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 $X$ 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

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

  commutes.

Definition

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

Write

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

for the (full) subcategory on the non-degenerate simplices. Notice that a morphism of $\Delta\downarrow X$ with source a non-degenerate simplex of $X$ is necessary a monomorphism.

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 $X$.

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

Properties

General

Proposition

If $X$ has the property that every face of every non-degenerate simplex is again non-degenerate, then the inclusion of the category of non-degenerate simplices $(\Delta \downarrow X)_{nondeg} \hookrightarrow (\Delta \downarrow X)$ has a left adjoint and is hence a reflective subcategory.

Proposition

The category of simplices is a Reedy category.

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

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

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. ,

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

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

The terminology “category of simplices” is attributed by Hovey 1999 to:

• William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, Section 5.9 of: Model Categories and More General Abstract Homotopy Theory (1997) [pdf, pdf]

Textbook account:

• Mark Hovey, Section 3.1 (specifically p. 75) in: Model Categories, Mathematical Surveys and Monographs, 63 AMS (1999) [ISBN:978-0-8218-4361-1, doi:10.1090/surv/063, pdf, Google books]

Homotopy finality of the non-degenerate simplices:

• Jacob Lurie, Prop. 4.2.3.14 (p. 254) of: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press (2009) [pup:8957, pdf]

More on barycentric subdivision:

• J. F. Jardine, Section 2 of: Simplicial approximation, Theory and Applications of Categories, 12 2 (2004) 34-72 [tac:12-02]