Author Archives: Mike Shulman

HoTT 2019 Call for Submissions

Submissions of talks are now open for the International Homotopy Type Theory conference (HoTT 2019), to be held from August 12th to 17th, 2019, at Carnegie Mellon University in Pittsburgh, USA.  Contributions are welcome in all areas related to homotopy type … Continue reading

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Impredicative Encodings, Part 3

In this post I will argue that, improving on previous work of Awodey-Frey-Speight, (higher) inductive types can be defined using impredicative encodings with their full dependent induction principles — in particular, eliminating into all type families without any truncation hypotheses … Continue reading

Posted in Foundations, Higher Inductive Types | 50 Comments

UF-IAS-2012 wiki archived

The wiki used for the 2012-2013 Univalent Foundations program at the Institute for Advanced Study was hosted at a provider called Wikispaces. After the program was over, the wiki was no longer used, but was kept around for historical and … Continue reading

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At the 2018 U.S. Joint Mathematics Meetings in San Diego, there will be an AMS special session about homotopy type theory. It’s a continuation of the HoTT MRC that took place this summer, organized by some of the participants to … Continue reading

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From June 4 — 10, 2017, there will be a workshop on homotopy type theory as one of the AMS’s Mathematical Research Communities (MRCs).

Posted in News, Publicity, Uncategorized | 1 Comment

Real-cohesive homotopy type theory

Two new papers have recently appeared online: Brouwer’s fixed-point theorem in real-cohesive homotopy type theory by me, and Adjoint logic with a 2-category of modes, by Dan Licata with a bit of help from me. Both of them have fairly … Continue reading

Posted in Applications, Foundations, Paper | 13 Comments

A new class of models for the univalence axiom

First of all, in case anyone missed it, Chris Kapulkin recently wrote a guest post at the n-category cafe summarizing the current state of the art regarding “homotopy type theory as the internal language of higher categories”. I’ve just posted … Continue reading

Posted in Models, Paper, Univalence | 5 Comments

Modules for Modalities

As defined in chapter 7 of the book, a modality is an operation on types that behaves somewhat like the n-truncation. Specifically, it consists of a collection of types, called the modal ones, together with a way to turn any … Continue reading

Posted in Code, Programming | 19 Comments

Not every weakly constant function is conditionally constant

As discussed at length on the mailing list some time ago, there are several different things that one might mean by saying that a function is “constant”. Here is my preferred terminology: is constant if we have such that for … Continue reading

Posted in Univalence | 10 Comments

The HoTT Book does not define HoTT

The intent of this post is to address certain misconceptions that I’ve noticed regarding the HoTT Book and its role in relation to HoTT. (At least, I consider them misconceptions.) Overall, I think the HoTT Book project has been successful … Continue reading

Posted in Uncategorized | 6 Comments

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

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Splitting Idempotents

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

Posted in Code, Homotopy Theory, Univalence | 13 Comments

Universal properties without function extensionality

A universal property, in the sense of category theory, generally expresses that a map involving hom-sets, induced by composition with some canonical map(s), is an isomorphism. In type theory we express this using equivalences of hom-types. For instance, the universal … Continue reading

Posted in Code, Foundations, Higher Inductive Types | 16 Comments

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

Posted in Homotopy Theory | 2 Comments

Higher inductive-recursive univalence and type-directed definitions

In chapter 2 of the HoTT book, we prove theorems (or, in a couple of cases, assert axioms) characterizing the equality types of all the standard type formers. For instance, we have and (that’s function extensionality) and (that’s univalence). However, … Continue reading

Posted in Code, Foundations, Higher Inductive Types, Univalence | 7 Comments

Homotopy Type Theory should eat itself (but so far, it’s too big to swallow)

The title of this post is an homage to a well-known paper by James Chapman called Type theory should eat itself. I also considered titling the post How I spent my Christmas vacation trying to construct semisimplicial types in a … Continue reading

Posted in Foundations, Models | 251 Comments

The surreals contain the plump ordinals

The very last section of the book presents a constructive definition of Conway’s surreal numbers as a higher inductive-inductive type. Conway’s classical surreals include the ordinal numbers; so it’s natural to wonder whether, or in what sense, this may be … Continue reading

