Questions tagged [homotopy-type-theory]
Homotopy Type Theory refers to a new interpretation of Martin-Löf’s system of intensional, constructive type theory into abstract homotopy theory.
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Identity types and universes
Let us consider Martin-Löf type theory with a cumulative hierarchy of universes
$$
\mathcal{U}_0\colon\mathcal{U}_1\colon\ldots
$$
If $A, B\colon \mathcal{U}_i$, we can form an identity type $A=_{\...
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0answers
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If you can have cyclic base types, or if they need to be infinite types
I am confused how to properly think about classes of classes. Basically, you can have a dog "filo" which is an instance of the dog "class". But the dog class is itself not an instanceof of "animal", ...
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1answer
228 views
What should we return when pattern matching on a Higher Indutive Type and the case is a Path?
Context: Cubical Type Theory
Consider a simple HIT, say, an HitInt:
...
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1answer
47 views
Interval extensionality?
For example, in the proof of lemma 6.4.1 in the HoTT book, a function inductively defined over a function is simply applied on paths loop and ...
5
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1answer
50 views
Why is reflexivity enough in this HoTT formulation of quotient types?
In this formulation of quotient types in cubical type theory I was able to implement an eliminator and use that to implement basically curried function application. However, in all this, the only ...
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Flawed argument in the proof of function extensionality in cubical type theory?
I am reading the lectures about cubical type theory in this github repo. In lecture 1 the author defines function extensionality the following way:
...
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1answer
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Can you automatically generate a parser for a type using type theory some how?
Was wondering since all the types are spelled out constructively, and the constructions can all be reflected symbolically on a computer, if you can automatically parse expressions in a type?
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1answer
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Show how lack of universe levels would create contradiction in homotopy type theory (in Agda)
The homotopy type theory book claims in section 1.3 that "As in naive set theory, we might wish for a universe of all types" but from this one could "deduce from it that every type, including the ...
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1answer
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When do we need U(n+2) to solve a problem that can be formulated in U(n)?
I understand the need for a universe hierarchy, and that each new level brings additionnal proof-theoretic strength.
In the HoTT book there are examples of proofs that need to use the next level in ...
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univalence and decidability
I wonder if the following theorem is correct.
Let T be a type universe with the univalence axiom.
Then one has the following theorem:
If it is decidable in T if for each A,B:T, a map f:A-> B is an ...
5
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1answer
133 views
Continuing the example of why it's hard to compute with univalence?
In "Towards a cubical type theory without an interval" Altenkirch and Kaposi motivate why it is hard to compute with univalence. They say:
For example, we can define the equivalence ...
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2answers
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Reducing products in HoTT to church/scott encodings
So I am currently going though the HoTT book with some people. I made the claim that most inductive types we will see can be reduced to types containing only dependent function types and universes by ...
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Cubical type theory for dummies?
I read one of those popular papers on cubical type theory, but no wonder I could only see formulas and diagrams without being able to recognize them at all.
So here's what I want. I want a deep ...
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1answer
47 views
How do the unit intervals compose when generating higher order homotopies?
I'm trying to apply homotopy type theory as a theoretical foundation for describing how database query plans solve relational algebra queries. I'm modeling the problem as, given a set of input ...
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1answer
126 views
Is it possible that the universe of types could be closed?
I asked a pretty vague question. I wasn't able to make it precise, but I can now. It seems to be out of the scope of the previous question, so I open another one.
In dependently-typed languages such ...
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1answer
128 views
Why is `map insertionsort` not to equal to`map mergesort`?
In the type theory podcast ep. 3, Dan Licata claims that the fact that for every input, insertionsort and mergesort give the same result does not imply that the result would be equal when used as ...
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Meaning of the univalence axiom on plain data types
I'm trying to wrap my head around HoTT and can't figure out the intuitive meaning and validity of the univalence axiom. IIUC this axiom says:
...
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1answer
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How to understand equivalence of indexes of a family of types that are not definitionally equal
So I've been reading things about HoTT and trying to get solid on the foundations before getting too much further into the book. I am confused by a certain point; maybe I just haven't read far enough ...
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1answer
86 views
Constructing a sphere ($S^2$) in HoTT directly?
(this is a repost of https://stackoverflow.com/questions/29802501/constructing-a-sphere-s2-in-hott-directly, which was voted out of SO)
I understand the construction of $S^2$ as a suspension of $S^1$ ...
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4answers
691 views
Is path induction constructive?
I'm reading through the HoTT book and I have a hard time with path induction.
When I look at the type in the section 1.12.1:
$$\text{ind}_{=_A}:\prod_{C:\prod\limits_{x,y:A}(x=_Ay)\to \mathcal{U}}
\...
11
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1answer
370 views
Can I have a “dependent coproduct type”?
I'm reading through the HoTT book and I have a (probably very naive) question about the stuff in the chapter one.
The chapter introduces the function type
$$ f:A\to B $$
and then generalizes it by ...
7
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1answer
605 views
What were the research outcomes of the Univalent Foundations Program year (Homotopy Type Theory)
The Institute for Advanced Study has had a year-long special program devoted to the Univalent Foundations Program.
At the end of this they have produced a book and a code repository.
At the end of ...
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2answers
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Universes in dependent type theory
I am reading about dependent types theory in the Homotopy Type Theory online book.
In section 1.3 of the Type Theory chapter, it introduces the notion of hierarchy of Universes: $\mathcal{U}_0 : \...
