category theory

Categories of simplices

Idea

For $X$ a simplicial set its category of simplices is the category whose objects are the simplices in $X$ and whose morphisms are maps between these, as simplices in $X$.

In particular the subcategory on the non-degenerate simplices has a useful interpretation: it is the poset of subsimplex inclusions whose nerve is the barycentric subdivision of $X$, at least if every non-degenerate simplex in $X$ comes from a monomorphism $\Delta^n \to X$, as for a simplicial complex.

Definition

Let $X \in$ sSet be a simplicial set.

Definition

The category of simplices of is equivalently (in increasing order of explicitness)

• the category of elements of the presheaf $X_\bullet : \Delta^{op} \to Set$;

• the comma category $(\Delta\downarrow X)$, where $\Delta$ denotes the Yoneda embedding $[n] \mapsto \Delta^n$.

• the category whose objects are homomorphisms of simplicial sets $c : \Delta^n \to X$ from a standard simplicial simplex $\Delta^n$ to $X$, and whose morphisms $c \to c'$ are morphisms $f : \Delta^n \to \Delta^{n'}$ in the simplex category $\Delta$ such that the diagram

$\array{ \Delta^n &&\stackrel{f}{\to}&& \Delta^{n'} \\ & {}_{c}\searrow && \swarrow_{c'} \\ && X }$
Definition

An $n$-simplex $x\in X_n$ is said to be nondegenerate if it is not in the image of any degeneracy map.

Write

$(\Delta\downarrow X)_{nondeg}\hookrightarrow (\Delta\downarrow X)$

for the subcategory on the nondegenerate simplices with monomorphisms between them.

This is called the category of non-degenerate simplices.

Remark

If every non-degenerate simplex in $X$ comes from a monomorphism $\Delta^n \to X$, then the nerve $N((\Delta \downarrow X)_{nondeg})$ is also called the barycentric subdivision of $X$.

Properties

General

Proposition

The category of simplices is a Reedy category.

Proposition

The inclusion of the non-generate simplices $(\Delta \downarrow X)_{nondeg} \hookrightarrow (\Delta \downarrow X)$ has a left adjoint and is hence a reflective subcategory.

Colimits

Write $(\Delta \downarrow X) \to sSet$ for the canonical functor that sends $(\Delta^n \to X)$ to $\Delta^n$.

Proposition

The colimit over the functor $(\Delta \downarrow X) \to sSet$ is $X$ itself

$X \simeq \underset{\to}{\lim}((\Delta \downarrow X) \to sSet)$
Proof

By the co-Yoneda lemma.

In the textbook literature this appears for instance as (Hovey, lemma 3.1.3).

Corollary

A colimit-preserving functor $F\colon sSet \to C$ is uniquely determined by its action on the standard simplices:

$F(X) \cong colim_{(\Delta\downarrow X)} F(\Delta^\bullet).$
Example

Important colimit-preserving functors out of sSet include

The nerve and subdivision

Let $N\colon$ Cat $\to$ sSet denote the simplicial nerve functor on categories.

Theorem

The functor $sSet \to sSet$ that assigns barycentric subdivision, def. 2,

$X\mapsto N(\Delta\downarrow X)$

preserves colimits.

Proof

An $n$-simplex of $N(\Delta\downarrow X)$ is determined by a string of $n+1$ composable morphisms

$\Delta^{k_n} \to \dots\to \Delta^{k_0}$

along with a map $\Delta^{k_0} \to X$, i.e. an element of $X_{k_0}$ Thus, each the functor $X\mapsto N(\Delta\downarrow X)_n$ from $SSet \to Set$ is a coproduct of a family of “evaluation” functors. Since evaluation preserve colimits, coproducts commute with colimits, and colimits in $SSet$ are levelwise, the statement follows.

Therefore, the simplicial set $N(\Delta\downarrow X)$ itself can be computed as a colimit over the category $(\Delta\downarrow X)$ of the simplicial sets $N(\Delta\downarrow \Delta^n)$.

References

A basic disussion is for instance in section 3.1 of

• Mark Hovey, Model categories, Mathematical surveys and monographs volume 63, American Mathematical Society

Homotopy finality of the non-degenerate simplices is discussed in section 4.1 of