## Label structures - another kind of parity structure

### Christopher Nguyen – 9 November 2016

A parity structure is a name given to various ways of formalising the idea of pasting diagrams, among these are; parity complexes (Street), pasting schemes (Johnson), and directed complexes (Steiner). Roughly speaking, the intuition behind these formalisms is to take a polytope-like object and attach orientation in a special way to obtain a presentation of a free omega category. Although these formalisms are powerful enough to capture the main examples (globes, simplexes and cubes), they tend to be too restrictive to cover various other examples. An example of interest is the associahedra, for which I have constructed an orientation which is not a parity complex. Nevertheless I am able to show the associahedra satisfy a weaker set of axioms, furthermore this set of axioms still gives a presentation for a free omega category. I will introduce a new kind of parity structure called label structures, which turns out to be useful for proving these axioms.

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