As of May 31, 2023, we have updated our Code of Conduct.

# Questions tagged [category-theory]

Category theory is used to formalize mathematics and its concepts as a collection of objects and arrows (also called morphisms). Category theory can be used to formalize concepts of other high-level abstractions such as set theory, ring theory, and group theory. (By Steve Awodey)

120 questions
Filter by
Sorted by
Tagged with
1 vote
30 views

### Is there some math problem that must use category theory to solve? [closed]

If there is no calculus，we can't solve a problem like “surface area”. Is there some math problem that can be solved thanks to category theory? If there is no such math problem, it means that category ...
78 views

### Resources for connections between dependent type theory and LCCC

Can someone recommend introductory articles/papers on the connections between dependent type theory and locally cartesian closed category? Many Thanks!
66 views

### $\beta$ reduction equational equality

In Categories, Types and Structures, authors talk about exponential objects in section 2.3.1. Let $C$ be a Cartesian category, and $a,b \in Ob_C$. The exponent of $a$ and $b$ is an object $b^a$ ...
884 views

### What exactly is the relation between Haskell and category theory?

In articles or Quora posts about category theory, I often find mentions of the programming language Haskell. I have little knowledge of category theory and even less of programming. Could someone ...
132 views

### Applying FP/Categorical terminology to non-FP languages

In my continuing effort to finally wrap my brain around advanced FP/categorical concepts, I've been reading dozens of articles and tutorials; what I have concluded is that: 1) Category Theory and ...
489 views

### Functional Programming and Category Theory

I'm a math Ph.D. having done research in Algebraic Geometry and Algebraic Topology in grad school for my thesis and I've studied a fair amount of category theory in the process (e.g. having worked ...
39k views

### Is Category Theory useful for learning functional programming?

I'm learning Haskell and I'm fascinated by the language. However I have no serious math or CS background. But I am an experienced software programmer. I want to learn category theory so I can become ...
52 views

### Goedel's theorem, halting problem and irreducible complexity

I have a vague idea on the tip of my mind that I can only convey through examples. Gödel's theorem states that some systems (ZFC, for example) are always incomplete in the sense that new axioms (which ...
31 views

### Ambiguous type of "triangle" operator for sum types

In Meijer, Fokkinga and Patersons "Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire" the ∇ operator for sum types is introduced which removes the tags from its ...
1 vote
46 views

### On the logical and categorical interpretation of lambda calculi and type systems

There is a well-known Curry-Howard-Lambek correspondence between certain type systems, proof calculi and categories. Some variants of Barendregt's pure type systems have the property of strong ...
328 views

### Rigorous proof that parametric polymorphism implies naturality using parametricity?

This question asks for an informal explanation of why all polymorphic functions between functors are natural transformations (This is a claim made by Bartosz Milewski). One answer to that question ...
638 views

### Functor laws and natural transformations in Haskell

As I've been struggling to get a deeper understanding of monads in Haskell, I started reading about functors and their counterparts in category theory. Keep in mind that I have no background in the ...
46 views

### what are all the ways of delimiting blocks

To my knowledge, in block-structured programming languages, there are 2, maybe 3 main ways of delimiting a block. Using start and end tokens, this can be brackets or reserved words etc Using ...
75 views

### Can we somehow get functoriality from purely type-theoretic reasoning?

In this question, I asked about how to prove naturality from parametric polymorphism, using parametricity. The current answer to that question simply assumes that the functors in question satisfy the ...
485 views

### Why are the laws of an applicative functor defined the way they are?

Let's recall the definition of an applicative functor. Throughout this question, I write $x: T$ to denote that the value $x$ has type $T$. Definition: An applicative functor consists of a type ...
150 views

### The second Functor law is redundant, but I don't understand the proof

When we defining a Functor instance in Haskell, it should satisfy the following two laws: fmap id = id ...
63 views

### Does category theory only deal with immutable objects? If so, why?

IIUC, category theory only applies to immutable objects, and mutability is modelled within that using e.g. functors, monads. Is that true? If so, why doesn't category theory include immutability? Has ...
5k views

### What exactly is the semantic difference between category and set?

In this question, I asked what the difference is between set and type. These answers have been really clarifying (e.g. @AndrejBauer), so in my thirst for knowledge, I submit to the temptation of ...
32 views

272 views

### Set theory pertaining to category theory and functional programming

I'm reading an unfinished Introduction to Category Theory/Products and Coproducts of Sets and have come across the following: A power set of a set is the set of all its subsets. A script 'P' is used ...
1 vote
35 views

### Interpreting Minimal STLC using a $\lambda 1$ Category

On page 139, example 2.4.5 of "Categorical Logic and Type Theory" by Bart Jacobs demonstrates the interpretation of the abstraction typing rule with respect to a $\lambda 1$ category. ...
90 views

