nLab
weak complicial set

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 $\omega$-category” to start from, is called simplicial weak ω-category theory. It is expected that the (nerves of) weak $\omega$-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 $\omega$-categories.

Weak complicial sets are a joint generalization of

• strict ω-categories;

• Kan complexes;

• quasi-categories.

Definition

Definition

Let

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

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

• ${\Lambda }^k[n]\prime$ be obtained from ${\Delta }^k[n]$ by making the $(k-1)$st $(n-1)$-face and the $(k+1)$st $(n-1)$ face thin;

• ${\Delta }^k[n]″$ be obtained from ${\Delta }^k[n]$ by making all $(n-1)$-faces thin.

An elementary anodyne extension in $\mathrm{Strat}$, the category stratified simplicial sets is

• a complicial horn extension ${\Lambda }^k[n]\stackrel{{\subset }_r}{\hookrightarrow }{\Delta }^k[n]$

or

• a complicial thinness extension ${\Delta }^k[n]\prime \stackrel{{\subset }_e}{\hookrightarrow }{\Delta }^k[n]″$

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

Definition

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

${\Lambda }^k[n]\stackrel{{\subset }_r}{\hookrightarrow }{\Delta }^k[n]$ and ${\Lambda }^k[n]\prime \stackrel{{\subset }_e}{\hookrightarrow }{\Delta }^k[n]″$

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 $\omega$-categories. See

• model structure for weak omega-categories?.

Examples

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

• There is also the stratification of $N(C)$ which regards each $\omega$-equivalence morphism as a thin cell. $N(C)$ 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 $\omega$-categories into weak ones.

• A simplicial set is a weak complicial set when equipped with its maximal stratification (every simplex of dimension $>0$ is thin) if and only if it is a Kan complex. This example is, of course, saturated, and is viewed as embedding $\omega$-groupoids into $\omega$-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 $>1$ 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 $(\mathrm{\infty },1)$-categories into weak $\omega$-categories.

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

References

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

• Dominic Verity, Weak complicial sets Part I: Basic homotopy theory (arXiv)

Further developments are in

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