Formalizing of Category Theory in Agda

The generality and pervasiness of category theory in modern mathematics makes it a frequent and useful target of formalization. It is however quite challenging to formalize, for a variety of reasons. Agda currently (i.e. in 2020) does not have a standard, working formalization of category theory. We document our work on solving this dilemma. The formalization revealed a number of potential design choices, and we present, motivate and explain the ones we picked. In particular, we find that alternative definitions or alternative proofs from those found in standard textbooks can be advantageous, as well as "fit" Agda's type theory more smoothly. Some definitions regarded as equivalent in standard textbooks turn out to make different "universe level" assumptions, with some being more polymorphic than others. We also pay close attention to engineering issues so that the library integrates well with Agda's own standard library, as well as being compatible with as many of supported type theories in Agda as possible

Constructive sheaf models of type theory

We generalise sheaf models of intuitionistic logic to univalent type theory over a small category with a Grothendieck topology. We use in a crucial way that we have constructive models of univalence, that can then be relativized to any presheaf models, and these sheaf models are obtained by localisation for a left exact modality. We provide first an abstract notion of descent data which can be thought of as a higher version of the notion of prenucleus on frames, from which can be generated a nucleus (left exact modality) by transfinite iteration. We then provide several examples.Comment: Simplified the definition of lex operation, simplified the encoding of the homotopy limit and remark that the homotopy descent data is a lex modality without using higher inductive type

31ème Journées Francophones des Langages Applicatifs

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