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) -categoryfreely generated from the shape.
A parity structure is a graded set together with, for each , functions
we assume throughout this article that , are finite, nonempty, and disjoint.
Following Street, we abbreviate to , and to . The Greek letters , refer to values in the set .
A parity structure is a parity complex if it satisfies the following axioms:
If , then and are both singletons.
If are distinct -cells, then and .
Define a relation by whenever , and let be the reflexive transitive closure of . Then is antisymmetric, and if for and , then .