Parity complexes redux

TL;DR

This paper refines the concept of parity complexes, demonstrating their categorical equivalence to strong Steiner complexes and highlighting the combinatorial advantages of Steiner's formalism.

Contribution

It redefines parity complexes with a specific set of axioms and proves their categorical equivalence to strong Steiner complexes, emphasizing the combinatorial structure of bases.

Findings

Parity complexes form a category equivalent to strong Steiner complexes.

The combinatorial structure of bases in free augmented directed complexes is characterized.

Steiner's formalism uses multisets instead of subsets, offering advantages.

Abstract

We fix the notion of parity complex by a judicious selection from among the axioms originally considered by Street. We show that parity complexes so defined, together with the morphisms of parity complexes defined by Verity, form a category equivalent to the category of strong Steiner complexes (n\'es augmented directed complexes with strongly loop-free unital bases). To this end, we isolate the purely combinatorial structure possessed by the bases of free augmented directed complexes. This analysis reveals the essential advantage of Steiner's formalism to be that the role of subsets in Street's formalism is played instead by multisets.