nLab
category 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 $X_{•}:{\Delta }^{\mathrm{op}}\to \mathrm{Set}$;

• 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 $c\to c\prime$ are morphisms $f:{\Delta }^n\to {\Delta }^{n\prime }$ in the simplex category $\Delta$ such that the diagram

  $$\begin{array}{ccccc}{\Delta }^n& & \stackrel{f}{\to }& & {\Delta }^{n\prime }\\ & {}_c\searrow & & {\swarrow }_{c\prime }\\ & & X\end{array}$$\array{ \Delta^n &&\stackrel{f}{\to}&& \Delta^{n'} \\ & {}_{c}\searrow && \swarrow_{c'} \\ && X }

  commutes.

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

$$(\Delta \downarrow X{)}_{\mathrm{nondeg}}\hookrightarrow (\Delta \downarrow X)$$(\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 $N((\Delta \downarrow X{)}_{\mathrm{nondeg}})$ is also called the barycentric subdivision of $X$.

See at barycentric subdivision – Relation to the category of simplices.

Properties

General

Proposition

The category of simplices is a Reedy category.

Proposition

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

Colimits

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

Proposition

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

$$X\simeq \underrightarrow{\lim }((\Delta \downarrow X)\to \mathrm{sSet})$$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 $F:\mathrm{sSet}\to C$ is uniquely determined by its action on the standard simplices:

$$F(X)\cong {\mathrm{colim}}_{(\Delta \downarrow X)}F({\Delta }^{•}).$$F(X) \cong colim_{(\Delta\downarrow X)} F(\Delta^\bullet).

Example

Important colimit-preserving functors out of sSet include

• geometric realization of simplicial sets
• subdivision of simplicial sets
• the left adjoint to the nerve functor $N:\mathrm{Cat}\to \mathrm{SSet}$
• the category of simplices itself, as a functor $\mathrm{SSet}\to \mathrm{Cat}$; see category of elements – colimit preserving.

The nerve and subdivision

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

Theorem

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

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

$${\Delta }^{k_n}\to \dots \to {\Delta }^{k_0}$$\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 $\mathrm{SSet}\to \mathrm{Set}$ is a coproduct of a family of “evaluation” functors. Since evaluation preserve colimits, coproducts commute with colimits, and colimits in $\mathrm{SSet}$ 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

• Jacob Lurie, Higher Topos Theory

For more on barycentric subdivision see also section 2 of

• Rick Jardine, Simplicial approximation, Theory and Applications of Categories, Vol. 12, 2004, No. 2, pp 34-72. (web)