Posted in Higher Inductive Types | 3 Comments

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

Posted in Homotopy Theory | 19 Comments


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

Posted in Homotopy Theory, Models | 18 Comments

On Heterogeneous Equality

(guest post by Jesse McKeown) A short narative of a brief confusion, leading to yet-another-reason-to-think-about-univalence, after which the Author exposes his vaguer thinking to derision. The Back-story In the comments to Abstract Types with Isomorphic Types, Dan Licata mentioned O(bservational)TT, … Continue reading

Posted in Uncategorized | 19 Comments

All Modalities are HITs

Last Friday at IAS, Guillaume Brunerie presented a very nice proof that for all . I hope he will write it up and blog about it himself; I want to talk instead about a question regarding modalities that was raised … Continue reading

Posted in Higher Inductive Types | 5 Comments

The Simplex Category

I recently had occasion to define the simplex category inside of type theory. For some reason, I decided to use an inductive definition of each type of simplicial operators , rather than defining them as order-preserving maps of finite totally … Continue reading

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Homotopy equivalences are equivalences: take 3

A basic fact in homotopy type theory is that homotopy equivalences are (coherent) equivalences. This is important because on the one hand, homotopy equivalences are the “correct” sort of equivalence, but on the other hand, for a function f the … Continue reading

Posted in Code | 10 Comments

Modeling Univalence in Inverse Diagrams

I have just posted the following preprint, which presents new set-theoretic models of univalence in categories of simplicial diagrams over inverse categories (or, more generally, diagrams over inverse categories starting from any existing model of univalence). The univalence axiom for … Continue reading

Posted in Models, Paper, Univalence | 5 Comments

Univalence versus Extraction

From a homotopical perspective, Coq’s built-in sort Prop is like an undecided voter, wooed by both the extensional and the intensional parties. Sometimes it leans one way, sometimes the other, at times flirting with inconsistency.

Posted in Uncategorized | 18 Comments

Localization as an Inductive Definition

I’ve been talking a lot about reflective subcategories (or more precisely, reflective subfibrations) in type theory lately (here and here and here), so I started to wonder about general ways to construct them inside type theory. There are some simple … Continue reading

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Modeling Univalence in Subtoposes

In my recent post at the n-Category Café, I described a notion of “higher modality” in type theory, which semantically ought to represent a left-exact-reflective sub–category of an -topos — once we can prove that homotopy type theory has models … Continue reading

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Axiomatic cohesion in HoTT

This post is to alert the members of the HoTT community to some exciting recent developments over at the n-Category Cafe. First, some background. Some of us (perhaps many) believe that HoTT should eventually be able to function as the … Continue reading

Posted in Uncategorized | 11 Comments

A formal proof that π₁(S¹)=Z

The idea of higher inductive types, as described here and here, purports (among other things) to give us objects in type theory which represent familiar homotopy types from topology. Perhaps the simplest nontrivial such type is the circle, , which … Continue reading

Posted in Code, Higher Inductive Types | 2 Comments

Higher Inductive Types via Impredicative Polymorphism

The proof assistant Coq is based on a formal system called the “Predicative Calculus of (Co)Inductive Constructions” (pCiC). But before pCiC, there was the “Calculus of Constructions” (CoC), in which inductive types were not a basic object, but could be … Continue reading

Posted in Code, Foundations, Higher Inductive Types | 28 Comments

What’s Special About Identity Types

From a homotopy theorist’s point of view, identity types and their connection to homotopy theory are perfectly natural: they are “path objects” in the category of types. However, from a type theorist’s point of view, they are somewhat more mysterious. … Continue reading

Posted in Uncategorized | 9 Comments

An Interval Type Implies Function Extensionality

One of the most important spaces in homotopy theory is the interval (be it the topological interval or the simplicial interval ). Thus, it is natural to ask whether there is, or can be, an “interval type” in homotopy type … Continue reading

Posted in Foundations | 53 Comments