Calum Hughes

Calum Hughes

PhD student studying category theory and logic at the University of Manchester.

Elementary axiomatisations of the $(2,1)$-category of groupoids

Date:

Axiomatising the notion of a set is not an entirely trivial matter, as shown by Russell’s paradox, for example. There are at least four approaches to this problem: Zermelo-Fraenkel set theory; the approach of Lawvere via the elementary theory of the category of sets which builds on the notion of an elementary topos; and Joyal and Moerdijk’s Algebraic Set Theory. The latter involves axiomatising the key properties of the category of classes and isolating sets as suitable “small” objects. All these settings relate naturally to extensional type theories.

In this talk, I will present some work aimed at axiomatising the notion of a groupoid, which is a higher dimensional analogue of a set. One approach is via what could be called the elementary theory of the (2,1)-category of small groupoids. Another approach is an extension of the ideas of Algebraic Set Theory to 2-dimensions, which allows us to deal with the important distinction between small (i.e. set-sized) and large (i.e. class-sized) groupoids, as well as to introduce a 2-dimensional counterpart of the notion of a subobject classifier. As I will explain, this setting relates naturally to intensional type theories.