Questions tagged [homotopy-type-theory]

The homotopy interpretation of constructive dependent type theory, the univalence axiom, higher inductive types, internal languages of higher toposes, univalent foundations for mathematics, and implementations of such theories in proof assistants.

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Vladimir Voevodsky's works

Vladimir Voevodsky has made several contributions in abstract algebraic geometry, focused on the homotopy theory of schemes, algebraic K-theory, and interrelations between algebraic geometry, and ...
66 votes
4 answers
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Why did Voevodsky consider categories "posets in the next dimension", and groupoids the correct generalisation of sets?

Earlier today, I stumbled upon this article written by V. Voevodsky about the "philosophy" behind the Univalent Foundations program. I had read it before around the time of his passing, and one ...
Soham Chowdhury's user avatar
44 votes
9 answers
3k views

Homotopy as a general organizing principle

One of the realizations that led to the development of Homotopy Type Theory (HoTT) is that the ideas of homotopy theory have very broad applicability in mathematics. Indeed, Quillen model categories ...
44 votes
1 answer
8k views

Homotopy Type Theory: What is it?

My question is, broadly, what is the main project of Homotopy Type Theory (HoTT). I asked a professor who is likely to be correct and he say the following: There are three directions: ...
James's user avatar
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38 votes
3 answers
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How do you define (infinity,1) categories in Homotopy Type Theory?

One of the major motivations of Homotopy Type Theory is that it naturally builds in higher coherences from the beginning. One important setting where higher coherence requirements get annoying is ...
Noah Snyder's user avatar
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35 votes
3 answers
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Defining $SU(n)$ in HoTT

From a recent answer by Mike Shulman, I read: "HoTT is (among other things) a foundational theory, on roughly the same ontological level as ZFC, whose basic objects can be regarded as $\infty$-...
André Henriques's user avatar
32 votes
1 answer
2k views

Deligne's doubt about Voevodsky's Univalent Foundations

In a recent lecture at the Vladimir Voevodsky's Memorial Conference, Deligne wondered whether "the axioms used are strong enough for the intended purpose." (See his remark from roughly minute 40 in ...
THC's user avatar
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25 votes
1 answer
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Coinduction for all?

Every undergraduate in mathematics learns about proofs by mathematical induction. Moreover, every undergraduate taking a course in theoretical computer science or logic learns about inductive ...
user984603's user avatar
25 votes
0 answers
4k views

What's the point of cubical type theory?

I have been following through the development of homotopy type theory since 2013 because I was really interested in the foundation of mathematics. The novel idea of combining programming with homotopy ...
Kaa1el's user avatar
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24 votes
3 answers
2k views

What technical and/or theoretical challenges are involved in automatically extracting proofs from books and papers into Coq code?

Over the years, advances in machine learning has allowed us to communicate and interact, using the same natural language, more and more semantically with computers, e.g. Google, Siri, Watson, etc. On ...
23 votes
2 answers
2k views

Prospects for reverse mathematics in Homotopy Type Theory

Reverse mathematics, as I mean here, is the study of which theorems/axioms can be used to prove other theorems/axioms over a weak base theory. Examples include Subsystems of Second Order Arithmetic (...
Jason Rute's user avatar
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What is the most transparent, rigorous definition of the Univalence Axiom?

I've been studying homotopy type theory and trying to grasp the Univalence Axiom. I have yet to find a concise, accessible, rigorous definition of Univalence. I have several excellent survey papers ...
antianticamper's user avatar
21 votes
2 answers
1k views

$\infty$-categorical interpretation of type theory

One can read at several places that Martin-löf type theory should be the internal language of a locally Cartesian closed infinity category, and that the univalence axiom should distinguished infinity ...
Simon Henry's user avatar
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20 votes
2 answers
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Stable homotopy type theory?

This is a combined question, strictly speaking I am asking three questions concerning, respectively, homotopy type theory, stable homotopy theory and Yetter-Drinfeld modules. But I believe in the ...
მამუკა ჯიბლაძე's user avatar
20 votes
2 answers
739 views

Small complete categories in HoTT+LEM

Freyd's theorem in classical category theory says that any small category $\mathcal{C}$ admitting products indexed by the set $\mathcal{C}_1$ of all its arrows is a preorder. I'm interested in whether ...
Amélia Liao's user avatar
19 votes
5 answers
2k views

How do you define the strict infinity groupoids in Homotopy Type Theory?

In the setting of Homotopy Type Theory, how would you construct $\mathrm{isStrict} : U \rightarrow U$ which is inhabited exactly when the first type is (equivalent to?) a strict $\infty$-groupoid? ...
Noah Snyder's user avatar
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19 votes
3 answers
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What are finite homotopy types?

