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Questions tagged [homotopy-type-theory]

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1
vote
1answer
129 views

What is the coproduct: A + A? [closed]

In the HoTT book, it is said The type of booleans 2 : U is intended to have exactly two elements. It is clear that we could construct this type out of coproduct and unit types as 1 + 1. I don't ...
15
votes
2answers
515 views

In the Hott book, are the most of the type formers redundant? And if so, why?

In chapter 1 and Appendix A of the Hott book, several primitive type families are presented (universe types, dependent function types, dependent pair types, Coproduct types, Empty Type, Unit type, ...
6
votes
4answers
484 views

Do I have to give up the Law of the Excluded Middle in order to Learn $\lambda$-Calculus?

I know very little about what I am talking about in what follows, so I appreciate any all help in pointing out all of my mistakes -- otherwise I won't be able to learn more and advance in my knowledge ...
13
votes
3answers
569 views

What are the negative consequences of extending CIC with axioms?

Is it true that adding axioms to the CIC might have negative influences in the computational content of definitions and theorems? I understand that, in the theory's normal behavior, any closed term ...
5
votes
2answers
231 views

In homotopy type theory, is there a IsManifold predicate?

Any type $A: \mathcal{U}$ can be thought of as a homotopy type, or a sufficiently nice topological space up to homotopy equivalence. Now, manifolds are topological spaces with some extra structure. ...
4
votes
0answers
291 views

Category theory in plain MLTT

I want to define a category in simple MLTT (not in HoTT). I defined it with the help of setoids. I.e. category consists of: a type of objects with equivalence relation (Obj : Set) a type of arrows ...
7
votes
1answer
522 views

Squash type vs Propositional truncation type

Homotopy type theory has a notion of propositional truncation type. It seems to me that it's strongly related to a notion of squash types. (See https://www.cs.kent.ac.uk/people/staff/sjt/TTFP/ttfp.pdf ...
16
votes
1answer
1k views

Formalizing Homotopy Type theory in Idris

Looking at the homotopy type theory blog one can easily find a lot of library formalizing most of Homotopy Type Theory in Agda and Coq. Is there anyone aware if there is any similar attempt to ...
6
votes
1answer
268 views

How Univalence can be used for proofs about algorithm correctness

I read a book on homotopy type theory. HoTT has the univalence axiom. This axiom seems to simplify working in category theory, but which other fields of mathematics it simplifies? I.e. how can I use ...
9
votes
1answer
122 views

Relating univalence for a theory of cateogries to the skeleton concept

Say I work in homotopy type theory and my sole objects of study are conventional categories. Equivalences here are given by functors $F:{\bf D}\longrightarrow{\bf C}$ and $G:{\bf C}\longrightarrow{\...
10
votes
1answer
1k views

Homotopy type theory and Gödel's incompleteness theorems

Kurt Gödel's incompleteness theorems establish the "inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic". Homotopy Type Theory provides an alternative ...