nLab
parity complex

Idea

The notion of parity complex, introduced by Ross Street, is a notion of pasting diagram shape. It is based on some combinatorial axioms on subshapes of codimension at most 2 which permit the construction of a (strict) $\omega$-category freely generated from the shape.

Definition

Definition

A parity structure is a graded set $\{C_n{\}}_{n\ge 0}$ together with, for each $n\ge 0$, functions

$${\partial }_n^{+}:C_{n+1}\to P(C_n),{\partial }_n^{-}:C_{n+1}\to P(C_n);$$\partial^+_n \colon C_{n+1} \to P(C_n), \qquad \partial^-_n \colon C_{n+1} \to P(C_n);

we assume throughout this article that ${\partial }_n^{+}(c)$, ${\partial }_n^{-}(c)$ are finite, nonempty, and disjoint.

Following Street, we abbreviate ${\partial }_n^{+}(c)$ to $c^{+}$, and ${\partial }_n^{-}(c)$ to $c^{-}$. The Greek letters $\epsilon$, $\eta$ refer to values in the set $\{+,-\}$.

Definition

A parity structure is a parity complex if it satisfies the following axioms:

1. $c^{--}\cup c^{++}=c^{-+}\cup c^{+-}$

2. If $c\in C_1$, then $c^{-}$ and $c^{+}$ are both singletons.

3. If $x,y\in c^{\eta }$ are distinct $n$-cells, then $x^{+}\cap y^{+}=\varnothing$ and $x^{-}\cap y^{-}=\varnothing$.

4. Define a relation $<$ by $x<y$ whenever $x^{+}\cap y^{-}\ne \varnothing$, and let $\prec$ be the reflexive transitive closure of $<$. Then $\prec$ is antisymmetric, and if $x\prec y$ for $x\in c^{\epsilon }$ and $y\in c^{\eta }$, then $\epsilon =\eta$.

Examples

• Simplexes

• Globes

• Cubes

• Products of parity complexes

• Polyhedra

Basic results

(…)

Related notions

• directed complex?

• direct category

• test category

• oriental

References

• Street, Parity complexes, Cahiers Top. Géom Diff. Catégoriques 32 (1991), 315-343. (link) Corrigenda, Cahiers Top. Géom Diff. Catégoriques 35 (1994), 359-361. (link)