Starting from an ordinary 1-categorical point of view, there are various obvious candidate definitions for ‘finite homotopy type’: The homotopy type of a simplicial set that has only finitely many ...
Zhen Lin's user avatar
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17 votes
4 answers
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Practical example in using (homotopy) type theory

I have just read Grayson's introduction on homotopy type theory as a possible foundation for mathematics. It is very enlightening about what all the fuss is about, but I am left with some doubts. ...
Andrea Ferretti's user avatar
17 votes
2 answers
1k views

What kind of category is generated by Cubical type theory?

What kind of ‘category’ is Cubical type theory the internal language of? Its known that Martin-Löf type theories are the internal language of Locally cartesian closed categories, adding higher ...
Ali Caglayan's user avatar
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17 votes
2 answers
1k views

Constructive homological algebra in HoTT

I'm curious how much of homological algebra carries over to a constructive setting, like say HoTT (or some other variety of intensional type theory) without AC or excluded middle. There doesn't seem ...
ಠ_ಠ's user avatar
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17 votes
1 answer
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Forcing in Homotopy Type Theory

I apologize if this question doesn't make any sense. I'll just go ahead and delete it if that's the case. But the question is just the title. Is there a notion of forcing in homotopy type theory? ...
Jonathan Beardsley's user avatar
17 votes
1 answer
1k views

What would be an infinity-groupoid analogue of the duality between sets and complete atomic boolean algebras?

Consider the object classifier of the $\infty$-topos of $\infty$-groupoids. For the role it plays in homotopy type theory as the type of types, let’s denote it as $Type = \coprod_{[F]} B Aut(F)$, the ...
David Corfield's user avatar
16 votes
2 answers
796 views

Appearance of proof relevance in "ordinary mathematics?"

I've been wondering recently what—if any—applications proof theory has to ordinary mathematics (by which I mean algebra, analysis, topology, and so on). In particular, I'd be fascinated to see a proof ...
mcncm's user avatar
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15 votes
3 answers
911 views

Role of univalence in homotopy group calculations

This book has a section with proofs of the fact $\pi_1(S^1)=\mathbb Z$ using the univalence axiom. They are a bit too technical for me at the moment to read, but I want to understand the following (...
user336697's user avatar
15 votes
2 answers
2k views

Formal definition of homotopy type theory

The HoTT community is quite friendly, and produces many motivational introductions to HoTT. The blog and the HoTT book are quite helpful. However, I want to get my hands directly onto that, and am ...
Student's user avatar
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15 votes
2 answers
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What is the status of (universal) algebra in type theory?

With the recent interest in homotopy type theory as a foundation for mathematics, it seems natural to develop algebra within the framework of type theory. So far, I can't find much literature ...
Cory Knapp's user avatar
15 votes
2 answers
633 views

How to formulate the univalence axiom without universes?

The standard formulation of the univalence axiom for a universe type $U$ is that, for all $X : U$ and $Y : U$, the canonical map $(X =_U Y) \to (X \simeq Y)$ is an equivalence. As we (usually) cannot ...
Zhen Lin's user avatar
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14 votes
1 answer
614 views

Why the reflection rule trivializes higher paths in Martin-Löf Extensional Type theory?

Martin-Löf Extensional Type theory differs from its intensional counterpart in that it contains the so-called reflection rule that says that if $p : x = y$, then actually $x \equiv y$ (i.e. $x$ and $y$...
StudentType's user avatar
13 votes
4 answers
839 views

The groupoid of algebraic expressions and proofs

Fix a set of variables $V$, and suppose we're given a presentation of a monosorted algebraic theory, with variable symbols taken from $V$. For the sake of example, suppose the presentation consists of ...
goblin GONE's user avatar
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13 votes
1 answer
633 views

What is meant by a computational interpretation of univalence?

In homotopy type theory the univalence axiom implies function extensionality. Suppose we have a recursive set we are not sure is empty (e.g. the set of even integers$\geq 4$ that are not a sum of two ...
npq's user avatar
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1 answer
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Practical advantages of univalent foundations

I'm interested in the machine translation of mathematics from informal to formal (a la Ganesalingam in The Language of Mathematics). As a first step, I am designing a computer language for expressing ...
Alexander Smith's user avatar
12 votes
1 answer
643 views

3 questions about basics of Martin-Löf type theory

I started to read the HoTT book. I'm now on chapter 1 and I have several questions concerning not even homotopical, but "regular" type theory. On page 24, where the universes are introduced,...
Grisha Taroyan's user avatar
12 votes
1 answer
867 views

Are the “topologies” arising from constructive type theories with quotients actually condensed sets?

This is the second in a pair of questions. For the other see Are representations in computable analysis the equivalent to countably-generated condensed sets?. Dustin Clausen and Peter Scholze have a ...
Jason Rute's user avatar
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11 votes
3 answers
906 views

Can Homotopy Type Theory or algebraic geometry deal with homotopy fibers in terms of families?

