#categorytheory — Public Fediverse posts

One of the miraculous proofs truly foundational of #categoryTheory - the proof that product (and any other object that is defined using universal property) is unique up to a (unique) isomorphism.

I have to remember to include it in #categorytheoryillustrated

https://proofwiki.org/wiki/Product_%28Category_Theory%29_is_Unique

New paper: "Valuation Without ℝ: A Category-Theoretic Foundation for Finite Measurement"

Can measurement be formalized independently of real numbers AND set theory?

We axiomatize "valuation" as a graded distinguishability map satisfying the ultrametric inequality. Dimension emerges as the growth exponent of the distinguishability graph.

Full paper: https://doi.org/10.5281/zenodo.21795656

#pAdic #QuantumFoundations #CategoryTheory #Measurement

A final monad theory blog post in the current series about transferring nice properties of monads along strong monad morphisms. Today, quotients:

https://stringdiagram.com/2026/08/07/quotients-of-commutative-affine-and-relevant-monads/

#categorytheory

Just a reminder, if you are about to take a category theory course next term/semester.

There is a page of links to reasonably accessible and (legally!) freely downloadable lecture notes etc. at https://logicmatters.net/categories. You could very well find there some helpful preliminary/parallel reading at a level that suits you.

At the gentler end, there are my own notes *Introducing Category Theory*, freely downloadable of course but also available as a minimum cost paperback. Some students do like the book!

If you giving a category theory course, please do alert your students to the links page https://logicmatters.net/categories -- they may well find it useful.

#CategoryTheory

Just learnt that our paper "A study of Kock's fat Delta" (https://arxiv.org/abs/2503.10963) with Nicolai Kraus (@Nicolai_Kraus), Simona Paoli and Stiéphen Pradal (@Stiephen) was accepted for publication in the Journal of Pure and Applied Algebra (JPAA).

I'm quite happy about this, but I'm especially pleased for Stiéphen who's currently working hard to write up his PhD thesis!

#categorytheory

While teaching a course on algebraic theories last semester I mentioned at some point that the familiar bicomplete categories Top and Poset were not algebraic, but were topological. A student asked about this later on and I found that the #Wikipedia page on «topological category» pointed to categories enriched in Top, not categories with a forgetful functor to Set which is topological. While I have lost a lot of faith in Wikipedia recently, I am pleased to report that https://en.wikipedia.org/wiki/Topological_category is now a disambiguation page.

#math #mathematics #topology #algebra #CategoryTheory

Cool writeup presenting universal properties as natural isomorphisms:

https://math.stackexchange.com/questions/3688510/understanding-universal-property-and-universal-element-from-category-theory-in

#categoryTheory

Random #categoryTheory fact: The Yoneda lemma can be seen (among 1000 other things) as a generalization of the definition that a set contains no structure, other than its elements.

Does anybody know how to typeset pullback/pushout corners in typst/fletcher?
The code provided at https://github.com/Jollywatt/typst-fletcher/issues/50#issuecomment-2851846670 does not quite work for me and even when it does, it requires experimenting with degrees which is bad.

[Update] I added my own workaround as a reply to the GitHub issue.

#typst #typesetting #categorytheory

🤦‍♂️ Ah yes, because what every overworked developer dreams of is an IDE that speaks the fluent gibberish of category theory. 🚀 Just sprinkle some "categorical magic dust" on your databases and watch them transform (or maybe just disappear into the void). Call Conexus AI for a personalized headache consultation. 📞🤯
https://categoricaldata.net/ #overworkeddevelopers #categorytheory #IDEs #ConexusAI #techhumor #HackerNews #ngated

📚🤖 "Behold! The magnum opus on #Rust and Category Theory that's sure to revolutionize your understanding of... nothing. 😂 Dive into this unfinished 'draft' if you're into the esoteric joy of turning simple concepts into convoluted gibberish. 🚀👨‍💻✨"
https://hghalebi.github.io/category_theory_transformer_rs/ #CategoryTheory #EsotericHumor #TechHumor #Programming #HackerNews #ngated

