# Questions tagged [ct.category-theory]

Questions in category theory

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### Functor Image example in Roman’s CT book

In his book, An Introduction to the Language of Category Theory,Steven Roman provide us with the following diagram: Then, he states: Note that in this example, the object part of F is not injective, ...
142 views

### What is the "standard" lambda-calculus model for bicartesian closed categories?

(I'm familiar with the lambda-calculus, less so with its categorical models.) It is well-known that cartesian-closed categories are in tight correspondence to the simply-typed lambda-calculus with ...
94 views

### Kleisli-like category for applicatives?

I am wondering if there is a good way to complete the following analogy: monad : Kleisli category :: applicative functor : ?? That is, a given monad T on a ...
84 views

### Relating functors to relational functors with the parametricity translation

$\newcommand{\Type}{\text{Type}}\newcommand{\id}{\text{id}}\newcommand{\map}{\text{map}}$ In attempting to answer the question: Rigorous proof that parametric polymorphism implies naturality using ...
94 views

### CCCs, computational calculi and point-surjectivity

The models of some computational calculi are in a correspondence with Cartesian Closed Categories with an object $U$ that has some relationship to its exponential object $U^U$ e.g. a retraction ...
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### "Interesting" categories whose internal logic is a dependent-linear type theory

Dependent-linear type theories may be a functional programmer's dream, but is it categorically interesting, i.e. is it the internal language of an "interesting" category? By "...
1k views

### Proof relevance vs. proof irrelevance

I want to use use Agda to help me write proofs, but I am getting contradictory feedback about the value of proof relevance. Jacques Carette wrote a Proof-relevant Category Theory in Agda library. But ...
124 views

### Given a domain, how do we build a language whose denotation is the domain?

Say we have an arbitrary domain $D$ with a countable basis $B$. Now, how do i build a "language" whose "denotation" lives in the domain? My understanding is that Dana Scott ...
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### Forming ordered pairs using monads and doing without the Kuratowski encoding of ordered pairs

Suppose we have a set $S$ of constants of the Simply-Typed Lambda Calculus (STLC) various types, and the operation of union $\cup$ which takes two constants and forms their union. For example, $S$ ...
78 views

I'm unsure whether the following three equations constitute a valid instance of a reader/environment monad on the simply-typed lambda calculus, where $\alpha$ is any type (I subscript some terms with ...
150 views

### Category theory lambda cube?

If simply typed lambda calculus corresponds to cartesian closed categories, what types of categories do other calculi in the lambda cube correspond to? https://en.m.wikipedia.org/wiki/Lambda_cube
209 views

### Categorical equivalent of higher order logic

From Simply typed lambda calculus and higher order logic, I get the impression that HOL is STLC + equality + equality axioms. I was wondering if there is a particular kind of category modelling this.
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### Does the Hindley-Milner type system (i.e. STLC with prenex polymorphism) have a category-theoretic model?

It is well known that any CCC (cartesian closed category) is a model of the simply-typed $\lambda$-calculus. It is less well known that System F admits a categorical model, but it is also well studied ...
111 views

### Are all computable functions monoidal from a category theory POV? [closed]

I have been studying some category theory only to understand the "why" behind the various categorical constructs we use ("borrow") in the world of functional programming. One of ...
151 views

### On the interpretation of coinduction in type theory

The notion of (co)datatype can be modeled satisfactorily in category theory as fixed-points of polynomial functors. Then, (co)induction principles are derived from initial algebras/terminal ...
281 views

### Can Category Theory help us prove P != NP?

Scott Aaronson, in this funny April Fools’ Day post, introduces a fictionalized $P \neq NP$ Proof and, among other things, he says that the proof make use of Higher topos theory to solve the biggest ...
540 views

### What category are Tagless Final Algebras final In?

The Haskell and Scala community have been very enamored recently with what they call tagless final 'pattern' of programming. These are referenced as dual to initial free algebras, so I was wondering ...
2k views

### Category-theoretic treatment of diffs, patches and merging?

Is there a category of patches that looks roughly like this: The objects are strings in some base alphabet The morphisms are edit scripts ("diffs" or "patches") between the strings I'm interested in ...
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### Implementing "Internal" Languages

One of the most practical consequences of the "Curry-Howard-Lambek" correspondence is that the syntax of many lambda-calucli/logics can be used to perform constructions in a sufficiently structured ...
305 views

### A bicartesian closed category of strict complete partial orders (Hask)

It seems to be well-known that programming languages can't have sums, products and nontermination together. Q1. Is this true? Below (or in the above link I gave) is a partial argument. However, ...
2k views

### Status quo of category theory and monads in theoretical computer science research?

Background. I am a bachelor student who is interested in research related to category theory, monads and Haskell, and I want to find a topic for my bachelor’s thesis in that area. I have looked at ...
83 views

### Categorical way of factoring out points

Major rewrite justifiably asked for: I'm currently trying to get a categorical way of doing something called the Gelfond-Lifschitz reduct on a set of single-headed Horn clauses. The semantics is the ...
321 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 ...
262 views

### How can I formalize key value stores with set theory? [closed]

I'm currently developing a simple key-value NoSQL store and want to build its formal model. I'm interested in knowing if there some work about formalization of key-value stores outside of category ...
250 views

### What are the morphisms of Adj(C,T) - the category whose objects are the adjunctions of a given monad?

The Wikipedia page for Monad says just that for a monad $(T,\eta,\mu)$ we can define the category of all adjunctions that define the monad: Let ￼$\textbf{Adj}(C,T)$ be the category whose objects ...
As I understand, a monoid $M$ is just a special kind of category; it has only one object and its morphisms can be composed through a composition operation. In principle, if $a$ is the object of the ...