# AustMS 2019 talks

Just so I have these somewhere, these are the notes/slides for the two talks I gave at the annual Australian Mathematical Society talk last week. The first is a revision of this talk.

# Cofibrations of internal groupoids and localisations by cospans

In my paper ICAL, I dealt extensively with internal categories and groupoids, and how to localise 2-categories of these at the fully faithful essentially ‘surjective’ functors (‘surjective’ in the sense there is some well-behaved class of regular epimorphisms playing the rôle of surjective maps). Even for 1-categories of ordinary small groupoids or categories one can consider this as a variant on the canonical model structure on groupoids or on categories, and formally invert the fully faithful, essentially surjective functors, which aren’t invertible in the 1-category sense. Actually it really only uses the category of fibrant objects structure, and ICAL (and more in generality, these notes) can be seen as a way to make this work in a way that doesn’t use the full category of fibrant objects structure, but the 2-category structure instead. There is another way to think about this process for the case of small groupoids (and more generally from some cofibration category), which I learned from a paper of Land, Nikolaus and Szumiło, Localization of Cofibration Categories and Groupoid C*-algebras (publisher, arXiv) (cofibrations of groupoids are functors that are injective on objects). I want to outline how one can repeat the 2-categorical, extra weak story for internal groupoids (at least; I haven’t thought about the case of internal categories) using this type of approach. In some respects, I can get more general results, in that I don’t require that I’m localising a small cofibration category with good cylinders, but also kinda less general in the sense that the universal property I get is only for bicategories, not more general $\infty$-categories (presented by something Quillen equivalent to a combinatorial model category). Below the fold I set up some basic assumptions, and then outline the constructions that will go into the hypotheses of the construction.