Building ML framework with Rust and Category Theory

https://hghalebi.github.io/category_theory_transformer_rs/

#HackerNews #ML #Rust #CategoryTheory #Framework #Development

Towards a Higher-Order Bialgebraic Denotational Semantics by Sergey Goncharov, @[email protected], @[email protected], Henning Urbat, and me has been (unconditionally) accepted at ICFP'26! Abstract below ​:Blobhaj_Read_Octopus:​

#icfp #functionalProgramming #programmingLanguages #semantics #coalgebra #categoryTheory

From 11:00 to 12:00 on Thursday, April 30, the PLUSLE reading group will discuss "Abstract Syntax and Variable Binding" by Marcelo Fiore, Gordon Plotkin, and Daniele Turi.

https://plsl.acp.sdu.dk/posts/2026-04-30-abstract-syntax-and-variable-binding/

#PLUSLE #syntax #programmingLanguages #categoryTheory #lambdaCalculus

Oh joy, another attempt at 'explaining' Category Theory using #orders, because clearly the first thing we all need is more #abstraction in our lives. 🎉 In a groundbreaking revelation, the article tells us orders are about #relationships. 🤯 Who knew binary relations could be so riveting? 🙄
https://abuseofnotation.github.io/category-theory-illustrated/04_order/ #CategoryTheory #HackerNews #HackerNews #ngated

Category Theory Illustrated – Orders

https://abuseofnotation.github.io/category-theory-illustrated/04_order/

#HackerNews #CategoryTheory #Orders #Illustration #Math #Education #Programming

Someone should make one of those "top 10 most satisfying" videos but for #CategoryTheory diagram chasing proofs

In the mountains, I love exploring nature and climbing any little hidden corner.

My academic interests include mathematics (#mathematics) and mathematical logic ( #mathematicallogic #logic), particularly model theory ( #modeltheory), category theory ( #categorytheory), higher-order logic and metalogic ( #metalogic), as well as the philosophy of mathematics and logic ( #philosophyofmathematics #philosophyoflogic), epistemology and ontology of them.

#introduction

Readings shared April 4, 2026. https://jaalonso.github.io/vestigium/posts/2026/04/04-readings_shared_04-04-26 #AI #AI4Math #ATP #Agda #AlphaProof #Autoformalization #CategoryTheory #CoqProver #FunctionalProgramming #ITP #IsabelleHOL #LLMs #LambdaCalculus #LeanProver #Lisp #Logic #LogicProgramming #LLMs #Math #Physics #Programming #Prolog #Racket #RocqProver #Vampire

Readings shared March 09, 2026. https://jaalonso.github.io/vestigium/posts/2026/03/10-readings_shared_02-09-26 #AI #AI4Math #CategoryTheory #CoqProver #Emacs #FunctionalProgramming #Haskell #ITP #IsabelleHOL #LambdaCalculus #LeanProver #Lisp #Math #Physics #RocqProver

Precursors Of Category Theory • 3
• https://inquiryintoinquiry.com/2024/05/27/precursors-of-category-theory-3-a/

❝Act only according to that maxim by which you can at the same time will that it should become a universal law.❞

— Immanuel Kant (1785)

C.S. Peirce • “On a New List of Categories” (1867)

❝§1. This paper is based upon the theory already established, that the function of conceptions is to reduce the manifold of sensuous impressions to unity, and that the validity of a conception consists in the impossibility of reducing the content of consciousness to unity without the introduction of it.❞ (CP 1.545).

❝§2. This theory gives rise to a conception of gradation among those conceptions which are universal. For one such conception may unite the manifold of sense and yet another may be required to unite the conception and the manifold to which it is applied; and so on.❞ (CP 1.546).

Cued by Kant's idea regarding the function of concepts in general, Peirce locates his categories on the highest levels of abstraction able to provide a meaningful measure of traction in practice. Whether successive grades of conceptions converge to an absolute unity or not is a question to be pursued as inquiry progresses and need not be answered in order to begin.

