2021-02-18: Nicolai Kraus (University of Nottingham)

Sat, 21 Oct 2023 21:24:08 +0100

Nicolai Kraus (University of Nottingham)

Internal ∞-Categories with Families

Abstract

It is natural to formalise the notion of a model of type theory (especially the syntax = intended initial model) inside type theory itself. This is often done by writing the definition of a category with families (CwF) as a generalised algebraic theory. What could the success criterion from a HoTT point of view be? The typical first goal is that the initial model is an h-set or even has decidable equality. Our second goal is to make the "standard model" work, i.e. the universe U should be a CwF in a straightforward way (cf. Mike Shulman's 2014 question whether the n-th universe in HoTT models HoTT with n-1 universes). Unfortunately, it is hard to combine these two goals. If we include set-truncatedness explicitly in the definition of a CwF, then the "standard model" is not a CwF. If we don't, then the initial model is not an h-set. The root of the problem is that 1-categories are not well-behaved concepts in an untruncated setting. The natural approach are higher categories, which corresponds to "equipping the syntax with all coherences" instead of truncating. In this talk, I will explain one approach to this based on a type-theoretic formulation of Segal spaces, expressed in HTS/2LTT. I will discuss what works and what is still open. The talk will be based on arXiv:2009.01883.

2021-02-18 16:00:00 UTC
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