+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Higher category theory +-- {: .hide} [[!include higher category theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea Complicial sets are precisely those [[stratified simplicial sets]] which arise as the [[oriental|omega nerve]] of a [[strict omega-category]]. ## Definition A $k$-**admissible** [[n-simplex|$n$-simplex]] in a [[stratified simplicial set]] $X$ is a [[stratified simplicial set|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 text: * [[Dominic Verity]], *Complicial Sets Characterising the Simplicial Nerves of Strict $\omega$-Categories*, Memoirs of the AMS, **193** 905 (2008) [[arXiv:math/0410412](http://arxiv.org/abs/math/0410412), [ams:memo-193-905](https://bookstore.ams.org/memo-193-905/)] Introduction: * [[Emily Riehl]]: _Complicial sets, an overture_, in *2016 MATRIX Annals*, MATRIX Book Series **1** Springer (2018) [[doi:10.1007/978-3-319-72299-3_4](https://doi.org/10.1007/978-3-319-72299-3_4), [arXiv:1610.06801](https://arxiv.org/abs/1610.06801)] [[!redirects complicial sets]]