Lexicographic sums and splayed categories
Nathanael Arkor
Tallinn University of Technology
Abstract:
Categories may be presented in several ways: the two most popular presentations involve (1) a set of objects and, for each pair of objects x and y, a set of morphisms C(x, y) from x to y; (2) a set of objects and a set of morphisms. These presentations lend themselves to different generalisations of categories: the former to enriched categories, and the latter to internal categories. In this talk, I will consider a third presentation of category involving (3) a set of objects and, for each object x, a set C[x] of morphisms from x. We show that this presentation lends itself to a new generalisation of category, which we call a splayed category. While enriched categories may be defined with respect to monoidal categories, and internal categories with respect to categories with pullbacks, splayed categories may be defined with respect to any category with lexicographic sums.