Normal isofibrations

Riehl and Verity [2] introduced the notion of $\mathrm{\infty }$-cosmos in order to develop a “synthetic” theory of $\mathrm{\infty }$-categories, analogous to the way in which a synthetic theory of ordinary categories can be developed in a 2-category. Every $\mathrm{\infty }$-cosmos has an associated “homotopy” 2-category, and a surprising amount of the theory depends only on this 2-category.

The structure of an $\mathrm{\infty }$-cosmos has three aspects: an enrichment over quasicategories, a class of maps called isofibrations, and certain (enriched) limits, including pullbacks of isofibrations along arbitrary maps.

The homotopy 2-category does not depend on the isofibrations (much as the homotopy category of a model category does not rely on the fibrations or cofibrations). Thus it is natural to wonder how much choice there might be in the isofibrations, and whether there is some sort of canonical choice.

John Bourke and I [1] studied this in the case of 2-categories, so that the enrichment is not just over quasicategories but over categories. We showed that (under very mild conditions) there is in fact a minimal choice of isofibrations, namely the normal isofibrations, and that any suitably complete 2-category becomes an $\mathrm{\infty }$-cosmos when one takes the isofibrations to be these normal isofibrations.

What then happens if the enrichment is over quasicategories? It turns out that, under an assumption on the quasicategorically-enriched category that holds in all the main cases of interest, the notion of normal isofibration makes sense and there are analogues of the various results mentioned above. Indeed in most important examples of $\mathrm{\infty }$-cosmoi, the chosen isofibrations are precisely the normal isofibrations. This forms part of the Master’s project currently being completed by Declan Zammit.

In this talk I will talk about both the 2-categorical case and the general one.