# Logic seminar: Adrian Miranda

Dates: | 6 December 2023 |
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Times: | 15:15 - 16:15 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, External researchers, Adults, Alumni, Current University students |

Title: The elementary theory of the 2-category of small categories

Abstract: The standard mathematical foundation system, ZFC, axiomatises sets and the membership relation. It can either be formulated either as a finite list of axioms in second order logic or with axiom schema in first order logic. However, a finitely axiomatised elementary subsystem RZFC (Restricted Zermelo Frankael with Choice) exists, in which most everyday mathematical constructions and proofs can be performed. If the axiom of replacement is added to RZFC then one recovers ZFC. Lawvere gave an elementary axiomatisation not of sets and the membership relation but of functions and their compositional structure. This theory became known as the elementary theory of the category of sets (ETCS), and it was later shown to be equiconsistent to RZFC.

Lawvere also posed the problem of axiomatising the 2-dimensional structure of categories, functors and natural transformations. The first real progress towards this was by Bourke in 2010 (Theorem 4.18 of 2), who characterised 2-categories Cat(E) of such data "inside" a category E with pullbacks. In this talk I will describe further progress extending Bourke's Theorem to the setting where E models Lawvere's ETCS. This is the elementary theory of the 2-category of small categories, of the title. It axiomatises the various composition structures of functors and natural transformations in a way that is bi-interpretable with ETCS. I will also describe how extra structure can be added to this theory to encode the axiom of replacement, giving a theory equiconsistent to ZFC albeit at the cost of no longer being elementary.

This talk is based on joint work with Calum Hughes (University of Manchester)

1 Lawvere, F. William (1964), ‘An elementary theory of the category of sets’, Proceedings of the National Academy of Science of the U.S.A 52, 1506–1511. 2 Bourke, John, (2010) Codescent Objects in two-dimensional Universal Algebra, PhD Thesis, University of Sydney, School of Mathematics and Statistics