Resources —

Precursors Of Category Theory
• https://oeis.org/wiki/Precursors_Of_Category_Theory

Propositions As Types Analogy
• https://oeis.org/wiki/Propositions_As_Types_Analogy

Survey of Precursors Of Category Theory
• https://inquiryintoinquiry.com/2024/05/24/survey-of-precursors-of-category-theory-5/

#Aristotle #Peirce #Kant #Carnap #Hilbert #Ackermann #SaundersMacLane
#Abstraction #Analogy #CategoryTheory #Diagrams #FoundationsOfMathematics
#FunctionalLogic #RelationTheory #ContinuousPredicate #HypostaticAbstraction
#CategoryTheory #PeircesCategories #PropositionsAsTypes #TypeTheory #Universals

Precursors Of Category Theory • 2.3
• https://inquiryintoinquiry.com/2024/05/26/precursors-of-category-theory-2-a/

In the logic of Aristotle categories are adjuncts to reasoning whose function is to resolve ambiguities and thus to prepare equivocal signs, otherwise recalcitrant to being ruled by logic, for the application of logical laws. The example of ζωον illustrates the fact that we don't need categories to “make” generalizations so much as to “control” generalizations, to reign in abstractions and analogies which have been stretched too far.

References —

• Aristotle, “The Categories”, Harold P. Cooke (trans.), pp. 1–109 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

• Karpeles, Eric (2008), Paintings in Proust, Thames and Hudson, London, UK.

Resources —

Precursors Of Category Theory
• https://oeis.org/wiki/Precursors_Of_Category_Theory

Propositions As Types Analogy
• https://oeis.org/wiki/Propositions_As_Types_Analogy

Survey of Precursors Of Category Theory
• https://inquiryintoinquiry.com/2024/05/24/survey-of-precursors-of-category-theory-5/

#Aristotle #Peirce #Kant #Carnap #Hilbert #Ackermann #SaundersMacLane
#Abstraction #Analogy #CategoryTheory #Diagrams #FoundationsOfMathematics
#FunctionalLogic #RelationTheory #ContinuousPredicate #HypostaticAbstraction
#CategoryTheory #PeircesCategories #PropositionsAsTypes #TypeTheory #Universals

Precursors Of Category Theory • 2.2
• https://inquiryintoinquiry.com/2024/05/26/precursors-of-category-theory-2-a/

Aristotle —

❝Things are equivocally named, when they have the name only in common, the definition (or statement of essence) corresponding with the name being different. For instance, while a man and a portrait can properly both be called animals (ζωον), these are equivocally named. For they have the name only in common, the definitions (or statements of essence) corresponding with the name being different. For if you are asked to define what the being an animal means in the case of the man and the portrait, you give in either case a definition appropriate to that case alone.

❝Things are univocally named, when not only they bear the same name but the name means the same in each case — has the same definition corresponding. Thus a man and an ox are called animals. The name is the same in both cases; so also the statement of essence. For if you are asked what is meant by their both of them being called animals, you give that particular name in both cases the same definition.❞ (Aristotle, Categories, 1.1a1–12).

Translator's Note. ❝Ζωον in Greek had two meanings, that is to say, living creature, and, secondly, a figure or image in painting, embroidery, sculpture. We have no ambiguous noun. However, we use the word ‘living’ of portraits to mean ‘true to life’.❞

#Aristotle #Peirce #Kant #Carnap #Hilbert #Ackermann #SaundersMacLane
#Abstraction #Analogy #CategoryTheory #Diagrams #FoundationsOfMathematics
#FunctionalLogic #RelationTheory #ContinuousPredicate #HypostaticAbstraction
#CategoryTheory #PeircesCategories #PropositionsAsTypes #TypeTheory #Universals

Precursors Of Category Theory • 2.1
• https://inquiryintoinquiry.com/2024/05/26/precursors-of-category-theory-2-a/

❝Thanks to art, instead of seeing one world only, our own, we see that world multiply itself and we have at our disposal as many worlds as there are original artists …❞

— Marcel Proust

When it comes to looking for the continuities of the category concept across different systems and systematizers, we don't expect to find their kinship in the names or numbers of categories, since those are legion and their divisions deployed on widely different planes of abstraction, but in their common function.

