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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1answer
88 views

Assuming decidable equality but not LEM in HoTT

The law of excluded middle in homotopy type theory is a term of $$\prod_{A:\mathcal{U}}\Big(\mathrm{isProp}(A)\to(A+\neg A)\Big).$$What if we assume a term of$$\prod_{A:\mathcal{U}}\Big(\mathrm{isSet}(...
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The difference between Agda and Idris for programming using Homotopy type theory [closed]

Which is better for programming benifiting by Homotopy type theory(HoTT),Idris or Agda.compare them.
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213 views

Intuitive (topological) explanation of a proof from the HoTT book [closed]

My friends and I are struggling with understanding intuitively the proof of equivalence between based and free path inductions (HoTT book 1.12.2) The first major problem is understanding the meaning ...
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1answer
293 views

Using HoTT, why is twisted cohomology of BG group cohomology?

I've been reading Michael Shulman's blog posts defining cohomology in homotopy type theory, and I'd like to understand (using HoTT) why cohomology of BG is group cohomology. if I understand correctly, ...
2
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1answer
119 views

Higher-dimensional paths as parametrizations of 1-dimensional paths

Sect. 6.7 of the HoTT Book establishes in the context of CW complexes that "we can obtain an n-dimensional path as a continuous family of 1-dimensional paths parametrized by an (n − 1)-...
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2answers
243 views

Construction of Dedekind reals using higher inductive-inductive types

In the textbook Homotopy Type Theory: Univalent Foundations of Mathematics, the authors give a predicative constructive construction of the initial Cauchy complete reals $\mathbb{R}_C$ in terms of a ...
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2answers
213 views

Explicit different proofs of the same identity type in MLTT

This question is similar to (but more specific than) this one: When are two proofs of the same theorem really different proofs I do not know very much about homotopy type theory, but I am trying to ...
8
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1answer
336 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,...
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1answer
196 views

Cohomology with local coefficients in homotopy type theory

I was just reading Mike Shulman's blog post on how to define cohomology in homotopy type theory (HoTT), and I was curious if we can similarly define cohomology with local coefficients in HoTT as well? ...
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1answer
339 views

What do UF and ZF do to each other?

(By request from a comment: UF stands for Univalent Foundations) Correct me if I'm wrong, but in a model $M$ of ZF each element $x$ of $M$ should produce a directed-graph-with-a-marked-sink $G_x$ ...
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1answer
177 views

Applications of opetopes

I've been reading about coherence problems in homotopy type theory (regarding semisimplicial sets and a raw syntax interpreter), and I've seen a remark about higher-dimensional operads perhaps being ...
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2answers
1k 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 ...
3
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1answer
236 views

Defining (infinity,1)-categories in HoTT using only an interval type

In this article, Emily Riehl and Michael Shulman describe a type theory in which one can do $\infty$-category theory synthetically. Their framework allows them to define simplices $\Delta^n$, and a ...
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495 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. ...
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2answers
617 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 ...
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95 views

Inductive type constructors with the defined type appearing in non-strictly positive position

In the HoTT book §5.6 ‘The general syntax of inductive definitions’ there is a proof that the existence of an inductive type $T$ with a constructor $t : ((T \to \mathsf{Prop}) \to \mathsf{Prop}) \to T$...
5
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1answer
112 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
150 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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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 ...
4
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1answer
248 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
370 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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164 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. ...
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1answer
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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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2answers
637 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 ...
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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
324 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
336 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
946 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
1k views

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
414 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
628 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
750 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
140 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
1k 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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0answers
127 views

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
594 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
344 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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0answers
1k 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
261 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
420 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
285 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 ...
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1answer
137 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
183 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
288 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
224 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
254 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
415 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 ...