## Final Activity Report Summary - NEAL-TYPE (Non-extensional and linear models for Dependent Type Theory)

This project has established, for the first time, a link between two previously unconnected areas of mathematics: dependent type theory and higher-dimensional category theory. Dependent type theory is a branch of mathematics which deals with the following question: to what extent can the proofs that mathematicians work with be carried out by a computer? It does so by considering a mathematical statement as a specification for computer programs, and a proof of that statement as an implementation of the corresponding specification. For example, here is a typical mathematical statement: 'There exist infinitely many primes.' Through the lens of dependent type theory, this is translated into the following specification: 'A program which, when you give it a number, computes another number which is larger than the input and is prime.' From this perspective, a proof that there are infinitely many primes is just a program implementing this specification.

On the other hand, higher-dimensional category theory is a kind of meta-mathematics. Let us first note that the objects which are studied in pure mathematics generally represent real-world phenomena. For example, the collection of ways in which you can rotate a Rubik's cube is represented by the mathematical notion of a 'group', which is defined to be any collection of operations which can be composed together and also undone; whilst the everyday notion of a shape is captured by the mathematical notion of 'topological space'. Category theory is a little different, because the notion of 'category' does not represent real-world phenomena, but instead mathematical ones. Informally, a category is a kind of mathematical universe. Higher-dimensional categories are a special kind of mathematical universe particularly suitable for representing mathematical objects which 'look like topological spaces'.

Intuitively, it is clear that there is a mathematical universe which is 'the universe of dependent type theory', or 'the universe of mathematics as done by a computer', and so we should expect to be able to build a category which represents this universe. This is indeed the case; but the novelty of this project has been to show that what we obtain is not only a category but also a higher-dimensional category. On the one hand, this tells us something about dependent type theory: that its objects of study in some sense 'look like topological spaces'. On the other, it tells us something about higher-dimensional category theory, by allowing us to talk about it using the language of intensional type theory.

On the other hand, higher-dimensional category theory is a kind of meta-mathematics. Let us first note that the objects which are studied in pure mathematics generally represent real-world phenomena. For example, the collection of ways in which you can rotate a Rubik's cube is represented by the mathematical notion of a 'group', which is defined to be any collection of operations which can be composed together and also undone; whilst the everyday notion of a shape is captured by the mathematical notion of 'topological space'. Category theory is a little different, because the notion of 'category' does not represent real-world phenomena, but instead mathematical ones. Informally, a category is a kind of mathematical universe. Higher-dimensional categories are a special kind of mathematical universe particularly suitable for representing mathematical objects which 'look like topological spaces'.

Intuitively, it is clear that there is a mathematical universe which is 'the universe of dependent type theory', or 'the universe of mathematics as done by a computer', and so we should expect to be able to build a category which represents this universe. This is indeed the case; but the novelty of this project has been to show that what we obtain is not only a category but also a higher-dimensional category. On the one hand, this tells us something about dependent type theory: that its objects of study in some sense 'look like topological spaces'. On the other, it tells us something about higher-dimensional category theory, by allowing us to talk about it using the language of intensional type theory.