#Aristotle #Peirce #Kant #Carnap #Hilbert #Ackermann #SaundersMacLane
#Abstraction #Analogy #CategoryTheory #Diagrams #FoundationsOfMathematics
#FunctionalLogic #RelationTheory #ContinuousPredicate #HypostaticAbstraction
#CategoryTheory #PeircesCategories #PropositionsAsTypes #TypeTheory #Universals

Precursors Of Category Theory • 1
• https://inquiryintoinquiry.com/2024/05/25/precursors-of-category-theory-1-a/

A few years ago I began a sketch on the “Precursors of Category Theory”, tracing the continuities of the category concept from Aristotle, to Kant and Peirce, through Hilbert and Ackermann, to contemporary mathematical practice. My notes on the project are still very rough and incomplete but I find myself returning to them from time to time.

Preamble —

❝Now the discovery of ideas as general as these is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from the philosophers: “Category” from Aristotle and Kant, “Functor” from Carnap (“Logische Syntax der Sprache”), and “natural transformation” from then current informal parlance.❞

— Saunders Mac Lane • “Categories for the Working Mathematician”

Resources —

Precursors Of Category Theory
• https://oeis.org/wiki/Precursors_Of_Category_Theory

Propositions As Types Analogy
• https://oeis.org/wiki/Propositions_As_Types_Analogy

Survey of Precursors Of Category Theory
• https://inquiryintoinquiry.com/2025/05/05/survey-of-precursors-of-category-theory-6/

#Aristotle #Peirce #Kant #Carnap #Hilbert #Ackermann #SaundersMacLane
#Abstraction #Analogy #CategoryTheory #Diagrams #FoundationsOfMathematics
#FunctionalLogic #RelationTheory #ContinuousPredicate #HypostaticAbstraction
#CategoryTheory #PeircesCategories #PropositionsAsTypes #TypeTheory #Universals

Survey of Precursors Of Category Theory • 5
• https://inquiryintoinquiry.com/2024/05/24/survey-of-precursors-of-category-theory-5/

A few years ago I began a sketch on the “Precursors of Category Theory”, tracing the continuities of the category concept from Aristotle, to Kant and Peirce, through Hilbert and Ackermann, to contemporary mathematical practice. A Survey of resources on the topic is given below, still very rough and incomplete, but perhaps a few will find it of use.

Background —

Precursors Of Category Theory
• https://oeis.org/wiki/Precursors_Of_Category_Theory

Propositions As Types Analogy
• https://oeis.org/wiki/Propositions_As_Types_Analogy

Blog Series —

Notes On Categories
• https://inquiryintoinquiry.com/2013/02/22/notes-on-categories-1/

Precursors Of Category Theory
1. https://inquiryintoinquiry.com/2013/12/20/precursors-of-category-theory-1/
2. https://inquiryintoinquiry.com/2013/12/30/precursors-of-category-theory-2/
3. https://inquiryintoinquiry.com/2014/01/03/precursors-of-category-theory-3/

Precursors Of Category Theory • Discussion
1. https://inquiryintoinquiry.com/2020/09/13/precursors-of-category-theory-discussion-1/
2. https://inquiryintoinquiry.com/2020/09/21/precursors-of-category-theory-discussion-2/
3. https://inquiryintoinquiry.com/2020/09/25/precursors-of-category-theory-discussion-3/

Categories à la Peirce —

C.S. Peirce • A Guess at the Riddle
• https://inquiryintoinquiry.com/2012/03/21/c-s-peirce-a-guess-at-the-riddle/

Peirce's Categories
1. https://inquiryintoinquiry.com/2015/10/30/peirces-categories-1/
2. https://inquiryintoinquiry.com/2015/10/31/peirces-categories-2/
3. https://inquiryintoinquiry.com/2015/11/04/peirces-categories-3/
•••
19. https://inquiryintoinquiry.com/2020/05/13/peirces-categories-19/
20. https://inquiryintoinquiry.com/2020/05/14/peirces-categories-20/
21. https://inquiryintoinquiry.com/2020/06/25/peirces-categories-21/

