Questions tagged [homotopy-type-theory]
For questions about homotopy type theory, including the homotopy theoretic interpretation of type theory or univalent type theory, univalence axiom, higher inductive types or related.
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Proof verification and hint request for HoTT book exercise 2.17
Exercise 2.17.
(i) Show that if $A \simeq A'$ and $B \simeq B'$, then $(A \times B) \simeq (A' \times B')$.
(ii) Give two proofs of this fact, one using univalence and one not using it, and show that ...
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Is building the universes for (∞-)toposes (of subsets of data points) a categorical generalization of theory induction in ML and algorithm?
I am reading https://arxiv.org/abs/1904.07004 "All (∞,1)-toposes have strict univalent universes" and I am thinking about its applications to the machine learning.
Can we recast the ...
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Is this the correct way to express primality in Martin-Löf type systems?
I am self studying type theory from the HoTT book. More or less as a way to test that I have the right idea I'm trying to see if I can express the idea of "Primes" in the theory. This is ...
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Definite description in homotopy type theory
I asked this question there and I have been suggested to ask it here.
In a paper by David Corfield, we have an account of definite description in homotopy type theory. The author gives the following ...
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Internal vs External in type theory
I'm learning type theory, and at one point in the HoTT book, is mentionned "external" constructions. I was wondering what precisely means internal/external in type thoery.
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What is a "type" in type theory?
Types are taken as atomic in type theories like homotopy type theory. But what is the best way to conceptualize what a type is? Is it appropriate to think of them as a property that defines a category?...
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Circularity in the definition of natural numbers with homotopy type theory
I am reading HoTT's book, I am interested in this theory because it is said that it works as a foundation of mathematics, so I want to see how this foundation works. I am interested in the definition ...
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For any module with vector set $V$ and scalar set $C$, must there exist a set $X$ such that $V$ and $(X → C)$ are isomorphic?
Here's some evidence that suggests the affirmative to my question. There exists an isomorphism between:
$\mathbb{R}^{n}$ and $(\mathbb{N}^{<n} \to \mathbb{R}$)
$\mathbb{R}^{\infty}$ and ($\mathbb{...
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HoTT and isomorphisms
I have heard that Homotopy Type Theory makes it so that isomorphic objects are “equal”. I wonder how this squares with a lot of mathematical examples from Algebra and Set Theory, where the nature of ...
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Why aren't $\infty$-groupoids commutative in HoTT?
I'm trying to read through HoTT, but I'm confused by the path induction principle, it seems too strong at the first glance. I tried "proving" that all suitable paths commute, and it looks ...
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Is the set-indexed wedge of connected $1$-types a $1$-type?
We are working in homotopy type theory.
Given a type $I$ and a family of pointed types $P : \prod i : I, \sum T : Type, T$, we can define the wedge product $\bigvee\limits_{i : I} P(i)$ as a certain ...
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What are the axioms of homotopy type theory?
The primitive notions of Zermelo-Fraenkel set theory are those of set and membership, i.e. we don't define what we mean by 'set' neither what we mean by '$\in$', rather, the axioms define what we mean ...
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LEM and the curry-howard correspondence
The curry-howard correspondence rests upon constructive/intuitionistic logic. Proof checkers only work because they are guaranteed to halt. Proof checkers are built on the simply typed lambda calculus ...
user590921
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Classicalities of Homotopy Type Theory
What are statements of HoTT that are not provable therein and thus may or may not be true in specific models, specifically models in $(\infty,1)$- topoi? I've also seen the term "classicalities&...
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What does it mean for a model category to present a higher category
A model category serves as an abstract/ axiomatic framework for homotopy theory. A higher category, in particular an $(\infty,1)$ category, I'm using the model of quasicategories, is a category with ...
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