nLab
complicial set

**higher category theory** * category theory * homotopy theory ## Basic concepts * k-morphism, coherence * looping and delooping * looping and suspension ## Basic theorems * homotopy hypothesis-theorem * delooping hypothesis-theorem * periodic table * stabilization hypothesis-theorem * exactness hypothesis * holographic principle ## Applications * applications of (higher) category theory * higher category theory and physics ## Models * (n,r)-category * Theta-space * ∞-category/ω-category * (∞,n)-category * n-fold complete Segal space * (∞,2)-category * (∞,1)-category * quasi-category * algebraic quasi-category * simplicially enriched category * complete Segal space * model category * (∞,0)-category/∞-groupoid * Kan complex * algebraic Kan complex * simplicial T-complex * n-category = (n,n)-category * 2-category, (2,1)-category * 1-category * 0-category * (−1)-category * (−2)-category * n-poset = (n-1,n)-category * poset = (0,1)-category * 2-poset = (1,2)-category * n-groupoid = (n,0)-category * 2-groupoid, 3-groupoid * categorification/decategorification * geometric definition of higher category * Kan complex * quasi-category * simplicial model for weak ω-categories * complicial set * weak complicial set * algebraic definition of higher category * bicategory * bigroupoid * tricategory * tetracategory * strict ω-category * Batanin ω-category * Trimble ω-category * Grothendieck-Maltsiniotis ∞-categories * stable homotopy theory * symmetric monoidal category * symmetric monoidal (∞,1)-category * stable (∞,1)-category * dg-category * A-∞ category * triangulated category ## Morphisms * k-morphism * 2-morphism * transfor * natural transformation * modification ## Functors * functor * 2-functor * pseudofunctor * lax functor * (∞,1)-functor ## Universal constructions * 2-limit * (∞,1)-adjunction * (∞,1)-Kan extension * (∞,1)-limit * (∞,1)-Grothendieck construction ## Extra properties and structure * cosmic cube * k-tuply monoidal n-category * strict ∞-category, strict ∞-groupoid * stable (∞,1)-category * (∞,1)-topos ## 1-categorical presentations * homotopical category * model category theory * enriched category theory

Edit this sidebar

Idea

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

Definition

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.

Generalizations

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

References

Revised on November 15, 2011 23:12:34 by Emily Riehl (140.247.39.189)