nLab
complicial set

Idea

Complicial sets are precisely those 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 ${\Delta }_k^a[n]\to X$. Explicitly, this consists of a (thin) $n$-simplex $x\in X$ such that $x\cdot \alpha$ is thin for every $\alpha :[m]\to [n]$ 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 $x_0,\dots ,x_{k-2}$ are $(k-1)$-admissible and whose faces $x_{k+2},\dots ,x_n$ 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 ${\Delta }_k^a[n]\prime \to {\Delta }_k^a[n]″$, where ${\Delta }_k^a[n]\prime$ has all $n-1$-simplices except ${\delta }^k:[n-1]\to [n]$ thin, ${\Delta }_k^a[n]″$ has all $n-1$-simplices thin, and both stratified sets have any simplex $\alpha :[m]\to [n]$ with $k-1,k,k+1\in$ im$(\alpha )$ thin;

2. the inclusions ${\Lambda }_k^a[n]\to {\Delta }_k^a[n]$ for all $n\ge 2$, $0<k<n$;

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.

References

• Dominic Verity, Complicial Sets (arXiv)