# Formalising Yoneda Ext in Univalent Foundations

## TL;DR

This paper formalises Yoneda Ext groups within homotopy type theory using Coq-HoTT, avoiding resolutions and providing new formal proofs of classical exact sequences and properties.

## Contribution

It introduces a resolution-free formalisation of Ext groups in HoTT, with novel proofs of exact sequences and implementation of Ext as a bifunctor.

## Key findings

- Formalisation of Ext groups in HoTT without resolutions
- Proof of the six-term exact sequence via fibre sequences
- Implementation of Ext as a bifunctor

## Abstract

Ext groups are fundamental objects from homological algebra which underlie important computations in homotopy theory. We formalise the theory of Yoneda Ext groups in homotopy type theory (HoTT) using the Coq-HoTT library. This is an approach to Ext which does not require projective or injective resolutions, though it produces large abelian groups. Using univalence, we show how these Ext groups can be naturally represented in HoTT. We give a novel proof and formalisation of the usual six-term exact sequence via a fibre sequence of 1-types (or groupoids), along with an application. In addition, we discuss our formalisation of the contravariant long exact sequence of Ext, an important computational tool. Along the way we implement and explain the Baer sum of extensions and how Ext is a bifunctor.