nLab
category of simplices

Context

Category theory

Homotopy theory

Categories of simplices

Idea

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

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 XX, at least if every non-degenerate simplex in XX comes from a monomorphism Δ nX\Delta^n \to X, as for a simplicial complex.

Definition

Let XX \in sSet be a simplicial set.

Definition

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

  • the category of elements of the presheaf X :Δ opSetX_\bullet : \Delta^{op} \to Set;

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

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

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

    commutes.

Definition

An nn-simplex xX nx\in X_n is said to be nondegenerate if it is not in the image of any degeneracy map.

Write

(ΔX) nondeg(ΔX) (\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 XX comes from a monomorphism Δ nX\Delta^n \to X, then the nerve N((ΔX) nondeg)N((\Delta \downarrow X)_{nondeg}) is also called the barycentric subdivision of XX.

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 (ΔX) nondeg(ΔX)(\Delta \downarrow X)_{nondeg} \hookrightarrow (\Delta \downarrow X) has a left adjoint and is hence a reflective subcategory.

Colimits

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

Proposition

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

Xlim((ΔX)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:sSetCF\colon sSet \to C is uniquely determined by its action on the standard simplices:

F(X)colim (ΔX)F(Δ ). 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:N\colon Cat \to sSet denote the simplicial nerve functor on categories.

Theorem

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

XN(ΔX) X\mapsto N(\Delta\downarrow X)

preserves colimits.

Proof

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

Δ k nΔ k 0 \Delta^{k_n} \to \dots\to \Delta^{k_0}

along with a map Δ k 0X\Delta^{k_0} \to X, i.e. an element of X k 0X_{k_0} Thus, each the functor XN(ΔX) nX\mapsto N(\Delta\downarrow X)_n from SSetSetSSet \to Set is a coproduct of a family of “evaluation” functors. Since evaluation preserve colimits, coproducts commute with colimits, and colimits in SSetSSet are levelwise, the statement follows.

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

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)

Revised on August 4, 2014 18:44:05 by Anonymous Coward (98.215.35.144)