nLab
weak complicial set

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

Weak complicial sets are simplicial sets with extra structure that are closely related to the ω-nerves of weak ω-categories.

The goal of characterizing such nerves, without an a priori definition of “weak ω\omega-category” to start from, is called simplicial weak ω-category theory. It is expected that the (nerves of) weak ω\omega-categories will be weak complicial sets satisfying an extra “saturation” condition ensuring that “every equivalence is thin.” General weak complicial sets can be regarded as “presentations” of weak ω\omega-categories.

Weak complicial sets are a joint generalization of

Definition

Definition

Let

  • Δ k[n]\Delta^k[n] be the stratified simplicial set whose underlying simplicial set is the nn-simplex Δ[n]\Delta[n], and whose marked cells are precisely those simplices [r][n][r] \to [n] that contain {k1,k,k+1}[n]\{k-1, k, k+1\} \cap [n];

  • Λ k[n]\Lambda^k[n] be the stratified simplicial set whose underlying simplicial set is the kk-horn of Δ[n]\Delta[n], with marked cells those that are marked in Δ k[n]\Delta^k[n];

  • Λ k[n]\Lambda^k[n]' be obtained from Δ k[n]\Delta^k[n] by making the (k1)(k-1)st (n1)(n-1)-face and the (k+1)(k+1)st (n1)(n-1) face thin;

  • Δ k[n]\Delta^k[n]'' be obtained from Δ k[n]\Delta^k[n] by making all (n1)(n-1)-faces thin.

An elementary anodyne extension in StratStrat, the category stratified simplicial sets is

  • a complicial horn extension Λ k[n] rΔ k[n]\Lambda^k[n] \stackrel{\subset_r}{\hookrightarrow} \Delta^k[n]

or

  • a complicial thinness extension Δ k[n] eΔ k[n]\Delta^k[n]' \stackrel{\subset_e}{\hookrightarrow} \Delta^k[n]''

for n=1,2,n = 1,2, \cdots and k[n]k \in [n].

Definition

A stratified simplicial set is a weak complicial set if it has the right lifting property with respect to all

Λ k[n] rΔ k[n]\Lambda^k[n] \stackrel{\subset_r}{\hookrightarrow} \Delta^k[n] and Λ k[n] eΔ k[n]\Lambda^k[n]' \stackrel{\subset_e}{\hookrightarrow} \Delta^k[n]''

A complicial set is a weak complicial set in which such liftings are unique.

Model structure

There is a model category structure that presents the (infinity,1)-category of weak complicial sets, hence that of weak ω\omega-categories. See

Examples

  • For CC a strict ω-category and N(C)N(C) its ω-nerve, the Roberts stratification which regards each identity morphism as a thin cell makes N(C)N(C) a strict complicial set, hence a weak complicial set. This example is not “saturated.”

  • There is also the stratification of N(C)N(C) which regards each ω\omega-equivalence morphism as a thin cell. N(C)N(C) with this stratification is a weak complicial set (example 17 of Ver06). This should be the “saturation” of the previous example, and exhibits the inclusion of strict ω\omega-categories into weak ones.

  • A simplicial set is a weak complicial set when equipped with its maximal stratification (every simplex of dimension >0\gt 0 is thin) if and only if it is a Kan complex. This example is, of course, saturated, and is viewed as embedding ω\omega-groupoids into ω\omega-categories.

  • A simplicial set is a quasi-category if and only if it is a weak complicial set when equipped with the stratification in which every simplex of dimension >1\gt 1 is thin, and only degenerate 1-simplices are thin. This example is not saturated; in its saturation the thin 1-simplices are the internal equivalences in a quasi-category (equivalently, those that become isomorphisms in its homotopy category). It presents the embedding of (,1)(\infty,1)-categories into weak ω\omega-categories.

    Note that 1-simplex equivalences in a quasi-category are automatically preserved by simplicial maps between quasi-categories; this is why QCatQCat can “correctly” be regarded as a full subcategory of sSetsSet. This is not true at higher levels; for instance not every simplicial map between nerves of strict ω\omega-categories necessarily preserves ω\omega-equivalence morphisms.

References

The definition of weak complicial sets is definition 14, page 9 of

Further developments are in

  • Dominic Verity, Weak complicial sets Part II: Nerves of complicial Gray-categories (arXiv)

Revised on November 10, 2014 20:40:07 by Urs Schreiber (86.212.18.175)