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6
votes
0answers
130 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 ...
5
votes
1answer
75 views

On an automatic translation of typed lambda calculus in untyped lambda calculus

I have a question regarding the "compilation" of typed lambda calculus in untyped lambda calculus. Take for example the inductive definition of lists, with introduction rules: and: We can ...
1
vote
0answers
62 views

Type theory: can multiple elimination rules be defined, in principle?

I'd like to ask a question on type theory: Consider the usual type theoretical definition of the natural numbers. We could give an elimination rule in the form: or in the form: I called the ...
7
votes
2answers
499 views

What does “simplification of proofs as evaluation of programs” mean?

I am currently going through Philip Walder's "Proposition as Types" and a passage of the introduction has struck me: for each way to simplify a proof there is a corresponding way to evaluate a ...
3
votes
0answers
78 views

Problem with a proof in Wellfounded trees in categories

I'm reading the paper Wellfounded trees in categories by Moerdijk and Palmgren. I'm having trouble understanding the proof of Theorem 7.2 (page 216), i.e. that in $\mathbf{ML}_{< \omega} ...
7
votes
1answer
310 views

Constructing unnatural transformations

In a nutshell, the question is: is it true that any explicit (not involving axiom of choice) pointwise transformation between sufficiently complicated functors is natural almost everywhere? Let $C$ ...
23
votes
1answer
791 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: ...
3
votes
0answers
94 views

Logical framework for type theories like ML and CIC

I'm looking for a logical framework in which it is possible to easily present both intensional and extensional theories of dependent types with a partially ordered set of universes à la Russell ...
6
votes
0answers
122 views

Preservation of universes in presheaves

In Lifting Grothendieck universes Hofmann and Streicher construct a universe in the category of presheaves over a small category given a Grothendieck universe in $\mathbf{Set}$. Suppose now I have ...
6
votes
3answers
397 views

Extensionality in HoTT versus extensionality in internal language of a category

What's the extension of judgmental identity in HoTT (homotopy type theory)? The Martin-Löf intensional dependent type theory with identity types is called (definitionally) extensional if the ...
4
votes
2answers
227 views

Question about higher inductive types and computational rules

I have been trying to make my way through the homotopy type theory book, slowly but surely, and I just finished reading this introductory series of 3 articles on hott on ScienceForAll. ...
8
votes
2answers
285 views

How should I be thinking about object classifiers / universal fibrations / universes?

I have been learning about homotopy type theory this summer. I am not a homotopy theorist but I am more comfortable with homotopy theory than I am with type theory, so the way I rationalize many of ...
13
votes
3answers
506 views

Formal/rigorous treatment of (im)predicativity/predicativism

There are several places on the web where one may find quite intuitively understandable accounts of (im)predicativity; here on MO I found two questions with very good detailed answers (Predicative ...
3
votes
1answer
189 views

Type with $X\rightarrow X\cong X + 1$

In the article Voevodsky’s Univalence Axiom in Homotopy Type Theory, an example is given of how types are not like sets: the existence of a nontrivial (nonzero) type $X$ such that $X\rightarrow ...
5
votes
2answers
632 views

Why is there no product type in simply typed lambda-calculus?

Consider simply typed $\lambda$-calculus that has only the unit type as primitive. We would like to encode the product and the sum types. An encoding of the product type in the untyped ...
4
votes
2answers
244 views

Is there a name for relations with this property, and the category of them?

The following math.stackexchange question asked whether there is a name for a certain sort of relation. I have become interested in the question, and no one suggested a name there, so I am asking ...
12
votes
2answers
920 views

What is the status of (universal) algebra in type theory?

With the recent interest in homotopy type theory as a foundation for mathematics, it seems natural to develop algebra within the framework of type theory. So far, I can't find much literature ...
0
votes
1answer
165 views

Does “Church's Law” really fail in Extensional Type Theory?

