nLab
fat simplex

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Contents

Idea

The fat simplex functor is a cosimplicial simplicial set

Δ:ΔsSet \mathbf{\Delta} : \Delta \to sSet

whose value Δ[n]\mathbf{\Delta}[n] at nn \in \mathbb{N} is a simplicial set that models the nn-simplex but is much bigger than the standard nn-simplex Δ[n]=Hom Δ(,[n])\Delta[n] = Hom_{\Delta}(-,[n]). This is such that Δ[]\mathbf{\Delta}[-] is a cofibrant replacement of ** and of Δ[]=Hom Δ(,)\Delta[-] = Hom_\Delta(-,-) in the projective model structure on functors ΔsSet Quillen\Delta \to sSet_{Quillen}.

The fat simplex can be used to express the homotopy colimit over simplicial diagrams in terms of coends of the form [n]ΔΔ[n]F n\int^{[n] \in \Delta} \mathbf{\Delta}[n] \cdot F_n. This construction is originally due to Bousfield and Kan.

Definition

Write Δ\Delta for the simplex category. For [n]Δ[n] \in \Delta write Δ/[n]\Delta/[n] for the corresponding overcategory. Finally write

Δ[n]:=N(Δ/[n]) \mathbf{\Delta}[n] := N(\Delta/[n])

(in sSet) for the nerve of this overcategory.

This construction is functorial in [n][n]:

Δ()=N(Δ/()):ΔsSet. \mathbf{\Delta}(-) = N(\Delta/(-)) : \Delta \to sSet \,.

Examples

Properties

There is a canonical morphism

ΔΔ \mathbf{\Delta} \to \Delta

of cosimplicial simplicial set, called the Bousfield-Kan map.

This exhibits Δ\mathbf{\Delta} as a cofibrant resolution of Δ\Delta and of ** in the projective model structure on functors on [Δ,sSet Quillen][\Delta, sSet_{Quillen}].

See the discussion at Reedy model structure and at Bousfield-Kan map for details.

Revised on February 13, 2011 00:16:33 by Urs Schreiber (62.28.152.34)