (∞,1)-categories (∞,1)Cat.
Like any elementary topos, $\SimpSet$ has an internal logic. Here we list some properties of this logic.
It is a two-valued topos, i.e. the only subobjects of $1 = \Delta^0$ are $0$ and $1$. (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 $A\subseteq B$ is the full simplicial subset on the vertices of $B$ not contained in $A$ (“full” meaning it contains a simplex of $B$ as soon as it contains all its vertices). Thus, $A\cup \neg A = B$ only if $A$ is a connected component of $B$, i.e. any simplex with at least one vertex in $A$ lies entirely in $A$.
By Diaconescu's theorem, $\SimpSet$ therefore does not satisfy the axiom of choice.
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.
Less obviously, it satisfies the Kreisel-Putnam axiom? that $(\neg p \to (q\vee r)) = ((\neg p \to q) \vee (\neg p \to r))$; see this MO question and answers.