complicial set

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


A kk-admissible nn-simplex in a stratified simplicial set XX is a map of stratified simplicial sets Δ k a[n]X\Delta^a_k[n] \to X. Explicitly, this consists of a (thin) nn-simplex xXx \in X such that xαx\cdot\alpha is thin for every α:[m][n]\alpha \colon [m] \to [n] whose image contains k1k-1, kk, and k+1k+1.

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

  1. if xXx \in X is a kk-admissible simplex whose k1k-1th and k+1k+1th faces are thin, then its kkth face is also thin;

  2. there is a unique thin filler for all (n1)(n-1)-dimensional inner kk-horns whose faces x 0,,x k2x_0,\ldots, x_{k-2} are (k1)(k-1)-admissible and whose faces x k+2,,x nx_{k+2},\ldots, x_n are kk-admissible (nb: there is no condition on x k1,x k+1X n1x_{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 tt-extensions Δ k a[n]Δ k a[n]\Delta^a_k[n]' \to \Delta^a_k[n]'', where Δ k a[n]\Delta^a_k[n]' has all n1n-1-simplices except δ k:[n1][n]\delta^k \colon [n-1] \to [n] thin, Δ k a[n]\Delta^a_k[n]'' has all n1n-1-simplices thin, and both stratified sets have any simplex α:[m][n]\alpha \colon [m] \to [n] with k1,k,k+1k-1,k,k+1 \in im(α)(\alpha) thin;

  2. the inclusions Λ k a[n]Δ k a[n]\Lambda^a_k[n] \to \Delta^a_k[n] for all n2n \geq 2, 0<k<n0 \lt k \lt n;

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


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


The original idea is developed in

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:

Revised on May 18, 2017 16:17:28 by Tim Porter (