((In conclusion) It was hard to choose which answer to accept. I decided for the one which addressed most of the various aspects of the question. ) (Later addon) I now decided to put a bounty on ...
მამუკა ჯიბლაძე's user avatar
11 votes
3 answers
1k views

Equivalent form of the Univalence Axiom

I'm reading the new HoTT book and I'm wondering about a potential equivalent form of the Univalence Axiom: $(A \simeq B) \simeq (A = B)$. For simplicity, I'm tacitly working in a fixed universe. It ...
François G. Dorais's user avatar
11 votes
1 answer
728 views

The role of univalence in the homotopy interpretation of type theory

In Martin-Löf type theory with identity eliminator $$ J : \prod_{B:\prod_{x,y:A}(x=y)\to\mathcal{U}}\left( \prod_{x:A}B(x,x,\mathrm{refl}_x)\to \prod_{x,y:A}\prod_{p:x=y}B(x,y,p) \right) $$ ...
coconut's user avatar
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11 votes
3 answers
692 views

natural metrics for proof length

I am trying to make my way into Homotopy Type Theory(HoTT) where a mathematician may view proofs as paths. Intuitively, this leads me to the idea of a metric on the space of mathematical propositions. ...
Aidan Rocke's user avatar
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11 votes
2 answers
1k views

Soft question: Deep learning and higher categories

Recently, I have stumbled upon certain articles and lecture videos that use category theory to explain certain aspects of machine learning or deep learning (e.g. Cats for AI and the paper An enriched ...
h3fr43nd's user avatar
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11 votes
1 answer
811 views

What is an Elementary "Homotopy, Model" Topos?

Context: Def (Rezk): A (Grothendieck) homotopy topos is a homotopy left exact Bousfield localization of the model category of simplicial presheaves sPsh(C) on a small simplicial category C. Thm (...
tttbase's user avatar
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10 votes
2 answers
991 views

Are simplicial sets the intended model of HoTT?

While thinking about Jason Rute's question, I wondered if there was an intended model for HoTT. The main candidate for the intended model are simplicial sets, where Vladimir Voevodsky first observed ...
François G. Dorais's user avatar
10 votes
1 answer
1k views

Why are W-types called "W"?

Why are W-types called "W"? Probably "W" means either "wellordered" or "wellfounded". (Martin-Löf uses the term "wellorder".) But these are notions ...
user347155's user avatar
10 votes
1 answer
573 views

How to proceed with a type-theoretic proof that $\Sigma \mathbb{S}^1 \simeq \mathbb{S}^2$?

The circle in homotopy type theory $\mathbb{S}^1$ is a higher inductive type freely generated by the following constructors: $\mathsf{b} : \mathbb{S}^1$ and $\mathsf{loop} : \mathsf{b} = \mathsf{b}$. ...
11Kilobytes's user avatar
10 votes
0 answers
309 views

Is there a definition of an $\infty$-groupoid in HoTT whose terms are $n$-manifolds and whose higher morphisms are diffeomorphisms/isotopies/etc?

Suppose you want to work with TQFTs in homotopy type theory (HoTT). Working with $(\infty,n)$-categories, or even $(\infty,1)$-categories, is something that I gather is too difficult for HoTT at the ...
Andy Manion's user avatar
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9 votes
2 answers
2k views

A (very naive) question about Homotopy Type Theory

In homotopy type theory, homotopy types can be viewed as logical types and it is possible to prove some theorems about them without using any underlying space (no simplicial set, no topological space)....
Philippe Gaucher's user avatar
9 votes
1 answer
769 views

Practical Benefits of HTT/univalent foundations for assisted proofs

I'm trying to understand what the claimed practical benefits of HTT/univalent foundations are for doing computer assisted proofs and while I've seen a lot of claims of benefits they all seem to be ...
Peter Gerdes's user avatar
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9 votes
2 answers
785 views

How should I be thinking about object classifiers / universal fibrations / universes?

I have been learning about homotopy type theory this summer. I am not a homotopy theorist but I am more comfortable with homotopy theory than I am with type theory, so the way I rationalize many of ...
Daniel Barter's user avatar
9 votes
2 answers
1k views

Progress towards a computational interpretation of the univalence axiom?

I will preface this by saying that I am not an expert on type theory. I am just a curious outsider slowly making my way through the HoTT book when I (rarely) have some spare time. I am just curious ...
ಠ_ಠ's user avatar
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9 votes
1 answer
442 views

Base change in homotopy type theory

Recall that with the internal language of 1-toposes, we have the nice, basic, and useful result that geometric sequents are stable under base change along geometric morphisms: If $\varphi$ and $\psi$ ...
Ingo Blechschmidt's user avatar
8 votes
2 answers
1k views

A pointless circle in HoTT

In the beginning of chapter two in The HoTT Book there is a discussion about synthetic vs. analytic geometry: An important difference between homotopy type theory and classical homotopy theory is ...
Kromanen's user avatar
8 votes
1 answer
619 views

The independence of path induction

In §1.12 of the Homotopy type theory book, it is mentioned that indiscernibility of identicals is a consequence of path induction. More precisely, for each type $C$ dependent over a type $A$, there is ...
Zhen Lin's user avatar
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