3
votes
0answers
32 views

What is the desirable function identification when setting up arrows in the category of types?

My question is which functions can not be allowed in a statically typed programming language, so that the "canonical" category is less coarse than what you get if you define it's arrows to be ...
0
votes
0answers
52 views

Continuations vs. Yoneda

There is this observation (worldpress blog) that the continuation monad (Wikipedia) stems from the Yoneda embedding. For fixed data types $R$, the monad maps data types $T$ to types $(T\to R)\to R$. ...
1
vote
1answer
64 views

Typed Category Theory?

This small book (or long paper) describes objects and morphisims in a cateogry as having types. He also talks about pre-categories as categories with typeless objects and morphisims. While I am ...
5
votes
3answers
90 views

Types, Sets and Categories

I am learning Category Theory and at first I just pictured a category like a class in object oriented programming: type definition + methods (morphisms). However the author I am reading uses maps ...
2
votes
1answer
80 views

Do we lose everything, if the natural transformations in a monad are exactly inverse?

I'm currently explaining monads $$T:{\bf C}\to{\bf C},\hspace{1cm}\eta:1_{\bf C}\to T,\hspace{1cm}\mu:T\circ T\to T,$$ to my brain and the "only" tricky thing are really the identity relations. I ...
3
votes
0answers
48 views

Logic in closed symmetric monoidal categories; reference request.

Suppose we want an algebraic theory $T$ to be interpretable in any closed symmetric monoidal category $\mathbf{C}.$ I am thinking in particular of the case where $\mathbf{C}$ is the category of models ...
0
votes
1answer
25 views

A class of “internal” endofunctors in cartesian closed categories

I'm interested in a class of endofunctors on cartesian closed categories with a quite natural definition, and am wondering whether/where this class has been studied so far (and how it is called). Fix ...
4
votes
1answer
86 views

How to describe free magmas in more structuralist terms?

Given a generating set $G$ (assume for simplicitly it consists entirely of urelements), the free magma on $G$ can be described concretely as follows. Its underlying set is the least $U \supseteq G$ ...
4
votes
2answers
192 views

How to understand the definition of sets in homotopy type theory and the role of univalence?

Bear with me, I'm a physicist. In homotopy type theory, as I understand it, a type $X$ is a set if all the morphisms over its terms $x:X$ are identies. When I say "morphisms", then I view the term as ...
5
votes
1answer
196 views

Foundational theories, their uses, interactions and comparisons?

Until now, I heard that there are some theories for building mathematical objects (or at least it is what it seems to my poor knowledge). Some of these are: Set theory; Logic; Category theory; Type ...
12
votes
3answers
465 views

Why is it worth spending time on type theory?

Looking around there are three candidates for "foundations of mathematics": set theory category theory type theory There is a seminal paper relating these three topics: From Sets to Types to ...
4
votes
0answers
102 views

W-types and inverse image functor

All sheaf topoi have W-types and in fact there's an explicit construction given by Benno van den Berg & Ieke Moerdijk, but the construction is quite involved. I would like to know whether the ...
4
votes
2answers
144 views

Is higher order type theory the same as higher order logic?

The internal language of a topos is higher order intuitionistic type theory (or logic). Here the higher order simply refers to allowing function types. In mathematical logic we have higher-order ...
4
votes
2answers
165 views

Eilenberg Moore category

I've been trying to code up the Eilenberg-Moore category for a monad in Haskell. As I understand it, given a category $C$ and a monad $(T,\eta,\mu)$ on $C$ we build the Eilenberg-Moore category $C^T$ ...
2
votes
2answers
246 views

Has a Dependent Type always a Type?

I am experimenting with dependent types. Lets assume the following short notation: ...
2
votes
2answers
631 views

Constructing dependent product (right adjoint to pullback) in a locally cartesian closed category

I've been trying to find a proof that the pullback functors in a locally cartesian closed category have right adjoints (used to model the notion of indexed product inside a category (rather than ...