nLab
SimpSet

Definition

The category $\mathrm{SimpSet}$, or $\mathrm{sSet}$ for short, is the category of simplicial sets.

This is the functor category from the opposite category ${\Delta }^{\mathrm{op}}$ of the simplex category $\Delta$ to the category Set of sets:

$$\mathrm{Simp}\mathrm{Set}:=[{\Delta }^{\mathrm{op}},\mathrm{Set}].$$Simp Set := [\Delta^{op}, Set] \,.

Its objects are simplicial sets.

Properties

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.

Monoidal structure

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

$$(S\otimes T):[n]\mapsto S_n\times T_n,$$(S \otimes T) : [n] \mapsto S_n \times T_n \,,

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 $\mid \cdot \mid :\mathrm{SSet}\to \mathrm{Top}$

Proposition

For $S$ and $T$ simplicial sets, we have

$$\mid S\times T\mid \simeq \mid S\mid \times \mid T\mid ,$$|S \times T| \simeq |S| \times |T| \,,

where on the right the cartesian product is in the nice category of compactly generated Hausdorff spaces.

See also products of simplices.

Closed structure

As described at closed monoidal structure on presheaves the internal hom $[S,T]$ of simplicial sets is the simplicial set

$$[S,T]:[n]\mapsto {\mathrm{Hom}}_{\mathrm{SSet}}(S\times \Delta [n],T),$$[S,T] : [n] \mapsto Hom_{SSet}(S \times \Delta[n], T) \,,

where $\Delta [n]={\mathrm{Hom}}_{\Delta }(-,[n])$ is the standard simplicial $n$-simplex, the image of $[n]\in \Delta$ under the Yoneda embedding. This formula is clearly representing a Kan extension.

Adjunctions

The maps $N:Cat\to SimpSet$ and $S:Top\to SimpSet$ 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 $F:\Delta \to E$ be a functor. We define the right adjoint $R:E\to SimpSet$ as follows. Given an object $e\in E$ the $n$-simplices of $\mathrm{Re}$ 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 $y:\Delta \to SimpSet$. Because the $y$ is full and faithful, we will have $\mathrm{Ly}=F$, 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 $SimpSet$. 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 $L⊣R$.

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 $\mid -\mid$ 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 $sd:SimpSet\leftrightarrow SimpSet:ex$.

• The homotopy coherent nerve functor and its left adjoint $SimpSet\leftrightarrow SimpCat$ where SimpCat? denotes the category of simplicially enriched categories, i.e., categories enriched in $SimpSet$.

• The adjunction $-\times X:SimpSet\leftrightarrow SimpSet:(-{)}^X$ between the product with a simplicial set $X$ and the internal-hom, which makes $SimpSet$ into a cartesian closed category.

Model category structures

There are important model category structures on $\mathrm{sSet}$.

• The standard model structure on simplicial sets presents the (∞,1)-category ∞Grpd of ∞-groupoids.

• The model structure for quasi-categories on $\mathrm{sSet}$ presents the (∞,2)-category of (∞,1)-categories (∞,1)Cat.

Related concepts

• sSet, asSet

• Ho(sSet)

category: category