David Kern: Codiscrete cofibrations vs iterated discrete fibrations for (∞,𝓁)-congruences

Abstract

The exactness properties putatively characterising \((\infty, \ell)\)-topoi should be phrased in terms of effectivity of higher congruences. In the \(2\)-dimensional case, it is known that \(2\)-congruences are internal categories whose underlying graph is a discrete two-sided fibration, but for higher values of \(\ell\) this admits two natural generalisations: internal categories whose underlying graph is an \((\ell - 2)\)-categorical two-sided fibration (recently studied by Loubaton), or internal \((\ell - 1)\)-categories whose underlying \((\ell - 1)\)-graph is an iterated discrete fibration.

I will explain how to compare both to a third notion suggested by formal enriched category theory: codiscrete two-sided cofibrations. While \((\ell - 2)\)-categorical fibrations require lax limits, the iterated discrete fibrations can be studied with very expressive weighted limits: the lost Australian folklore of generalised kernels, taking full advantage of the enrichment over \((\infty, \ell - 1)\text{-}\mathsf{Cat}\). If time allows, I will then use these tools to elucidate the structure of the fibration classifiers.