nLab complicial set

Idea

Complicial sets are precisely those stratified simplicial sets which arise as the omega nerve of a strict omega-category.

Definition

A $k$-admissible $n$-simplex in a stratified simplicial set $X$ is a map of stratified simplicial sets $\Delta^a_k[n] \to X$. Explicitly, this consists of a (thin) $n$-simplex $x \in X$ such that $x\cdot\alpha$ is thin for every $\alpha \colon [m] \to [n]$ whose image contains $k-1$, $k$, and $k+1$.

A complicial set is a stratified simplicial set satisfying the following three axioms:

1. if $x \in X$ is a $k$-admissible simplex whose $k-1$th and $k+1$th faces are thin, then its $k$th face is also thin;

2. there is a unique thin filler for all $(n-1)$-dimensional inner $k$-horns whose faces $x_0,\ldots, x_{k-2}$ are $(k-1)$-admissible and whose faces $x_{k+2},\ldots, x_n$ are $k$-admissible (nb: there is no condition on $x_{k-1}, x_{k+1} \in X_{n-1}$);

3. all thin 1-simplices are degenerate.

Equivalently, a complicial set is a stratified simplicial set that is (right) orthogonal to each of the following classes of stratified maps:

1. the primitive $t$-extensions $\Delta^a_k[n]' \to \Delta^a_k[n]''$, where $\Delta^a_k[n]'$ has all $n-1$-simplices except $\delta^k \colon [n-1] \to [n]$ thin, $\Delta^a_k[n]''$ has all $n-1$-simplices thin, and both stratified sets have any simplex $\alpha \colon [m] \to [n]$ with $k-1,k,k+1 \in$ im$(\alpha)$ thin;

2. the inclusions $\Lambda^a_k[n] \to \Delta^a_k[n]$ for all $n \geq 2$, $0 \lt k \lt n$;

3. the unique surjection $\Delta[1]_t \to \Delta[0]$, where every 1-simplex in $\Delta[1]_t$ is thin.

Generalizations

• Weakening the conditions on a stratified simplicial set to be a complicial set yields notions of simplicial weak omega-category.

• n-complicial set

References

The original idea is developed in

• Dominic Verity, Complicial Sets (arXiv)

There are also Lecture notes written by Emily Riehl to accompany a three-hour mini course entitled “Weak Complicial Sets” delivered at the Higher Structures in Geometry and Physics workshop at the MATRIX Institute from June 6-7, 2016. They are to be found at:

• Emily Riehl, Complicial sets, an overture, (arXiv)