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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4
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1answer
98 views

Considering each half of factorization of weak equivalence separately

I have been working through the proof of Theorem 3.4.1 in the article https://arxiv.org/pdf/1211.2851.pdf (pages 35-6), but there is one technical detail still unclear to me. Specifically, we have ...
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1answer
145 views

Showing that a certain simplicial set has levelwise small cardinality

My question concerns Section 2 of the article "The Simplicial Model of Univalent Foundations (after Voevodsky)" (https://arxiv.org/pdf/1211.2851.pdf). Let $\alpha$ be a strongly inaccessible cardinal....
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146 views

Is there a foundational approach that takes “structure” as primitive?

As per the title, I'd be curious to know if there have been attempts at constructing a foundation of mathematics taking, somehow, purely the notion of "structure" as primitive, maybe via a system of ...
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1answer
229 views

Checking the functoriality of an expression involving dependent sum and product

I'm unsure if my question is advanced enough for this site, but let's see. Let $\mathcal{C}$ be a locally cartesian closed category, so that it always has dependent products $\Pi_f$, i.e., right ...
4
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1answer
323 views

Uniqueness principle for functions types in the HoTT book

Chapter 1.2 of the HoTT book says this about eta-conversion: $$ f \equiv (\lambda x . f(x)). $$ This equality is the uniqueness principle for function types, because it shows that $f$ is uniquely ...
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148 views

Categorical semantics of the identity type

In Appendix B of the article Simplicial Model of Univ. Foundations, Definition B.1.3 specifies the structure for identity types in a contextual category $\mathcal{C}$ (which in particular is equipped ...
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4answers
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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. ...
28
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1answer
956 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 ...
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1answer
443 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 ...
58
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4answers
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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 ...
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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 ...
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1answer
289 views

Definition of $(\infty,1)$-category in HoTT [duplicate]

Reading Lurie's Higher topos theory got me thinking, if we can think of $(\infty,1)-$categories as (1-)categories enriched over $\infty$-groupoid, then wouldn't the internal definition of an $(\infty,...
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1answer
320 views

How much homotopy type theory should be modeled by the unstable motivic category?

It's often said that homotopy type theory should be interpretable in any $\infty$-topos. But really it should be interpretable in any "predicative $\infty$-topos". I'm not quite sure what this means, ...
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2answers
860 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 ...
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2answers
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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 $\infty$-...
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3answers
363 views

(Co)limits of locally cartesian closed categories

Do locally cartesian closed $\infty$-categories form a presentable $\infty$-category? It seems like they should, and that the inclusion $\text{LCC}\rightarrow\text{Cat}$ preserves limits and maybe ...
13
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1answer
546 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 ...
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3answers
671 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 ...
5
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1answer
136 views

Models for Higher Inductive Types in Homotopy Type Theory

Ordinary inductive types is initial algebras for free monads. However, HITs are not initial algebras for endofunctors but presented monads. From nLab, initial algebra of a presentable (infinity,1)-...
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2answers
926 views

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 ...
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Synthetic type theory for virtual double category and its higher categories

For some monad T on a virtual equipment, the paper A unified framework for generalized multicategories by Cruttwell and Shulman (arXiv:0907.2460) proposes the normalized T-monoid. Another paper, by ...
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2answers
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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 ...
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1answer
568 views

The role of univalence in the homotopy interpretation of type theory

In Martin-Löf type theory with identity eliminator $$ J : \prod_{B:\prod_{x,y:A}(x=y)\to\mathcal{U}}\left( \prod_{x:A}B(x,x,\mathrm{refl}_x)\to \prod_{x,y:A}\prod_{p:x=y}B(x,y,p) \right) $$ ...
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2answers
318 views

Type Theory to Study $(\infty,n)$-Categories and $(r,n)$-Categories

The recent paper "A Type Theory for Synthetic $\infty$-Categories" proposes the syntax as the theory of the strict interval. In principle, any other suitable theory could be used instead. For instance,...
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857 views

How can one define “punctured torus” in Homotopy Type Theory? Is its fundamental group the free product of the integers with themselves?

Questions. Has the beautiful old idea, in part already known to Gauß, of a punctured torus surface (take, if you will, the classical set-theoretic definition as the meaning of the latter three words)...
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1answer
251 views

Problems for a Homotopy excursion in HoTT

I would like to know some open problems in HoTT for Homotopy theorist (without knowledge of logic). With such a Homotopy theorist I mean someone who takes a serious reading ("at the level") of ...
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3answers
395 views

Homotopy type theory: Are the hierarchy of Type_k universes isomorphic?

