# Category Archives: Homotopy Theory

## Universal Properties of Truncations

Some days ago at the HoTT/UF workshop in Warsaw (which was a great event!), I have talked about functions out of truncations. I have focussed on the propositional truncation , and I want to write this blog post in case … Continue reading

## Double Groupoids and Crossed Modules in HoTT

The past eight months I spent at CMU for my master thesis project. I ended up formalizing some algebraic structures used in Ronald Brown’s book “Non-Abelian Algebraic Topology”: Double groupoids with thin structure and crossed modules over groupoids. As the … Continue reading

## The torus is the product of two circles, cubically

Back in the summer of 2012, emboldened by how nicely the calculation π₁(S¹) had gone, I asked a summer research intern, Joseph Lee, to work on formalizing a proof that the higher-inductive definition of the torus (see Section 6.6 of the HoTT book) … Continue reading

## Splitting Idempotents, II

I ended my last post about splitting idempotents with several open questions: If we have a map , a witness of idempotency , and a coherence datum , and we use them to split as in the previous post, do … Continue reading

## Splitting Idempotents

A few days ago Martin Escardo asked me “Do idempotents split in HoTT”? This post is an answer to that question.

## Fibrations with fiber an Eilenberg-MacLane space

One of the fundamental constructions of classical homotopy theory is the Postnikov tower of a space X. In homotopy type theory, this is just its tower of truncations: One thing that’s special about this tower is that each map has … Continue reading

## New writeup of πn(Sn)

I’m giving a talk this week at CPP. While I’m going to talk more broadly about applications of higher inductive types, for the proceedings, Guillaume Brunerie and I put together an “informalization” of πn(Sn), which you can find here. This is … Continue reading

## Spectral Sequences

Last time we defined cohomology in homotopy type theory; in this post I want to construct the cohomological Serre spectral sequence of a fibration (i.e. a type family). This is the second part of a two-part blog post. The first … Continue reading

## Cohomology

For people interested in doing homotopy theory in homotopy type theory, Chapter 8 of the HoTT Book is a pretty good record of a lot of what was accomplished during the IAS year. However, there are a few things it’s … Continue reading

## The HoTT Book

This posting is the official announcement of The HoTT Book, or more formally: Homotopy Type Theory: Univalent Foundations of Mathematics The Univalent Foundations Program, Institute for Advanced Study The book is the result of an amazing collaboration between virtually everyone involved … Continue reading

## Homotopy Theory in Type Theory: Progress Report

A little while ago, we gave an overview of the kinds of results in homotopy theory that we might try to prove in homotopy type theory (such as calculating homotopy groups of spheres), and the basic tools used in our synthetic approach … Continue reading

## Covering Spaces

Covering spaces are one of the important topics in classical homotopy theory, and this post summarizes what we have done in HoTT. We have formulated the covering spaces and (re)proved the classification theorem based on (right) -sets, i.e., sets equipped with … Continue reading

## Homotopy Theory in Homotopy Type Theory: Introduction

Many of us working on homotopy type theory believe that it will be a better framework for doing math, and in particular computer-checked math, than set theory or classical higher-order logic or non-univalent type theory. One reason we believe this … Continue reading