C.S. Peirce and Category Theory
1. https://inquiryintoinquiry.com/2021/06/23/c-s-peirce-and-category-theory-1/
2. https://inquiryintoinquiry.com/2021/06/24/c-s-peirce-and-category-theory-2/
3. https://inquiryintoinquiry.com/2021/06/27/c-s-peirce-and-category-theory-3/
4. https://inquiryintoinquiry.com/2021/06/28/c-s-peirce-and-category-theory-4/
5. https://inquiryintoinquiry.com/2021/06/29/c-s-peirce-and-category-theory-5/
6. https://inquiryintoinquiry.com/2021/06/30/c-s-peirce-and-category-theory-6/
7. https://inquiryintoinquiry.com/2021/07/01/c-s-peirce-and-category-theory-7/
8. https://inquiryintoinquiry.com/2021/07/02/c-s-peirce-and-category-theory-8/

#Aristotle #Peirce #Kant #Carnap #Hilbert #Ackermann #SaundersMacLane
#Abstraction #Analogy #CategoryTheory #Diagrams #FoundationsOfMathematics
#FunctionalLogic #RelationTheory #ContinuousPredicate #HypostaticAbstraction
#CategoryTheory #PeircesCategories #PropositionsAsTypes #TypeTheory #Universals

Peirce's 1885 “Algebra of Logic” • Discussion 2
• https://inquiryintoinquiry.com/2024/04/03/peirces-1885-algebra-of-logic-discussion-2/

Re: FB | Daniel Everett

One thing I've been trying to understand for a very long time is the changes in Peirce's writing about math and logic from 1865 to 1885. If there's anything I've learned from reading Peirce in the often dim light of intellectual history it is to be wary of progressivist assumptions — but unlike many of his other fans I apply that caution also within the body of his own work. Long story short, from 1865 to 1885 I see progress on several fronts but also bits of backsliding from his more prescient early insights. So it's a puzzle … and it will take more study to ravel out the reasons why.

Resources for reconciling Peirce's two accounts —
1. The 1870 account of logical involution
2. The 1885 account of universal quantification

Peirce's 1870 “Logic of Relatives” • Selection 12 • The Sign of Involution
• https://inquiryintoinquiry.com/2014/06/09/peirces-1870-logic-of-relatives-selection-12/
Comments —
(1) https://inquiryintoinquiry.com/2014/06/10/peirces-1870-logic-of-relatives-comment-12-1/
(2) https://inquiryintoinquiry.com/2014/06/11/peirces-1870-logic-of-relatives-comment-12-2/
(3) https://inquiryintoinquiry.com/2014/06/12/peirces-1870-logic-of-relatives-comment-12-3/
(4) https://inquiryintoinquiry.com/2014/06/14/peirces-1870-logic-of-relatives-comment-12-4/
(5) https://inquiryintoinquiry.com/2014/06/15/peirces-1870-logic-of-relatives-comment-12-5/

Peirce's 1885 “Algebra of Logic” • Selections
(1) https://inquiryintoinquiry.com/2024/03/24/peirces-1885-algebra-of-logic-selection-1/
(2) https://inquiryintoinquiry.com/2024/03/26/peirces-1885-algebra-of-logic-selection-2/
(3) https://inquiryintoinquiry.com/2024/03/30/peirces-1885-algebra-of-logic-selection-3/
(4) https://inquiryintoinquiry.com/2024/04/01/peirces-1885-algebra-of-logic-selection-4/

Peirce, C.S. (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, American Journal of Mathematics 7, 180–202.
• https://www.jstor.org/stable/2369451

#Peirce #Logic #AlgebraOfLogic #LogicOfRelatives #RelationTheory #CategoryTheory
#Semiotics #PredicateCalculus #Quantification #LogicalInvolution #ComputerScience

Peirce's 1885 “Algebra of Logic” • Discussion 1
• https://inquiryintoinquiry.com/2024/04/02/peirces-1885-algebra-of-logic-discussion-1/

Re: FB | Daniel Everett

DE:
❝One of the most important papers in the history of logic. “On the Algebra of Logic” was the first to introduce the term “quantifier”.