Last year Bob Harper wrote a blog post about the failure of "Church's Law" in Extensional Type Theory[1]. However his statement of the law looks to me more like an internal version of the statement ...
6
votes
1answer
319 views

The independence of path induction

In §1.12 of the Homotopy type theory book, it is mentioned that indiscernibility of identicals is a consequence of path induction. More precisely, for each type $C$ dependent over a type $A$, there is ...
8
votes
2answers
1k views

A (very naive) question about Homotopy Type Theory

In homotopy type theory, homotopy types can be viewed as logical types and it is possible to prove some theorems about them without using any underlying space (no simplicial set, no topological ...
13
votes
2answers
664 views

$\infty$-categorical interpretation of type theory

One can read at several places that Martin-löf type theory should be the internal language of a locally Cartesian closed infinity category, and that the univalence axiom should distinguished infinity ...
16
votes
2answers
1k views

Prospects for reverse mathematics in Homotopy Type Theory

Reverse mathematics, as I mean here, is the study of which theorems/axioms can be used to prove other theorems/axioms over a weak base theory. Examples include Subsystems of Second Order Arithmetic ...
5
votes
1answer
234 views

$\Pi$, $\Sigma$, and identity types without $\eta$ in comprehension categories

In comprehension categories, dependent sums are defined as a choice of left adjoints for all reindexing functors along display maps, satisfying a Beck-Chevalley condition. Dependent products are ...
5
votes
0answers
177 views

Feasible Type Theories

I am looking for references about efficient type theories, efficiency in the sense of computational complexity, and type theory in the sense of Martin-Lof's type theories. Has there been any studies ...
9
votes
2answers
831 views

Equivalent form of the Univalence Axiom

I'm reading the new HoTT book and I'm wondering about a potential equivalent form of the Univalence Axiom: $(A \simeq B) \simeq (A = B)$. For simplicity, I'm tacitly working in a fixed universe. It ...
14
votes
3answers
767 views

How do we express measurable spaces using type theory?

A measurable space $(X,\mathcal X)$ consists of a set $X$ equipped with a $\sigma$-algebra of subsets $\mathcal X$. I would like to write computer programs involving measurable spaces, but to the best ...
7
votes
2answers
505 views

Connection between codata and greatest fixed points

This is, I'm afraid, another question that MSE couldn't answer. It's easy to see how inductively-defined data types correspond to least fixed points. Let's take the natural numbers as an example, ...
11
votes
1answer
1k views

Forcing in Homotopy Type Theory

I apologize if this question doesn't make any sense. I'll just go ahead and delete it if that's the case. But the question is just the title. Is there a notion of forcing in homotopy type theory? ...
0
votes
1answer
290 views

prod and sig in COQ

Hello, Apparently in COQ the type prod (with one constructor pair) corresponds to cartesian product and the type sig (with one constructor exist) to dependent sum but how is described the fact that ...
2
votes
1answer
170 views

Is there research on the notion of co-accessibility?

I want to start off with a disclaimer that I am only a mathematical amateur. Please forgive me for ignorance or any non-standard nomenclature I use here :) Let's start off with some context. Let X ...
1
vote
0answers
71 views

Natural relations between substitutions

Consider two contexts $\Gamma,\Delta$ (from some background type theory), and substitutions $s_1,s_2:\Gamma\rightarrow \Delta$. In the case of $1$-element contexts, we get that a substitution is ...
9
votes
2answers
587 views

Category of Judgements?

I have been able to find a lot of information on the category of contexts -- for example, the page on syntactic categories at the nLab is a good starting point. However, when I try to find similar ...
5
votes
1answer
644 views

intensional equality in type theory

I want to know why we add an intensional equality in type theory to definitional equality ? What is the aim with this intensional equality ? thanks
6
votes
2answers
277 views

Why no morphisms from the contradictory proposition to the inconsistent context?

