nLab weak complicial set

Idea

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

The goal of characterizing such nerves, without an a priori definition of “weak $\omega$-category” to start from, is called simplicial weak ∞-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 ∞-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 $\Delta^k[n]$;

• $\Delta^k[n]'$ 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 $Strat$, 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 $\Delta^k[n]' \stackrel{\subset_e}{\hookrightarrow} \Delta^k[n]''$

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 $\Delta^k[n]' \stackrel{\subset_e}{\hookrightarrow} \Delta^k[n]''$

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 $\omega$-categories. See

• model structure for weak complicial sets.

Examples

• For $C$ a strict ∞-category and $N(C)$ its ∞-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 $\gt 0$ 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 $\gt 1$ 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 $(\infty,1)$-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 $QCat$ can “correctly” be regarded as a full subcategory of $sSet$. 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)

A model category structure on stratified simplicial sets modelling $(\infty,n)$-categories in the guise of $n$-complicial sets:

• Viktoriya Ozornova, Martina Rovelli, Model structures for (∞,n)-categories on (pre)stratified simplicial sets and prestratified simplicial spaces, Algebr. Geom. Topol. 20 (2020) 1543-1600 [arxiv:1809.10621, doi:10.2140/agt.2020.20.1543]

A Quillen adjunction relating $n$-complicial sets to $n$-fold complete Segal spaces:

• Viktoriya Ozornova, Martina Rovelli, A Quillen adjunction between globular and complicial approaches to $(\infty,n)$-categories, Advances in Mathematics

  421 (2023) 108980 [doi:10.1016/j.aim.2023.108980, arXiv:2206.02689]

Review:

• Martina Rovelli, $n$-Complicial sets as a model for $(\infty,n)$-categories, talk at CQTS (2023) [web, video: YT]