nLab
parity complex

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

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) ω\omega-category freely generated from the shape.

Definition

Definition

A parity structure is a graded set {C n} n0\{C_n\}_{n \geq 0} together with, for each n0n \geq 0, functions

n +:C n+1P(C n), n :C n+1P(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 n +(c)\partial^+_n(c), n (c)\partial^-_n(c) are finite, nonempty, and disjoint.

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

Definition

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

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

  2. If cC 1c \in C_1, then c c^- and c +c^+ are both singletons.

  3. If x,yc ηx, y \in c^\eta are distinct nn-cells, then x +y +=x^+ \cap y^+ = \emptyset and x y =x^- \cap y^- = \emptyset.

  4. Define a relation <\lt by x<yx \lt y whenever x +y x^+ \cap y^- \neq \emptyset, and let \prec be the reflexive transitive closure of <\lt. Then \prec is antisymmetric, and if xyx \prec y for xc εx \in c^\varepsilon and yc ηy \in c^\eta, then ε=η\varepsilon = \eta.

Examples

Basic results

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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)