❝Peirce, C.S. (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, American Journal of Mathematics 7, 180–202.
• https://www.jstor.org/stable/2369451 ❞

As far as quantification by any other word goes, Peirce had already introduced a more advanced and “functional” concept of quantification in his 1870 “Logic of Relatives”. The subsequent passage to Fregean styles of first order logic would turn out to be a retrograde movement toward syntacticism (a species of nominalism), as seen in the general run of what fol‑lowed in the fol‑lowing years.

See ☞ Peirce's 1870 “Logic of Relatives”
• https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-overview/

Especially ☞ “The Sign of Involution”
• https://inquiryintoinquiry.com/2014/06/09/peirces-1870-logic-of-relatives-selection-12/

The connection between logical involution and universal quantification which Peirce put to use in his 1870 Logic of Relatives will turn up again a century later with the application of category theory to computer science and both of those in turn to logic. Just one more time Peirce was that far ahead of it.

See ☞ Lambek and Scott (1986), Introduction to Higher Order Categorical Logic, Cambridge University Press.
• https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems#Lambek

#Peirce #Logic #AlgebraOfLogic #LogicOfRelatives #RelationTheory #CategoryTheory
#Semiotics #PredicateCalculus #Quantification #LogicalInvolution #ComputerScience

Survey of Precursors Of Category Theory • 4
• http://inquiryintoinquiry.com/2023/08/01/survey-of-precursors-of-category-theory-4/

A few years ago I began a sketch on the “Precursors of Category Theory”, tracing the continuities of the category concept from Aristotle, to Kant and Peirce, through Hilbert and Ackermann, to contemporary mathematical practice. A Survey of resources on the topic is given below, still very rough and incomplete, but perhaps a few will find it of use.

Background —

Precursors Of Category Theory
• https://oeis.org/wiki/Precursors_Of_Category_Theory

Propositions As Types Analogy
• https://oeis.org/wiki/Propositions_As_Types_Analogy

Blog Series —

Notes On Categories
• https://inquiryintoinquiry.com/2013/02/22/notes-on-categories-1/

Precursors Of Category Theory
1. https://inquiryintoinquiry.com/2013/12/20/precursors-of-category-theory-1/
2. https://inquiryintoinquiry.com/2013/12/30/precursors-of-category-theory-2/
3. https://inquiryintoinquiry.com/2014/01/03/precursors-of-category-theory-3/

Precursors Of Category Theory • Discussion
1. https://inquiryintoinquiry.com/2020/09/13/precursors-of-category-theory-discussion-1/
2. https://inquiryintoinquiry.com/2020/09/21/precursors-of-category-theory-discussion-2/
3. https://inquiryintoinquiry.com/2020/09/25/precursors-of-category-theory-discussion-3/

Categories à la Peirce —

C.S. Peirce • A Guess at the Riddle
• https://inquiryintoinquiry.com/2012/03/21/c-s-peirce-a-guess-at-the-riddle/

Peirce's Categories
1. https://inquiryintoinquiry.com/2015/10/30/peirces-categories-1/
2. https://inquiryintoinquiry.com/2015/10/31/peirces-categories-2/
3. https://inquiryintoinquiry.com/2015/11/04/peirces-categories-3/
•••
19. https://inquiryintoinquiry.com/2020/05/13/peirces-categories-19/
20. https://inquiryintoinquiry.com/2020/05/14/peirces-categories-20/
21. https://inquiryintoinquiry.com/2020/06/25/peirces-categories-21/

#Aristotle #Peirce #Kant #Carnap #Hilbert #Ackermann #SaundersMacLane
#Abstraction #Analogy #CategoryTheory #Diagrams #FoundationsOfMathematics
#FunctionalLogic #RelationTheory #ContinuousPredicate #HypostaticAbstraction
#CategoryTheory #PeircesCategories #PropositionsAsTypes #TypeTheory #Universals