### Meaning of Free (Arbitrary Abstract Algebra Term)

I'm currently learning abstract algebra and the word free appears (free monoid, free vector space) throughout different literatures. Is there a general (and simple) definition of the word (and ...
1 vote
49 views

### Is there a term for the inverse of a fixed-point operator?

When working with recursion it is often useful to find the least or greatest fixed points of a morphism, often using a fixed-point combinator. When working with recursion schemes, the inverse ...
75 views

### Mapping a free variable of type A to a cartesian closed category

In From Lambda Calculus to Cartesian Closed Categories, the author explains the interpretation of lambda calculus in cartesian closed category and at one point he explains how a term representing a ...
54 views

### What does point-free mean mathematically?

Point-free style is generally taken to mean a style of programming without explicit variables. I have some intuitions on point-free style but I want to know what the formal mathematical definition is. ...
71 views

### The word "algebra" in category theory

I am currently learning category theory and a saying that I see a lot is that X is the algebra of something (e.g. Monoid is an algebra of something). Can someone explain to me what that means? Thanks!
1k views

I have a question about concept of monad used in Haskell programming and category theory in math. Recall in Haskell a monad consists of following components: A type constructor that defines for each ...
82 views

### What is the category theory interpretation of higher order abstract syntax?

Suppose you have a simple sort of lambda calculus abstract syntax tree. The fine details don't really matter. ...
94 views

### Can we think of a non-symmetric product type in Haskell?

Meta note: I asked this question here a while ago. It got an answer: type a /\!! b = (a, ((b -> Void) -> Void)) Unfortunately, I do not reckon it to be ...
1 vote
232 views

### Has category theory had an impact on computer science?

I've only learned bit of category theory, but so far its relation to type systems seems mostly descriptive. For example, you really don't need to know about coproducts to come up with the idea of ...
163 views

### What's the internal language of the opposite of a Cartesian closed category?

I have heard the simply typed lambda calculus is the internal language of Cartesian closed categories. What's the internal language of the opposite type of category? The rules dual to currying and ...
1 vote
162 views

### Functor in category theory: The free theorem for fmap

According to nLab article: https://ncatlab.org/nlab/show/functor Definition External definition A functor $F$ from a category $C$ to a category $D$ is a map sending each object $x \in C$ to an object ...
1 vote
34 views

### Can I get the lambda calculus for free from Cartesian closed categories?

I have heard the simply typed lambda calculus is the internal language of closed Cartesian categories. I have a written a compiler from the STLC to CCCs but this involves a lot of ugly tuple shuffling....
196 views

### Can lists be defined in a special way so that they contain things of different type?

In https://www.seas.harvard.edu/courses/cs152/2019sp/lectures/lec18-monads.pdf it is written that A type $\tau$ list is the type of lists with elements of type $\tau$ Why must a list contain ...
1 vote
101 views

### Datatypes as initial algebras

I'm refining my understanding of the connection between initial algebras and datatypes. This paper suggests that one could even represent the categories corresponding to datatypes definitions: as ...
98 views

The bounty above should read 'I would like to know whether the example I discuss is a com-monad and why (why not).' Suppose we set $\mathbb{M} \alpha := r \to \alpha$, where $r$ is some fixed type, ...
1 vote
27 views

### Constructing a monad via type synonyms of a particular kind

We can define a reader/environment monad on the simply-typed lambda calculus, using the following three equations, where $r$ is some fixed type, $\alpha$ is any type (I subscript some terms with their ...
905 views

### A monad is just a monoid in the category of endofunctors, what's the enlightenment?

Pardon the word play. I'm a little confused about the implication of the claim and hence the question. Background: I ventured into Category Theory to understand the theoretical underpinnings of ...
524 views

### Given A to C, and B to C with known complexities, what can be said about A to B?

Say I have two sets of values $A$ and $B$ and for each set I have a computable function from that set to a third set $C$. Now suppose that I want to construct a function from $A$ to $B$, such that if ...
293 views

### Describing the Bool-And monoid in terms of categories

Normally I put the question before the context, but in this case I want to admit the possibility that the context and my understanding nullify the question. Plus it helps me think through my question. ...
1 vote
56 views

### What are $(S,\Sigma)$-CCCs?

I was reading this and I was trying to understand the definition of $(S,\Sigma)$-CCC. The first requirement says: a mapping [[_]] : S → |C|, associating some object [[s]] ∈ |C| to any s ∈ S; ...
1 vote
44 views

### Is the identity functor a kind of free object?

My understanding of free objects is: Free functors, free applicatives, free monads, free monoids, &c, give you more structure "for free", i.e. in general, or for all some thing with less structure,...