In homotopy type theory, or dependent type theories more generally, there is a "top-level" type called the universe, generally denoted $\newcommand{\type}{\mathtt{Type}}\type$. So for a concrete ...
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1answer
277 views

Why are types in type theory unordered collections?

Please excuse my naïveté, I have no higher math education, just a curious observer. In type theories, types are treated as set-like because they're unordered collections. Yet, the putative motive in ...
2
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1answer
131 views

Is the univalence of the canonical family over a universe small?

Working in Martin-Löf intensional type theory, $\mathsf{ITT}$, with a universe $U$ within the theory closed under all the usual constructors, the univalence of the canonical family $\mathsf{EI}$ over $...
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1answer
177 views

The axiom $\Xi : \prod_{A:\mathcal{U}} \|A\| \to A$ and the $n$-truncation of a type

My question is whether the following has been considered as an axiom, and if so, where I may find a discussion of it: $\Xi : \prod_{A : \mathcal{U}} \|A\| \to A$. For example, using this axiom, we ...
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1answer
271 views

Univalent Foundations and canonicity property are compatible?

I'm interested in HoTT, especially its application to foundations of mathematics. I believe strongly that Univalent Foundations is the very foundation of mathematics. So,I have a question. (Q) ...
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1answer
222 views

Help with simple homotopy type theory proof [closed]

I'm having some beginner problems understanding / proving simple facts about higher inductive type paths. If you take this higher inductive type for natural numbers modulo 1: ...
3
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1answer
240 views

Are there types with nontrivial paths in all dimensions? (HoTT)

I'm trying to construct a model of homotopy type theory, and in my development it seems like it would be helpful to assume that all types only have trivial paths below a given level (forgive my ...
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2answers
349 views

The homotopy pullback of a point along itself is the loop space

I have seen on the nLab that we can view the loop space as a particular homotopy pullback, and that it is even the way a "loop space object" is defined in general (when it exists). Can someone give ...
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0answers
432 views

New foundation in homotopy type theory

Is there any model of NF (New Foundations) on HoTT (homotopy type theory)? Because there is a model of ZF(C) on HoTT (The HoTT book, Section 10.5) and NF on ZFC (by this Wikipedia articile), I think ...
4
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1answer
494 views

HoTT without Funext, Univalence

Are there any models of Martin-Löf's intensional type theory in which univalence or function extensionality fails? In the HoTT book, axioms like $\mathsf{LEM}_{\infty}$ (in Section 3.4) are proved to ...
5
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1answer
525 views

Uniqueness Principle for function types

I am currently trying to understand the first chapter of the HoTT book and for 1.2 Functions Types of the book, Since it is by definition ``the function that applies $f$ to its argument'' we ...
2
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1answer
187 views

Set truncations in homotopy type theory

We can define set truncation as a higher inductive type with the following constructors: $|-|_0 : A \to ||A||_0$ trunc : $(a\ a' : ||A||_0)\ (p\ p' : a = a') \to p = p'$ If we replace the type of ...
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1answer
333 views

What constitutes “a path” in HoTT? [closed]

($I$ is the interval [0..1], the domain of all paths in HoTT.) ($2$ is the type of two elements, named here as $0_2$ and $1_2$.) Define a map $p : I \rightarrow 2$ by $p(0) = 0_2$ and $p(x) = 1_2$ ...
14
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1answer
496 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 $y$...
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1answer
150 views

Construction of the necklace by homotopy type theory [closed]

The necklace can be obtained from a circle by attaching $n$ 2-spheres $S^2$ along arcs, so the necklace $N(n,S^1,a_i)$ is homotopy equivalent to the space obtained by attaching $n$ 2-spheres $S^2$ to ...
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1answer
666 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 (...
13
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1answer
811 views

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 ...
9
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0answers
259 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 ...
3
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1answer
289 views

Defining equivalence of equivalences without assuming extensionality

In homotopy type theory, we say that a function $f$ is an equivalence between two types $A$ and $B$, if the inverse image of all points is contractible (as per say the n-lab or Paolo's Capriotti's ...
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3answers
1k 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 ...
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1answer
3k views

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 ...
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1answer
389 views

Functional programing and intensional type theory

I know very little about how computers work, so please excuse my ignorance! I think of the Glasgow Haskell Compiler as a program that eats up extensional type theory and spits out a program which ...
4
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1answer
350 views

Is the infinity-groupoid of a finite CW complex finitely-presented?

An infinity-groupoid is finitely-presented when it is equivalent to the free infinity-groupoid on a finite family of generators, possibly of different dimensions. Is the infinity-groupoid of a finite ...
37
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1answer
5k 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: ...