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)
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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 ...
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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 ...
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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 ...
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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 ...
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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
...
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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 ...
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Exercise from Algebra of Programming on catamorphisms
This is an exercise from Algebra of Programming which I'm reading for self-study.
Below, $T$ is the initial algebra of $F$.
What I've tried:
I can construct $h \circ F (\pi_2) : F(A \times B) \...
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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 ...
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Why does the CwF definition require a set of types under a context rather than a class of types?
In "Syntax and Semantics of Dependent Types" at the top of page 24, Martin Hoffman describes $\mathit{Ty}_{\mathcal C}(\Gamma)$ as the collection of semantic types under context $\Gamma$.
It ...
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Are monads a unification of a number of computer science concepts?
The following commentator writes:
Monads are a unification of a bunch of computer stuff, including sequencing, IO, non-determinism, state, concurrency and exceptions. When I say "unification&...
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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 ...
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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. ...
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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 ...
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What is name of type " Function->Value->Bool = if (Bool) Function (Value) " in Category theory?
I am very new to functional programming so sorry if the question is stupid.
Having this function
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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 ...
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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 ...
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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!
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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 ...
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Category Set language using simply typed lambda calculus
I am currently self learning Category Theory and Simply typed lambda calculus (STLC). For learning purposes, I have implemented an STLC interpreter as given in Types and Programming Languages book ...
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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.
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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!
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Monad in Haskell programming vs. Monad in category theory
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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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....
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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.
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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 ...
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Creating a large tuple from smaller tuples via a monad or applicative
Suppose I have a term $a :\alpha$ of the Simply-Typed Lambda Calculus (in the following, $\alpha, \beta, \gamma$ stand for arbitrary types) and I want to lift it to a term
$\lambda x_{\beta}. \;(x, \, ...
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(Co)-monads and terminating implementations
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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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.
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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,...
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Is there any correspondence between SUM type in type theory and arithmetical summation?
Is there any correspondence between the coproduct(sum) type in type theory and arithmetical summation?
For example what does 3+4 or x+6 mean in type theory?
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Is this possible: In OOP, private methods in a class form a F-coalgebra and public methods in a class form an F-algebra?
I recently found out that OOP classes turn out to be F-coalgebras:
https://www.semanticscholar.org/paper/Objects-and-Classes%2C-Co-Algebraically-Jacobs/c7c45abf7d99e0aef627fd5223023bf82e70dc71
The ...
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Why are all polymorphic functions between functors natural transformations?
Bartosz Milewski's Category Theory for Programmers says the following:
A parametrically polymorphic function between two functors (including the edge case of the Const functor) is always a natural ...
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What is the maths name for a set which contains the Domain and Codomain of a function? [closed]
Im interested in this so that I can name a type parameter in a program I'm writing.
There is function that that has three parameters.
D, Domain
C, Codomain
X, where D is a subset of X and C is a ...
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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 ...
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Can the most general unifier be defined categorically?
I remember from my reading of Category Theory for Computing Science that classical concepts like weakest preconditions can be seen as the categorical notion of pullback.
I was wondering if the same ...
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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 ...
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Why does the category of language types have morphisms, not functors?
I am probably not phrasing this question well, so please bear with me as I try to explain what I mean.
I am working on learning category theory, as applied to programming. So far, I understand that:
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Proof of the limit-colimit coincidince
Note: I figured this out, but haven't had the time to write an answer for it: see the comment.
For reference: the discussed material appears in http://www.cs.ru.nl/B.Jacobs/CLG/JacobsCoalgebraIntro....
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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 ...
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Understanding Isabelle's implementation of coinduction
I'm studying how coinduction was encoded in Isabelle. At page 7 of the attached document, the author describes how some datatypes can be encoded as initial algebras. Here is one example:
Finite lists
...
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How is substitution in type theory the composition of classifying morphisms in category theory?
In the article at nlab about relation between category theory and type theory, it is said that substitution in type theory is the same as composition of classifying morphisms in category theory.
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Is, beta reduction in type theory being considered as counit for hom-tensor adjunction in category theory, a denotational or operational semantic?
In the article at nlab about the relation between type theory and category theory, it is said that "beta reduction" in type theory corresponds to "counit for hom-tensor adjunction" in category theory ...
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