# nLab parity complex

### Context

#### Higher category theory

higher category theory

# Contents

## 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 $\left\{{C}_{n}{\right\}}_{n\ge 0}$ together with, for each $n\ge 0$, functions

${\partial }_{n}^{+}:{C}_{n+1}\to P\left({C}_{n}\right),\phantom{\rule{2em}{0ex}}{\partial }_{n}^{-}:{C}_{n+1}\to P\left({C}_{n}\right);$\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 ${\partial }_{n}^{+}\left(c\right)$, ${\partial }_{n}^{-}\left(c\right)$ are finite, nonempty, and disjoint.

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

###### 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 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 $\epsilon =\eta$.

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## 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)
Revised on February 12, 2012 01:21:29 by Urs Schreiber (89.204.138.114)