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 $\{C_n\}_{n \geq 0}$ together with, for each $n \geq 0$, functions

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

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 $\varepsilon$, $\eta$ 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 $x^+ \cap y^+ = \emptyset$ and $x^- \cap y^- = \emptyset$.

4. Define a relation $\lt$ by $x \lt y$ whenever $x^+ \cap y^- \neq \emptyset$, and let $\prec$ be the reflexive transitive closure of $\lt$. Then $\prec$ is antisymmetric, and if $x \prec y$ for $x \in c^\varepsilon$ and $y \in c^\eta$, then $\varepsilon = \eta$.

Examples

• Simplexes

• Globes

• Cubes

• Products of parity complexes

• Polyhedra

Basic results

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Related notions

• directed complex?

• direct category

• test category

• oriental

• pasting scheme

References

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