Consider Higher order predicate logic over dependent type theory (DPL) as defined in Chapter 11 of B. Jacobs's book "Categorical Logic and Type Theory" (though I think this question applies to ...
9
votes
2answers
277 views

Categorical semantics of W-types

Jacob's book titled "Categorical Logic and Type Theory" gives a nice description of Π and Σ types as adjunctions to substitution functors induced by display maps. Is there a similar ...
8
votes
2answers
324 views

Reduction rules for inductive types

(I'm not sure if I should post this here rather than at Theoretical Computer Science, I've found a lot of type theory related questions on MathOverflow) I'm working in Martin-Löf type theory with ...
4
votes
1answer
340 views

What are categorical models of W-types in intensional type theory?

I'm familiar with container functors and older work by Dybjer on categorical models for W-types in the extensional theory, but I was looking for some similar semantics in the intensional case.
14
votes
3answers
2k views

What is the theory of polynomials?

We all know what polynomials are, along with their elementary properties and many effective algorithms for different representations of polynomials. The question here is more of a universal algebra ...
5
votes
1answer
951 views

categorifying induction in homotopy type theory

In trying to understand homotopy type theory, I stumbled upon the following silly question, which is likely to be trivial for the experts. Let $B=\sqcup_{n\in\Bbb N} BS_n$, which I'd like to think of ...
17
votes
3answers
1k views

Surreal Numbers as Inductive Type?

Prompted by James Propp's recent question about surreal numbers, I was wondering whether anyone had investigated the idea of describing surreal numbers (as ordered class) in terms of a universal ...
23
votes
6answers
4k views

How true are theorems proved by Coq?

Less tongue in cheek, is it known what the relative consistency is for theorems proved with an automatic theorem prover? Of course this depends somewhat on what assumptions one makes with respect to ...
4
votes
1answer
553 views

What fails when using call/cc as realizer of the Peirce formula

Define the axiom constants $p_{A,B}^{((A\rightarrow B)\rightarrow A)\rightarrow A}$ as realizers of the Peirce formula, and $f_A^{\bot\rightarrow A}$ as realizers of the Ex Falso Quodlibet. Then ...
-1
votes
5answers
804 views

finding cutting edge papers and books

Hi all, what are the best strategies to find cutting edge papers and books on a field of mathematics? .. Example: 2-3 years ago I had to analyze a time series. I found a paper and showed that to ...
5
votes
3answers
578 views

What do I call type theory without Curry-Howard?

Is there a word I can say which will convey to type theorists that I am not thinking about types as propositions? Background: as a category theorist, I am mostly interested in type theories as a ...
4
votes
3answers
497 views

Can a typing judgment admit essentially different derivations?

In any sort of type theory, there are a bunch of rules for constructing derivations of typing judgments such as $x:A,\; y:B(x) \;\vdash\; z:C(x,y)$. (I intend to include also judgments of the form ...
3
votes
0answers
194 views

An elegant formulation for typed sets

Fix a poset $T$, which we'll think of as a set of "types," interpreting $a \leq b$ as "$a$ is more general than $b$." Construct a category of TSet as follows. Objects: Pairs ($X$, $\tau : X ...
2
votes
1answer
249 views

Relation between different definitions of types [closed]

Is there any connection between the definition of type in model theory and the definitions from type theory? Is there any explanation why the same term is used for these notions, maybe in the ...
6
votes
4answers
683 views

What is the intuitive meaning of star and box in a pure type system?

The systems of the λ-cube have the axiom $\star:\square$. I've listed a few meanings that the Curry-Howard isomorphism gives to $t : T$ below. What are the intuitive meanings of $\star$ and ...
1
vote
2answers
923 views

What is a semigroup or, what do I do with that associativity proof?

Mathematically, I know what a semigroup is: It is a set S along with an associative binary operation $* : S \times S \rightarrow S$. So far, so good. From a computational perspective, one can ...
11
votes
4answers
863 views

Reference request for type theory

I am interested in learning the theory of types, especially in how they can provide a foundation to mathematics different to sets and how they can avoid self-referential paradoxes by stipulating that ...