#categorytheory — Public Fediverse posts on home.social

JxHxBxR @[email protected] · 2026-04-05 · 07:30 UTC

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

#logic #philosophyoflogic #introduction #mathematicallogic #modeltheory #categorytheory

CubeRootOfTrue @[email protected] · 2025-05-03 · 16:32 UTC

ChatGPT on categorical logic again.

🧱 Level 1: Morphisms = Proofs, Typed with Validity

You can think of these arrows as typed by a truth value — i.e., each morphism has a color: valid, invalid, plausible, context-sensitive, contradictory-but-derivable, etc.

In this sense, truth is not binary, but becomes a fiber over each morphism: a coloring or modality.

So your category becomes a fibration over a poset of truth values, or a category enriched in truth values — maybe in a Heyting algebra or relevance lattice.

🧱 Level 2: 2-Cells = Laws, Derivations, Transformations

Now we raise it to a 2-category:

0-cells: Propositions (types)

1-cells: Deductions / proof structures f:A→B

2-cells: Proofs of equivalence between proofs (e.g., natural transformations, rewrite rules, context substitution, modality shifts)

This is where natural transformations live: between two different "routes" from A to B. They express meta-logical structure: laws, policies, meanings.

Let’s say you have:

One arrow f:A→B defined in deontic logic (permission-based)

Another arrow g:A→B in alethic logic (necessity-based)

A natural transformation η:f⇒g might be a social contract or legal interpretation that maps from a space of permitted inferences to necessary ones — or vice versa.

#categorytheory #logic #RelevanceLogic

#relevancelogic #logic #categorytheory

CubeRootOfTrue @[email protected] · 2025-04-06 · 23:51 UTC

ChatGPT has a new sister called Monday. I will let you find out about that. Meanwhile here is what ChatGPT says about using enriched categories to model relevance logic:

An Example Sketch

Let V=Pos be a poset-enriched monoidal category where each hom-object is a set of “proofs” or “derivations,” ordered by resource usage.

Then C(A,B) is itself an object in Pos, i.e., a poset of ways to prove B from A.

The product ⊗ inside C does not come with free projections, so there is no arrow from (A⊗B) to B in general.

If someone claims “Surely, we can discard A and prove B anyway,” the poset of proofs for C(A⊗B,B) is _empty_, or has no minimal element if your ordering demands using all resources.

Thus, the absence of a projection morphism is encoded in the structure of the hom-object: it simply does not contain a suitable proof.

--
here 'resource usage' is 'relevant stuff'

You can write (A⊗B) -> B, in a diagram. But that arrow is "False", so it doesn't really "exist". Enriched categories capture this concept.

#RelevanceLogic #categorytheory #enrichedcategory #rm3

#rm3 #enrichedcategory #categorytheory #relevancelogic

Inquiry Into Inquiry @[email protected] · 2024-12-09 · 13:12 UTC

Differential Propositional Calculus • 10

Special Classes of Propositions (cont.)

Let’s pause at this point and get a better sense of how our special classes of propositions are structured and how they relate to propositions in general. We can do this by recruiting our visual imaginations and drawing up a sufficient budget of venn diagrams for each family of propositions. The case for 3 variables is exemplary enough for a start.

Linear Propositions

The linear propositions, may be written as sums:

One thing to keep in mind about these sums is that the values in are added “modulo 2”, that is, in such a way that

In a universe of discourse based on three boolean variables, the linear propositions take the shapes shown in Figure 8.

At the top is the venn diagram for the linear proposition of rank 3, which may be expressed by any one of the following three forms.

Next are the venn diagrams for the three linear propositions of rank 2, which may be expressed by the following three forms, respectively.

Next are the three linear propositions of rank 1, which are none other than the three basic propositions,

At the bottom is the linear proposition of rank 0, the everywhere false proposition or the constant function, which may be expressed by the form or by a simple

Resources

cc: Academia.edu • Cybernetics • Structural Modeling • Systems Science
cc: Conceptual Graphs • Laws of Form • Mathstodon • Research Gate

#Amphecks #Animata #BooleanAlgebra #BooleanFunctions #CSPeirce #CactusGraphs #CategoryTheory #Change #Cybernetics #DifferentialAnalyticTuringAutomata #DifferentialCalculus #DifferentialLogic #DiscreteDynamics #EquationalInference #FunctionalLogic #GraphTheory #Hologrammautomaton #IndicatorFunctions #InquiryDrivenSystems #Leibniz #Logic #LogicalGraphs #Mathematics #MinimalNegationOperators #PropositionalCalculus #Time #Topology #Visualization

#amphecks #animata #booleanalgebra #booleanfunctions #cspeirce #cactusgraphs

Matt Austern @[email protected] · 2024-07-06 · 00:01 UTC

I came across a diagram illustrating transfer learning in the ML textbook I'm currently in the middle of, and it looked an awful lot like diagrams for natural transformations. I bet I'm not the first person to have noticed that, or the tenth. Any interesting papers I ought to read?

#MachineLearning #TransferLearning #CategoryTheory

#machinelearning #transferlearning #categorytheory

Charlotte Aten @[email protected] · 2023-12-01 · 20:35 UTC

As I prepare to head back home from #Cambridge, I'm reflecting on how much I've learned and on the hospitality shown to me by Jamie Vicary and his graduate students, @jonmsterling, and the computer lab as a whole. My first trip to the UK has been the experience of a lifetime and I look forward to seeing new friends I've made here again soon.

I was surprised to find that one person did a great deal to make me feel welcome despite the fact that I never met her. I noticed the pictured pride flag on my first day in the computer lab, and I assumed everyone signed it as part of some event. I don't talk much about being queer on Mastodon, but seeing this definitely helped me feel much more at ease in a place which was foreign to me, an American mathematician, in multiple senses.

(1/2)

#ComputerScience #CategoryTheory #AppliedCategoryTheory #queer

#queer #appliedcategorytheory #categorytheory #computerscience #cambridge

Charlotte Aten @[email protected] · 2023-12-01 · 20:35 UTC

As I prepare to head back home from #Cambridge, I'm reflecting on how much I've learned and on the hospitality shown to me by Jamie Vicary and his graduate students, @jonmsterling, and the computer lab as a whole. My first trip to the UK has been the experience of a lifetime and I look forward to seeing new friends I've made here again soon.

I was surprised to find that one person did a great deal to make me feel welcome despite the fact that I never met her. I noticed the pictured pride flag on my first day in the computer lab, and I assumed everyone signed it as part of some event. I don't talk much about being queer on Mastodon, but seeing this definitely helped me feel much more at ease in a place which was foreign to me, an American mathematician, in multiple senses.

(1/2)

#ComputerScience #CategoryTheory #AppliedCategoryTheory #queer

#queer #appliedcategorytheory #categorytheory #computerscience #cambridge

Charlotte Aten @[email protected] · 2023-12-01 · 20:35 UTC

As I prepare to head back home from #Cambridge, I'm reflecting on how much I've learned and on the hospitality shown to me by Jamie Vicary and his graduate students, @jonmsterling, and the computer lab as a whole. My first trip to the UK has been the experience of a lifetime and I look forward to seeing new friends I've made here again soon.

I was surprised to find that one person did a great deal to make me feel welcome despite the fact that I never met her. I noticed the pictured pride flag on my first day in the computer lab, and I assumed everyone signed it as part of some event. I don't talk much about being queer on Mastodon, but seeing this definitely helped me feel much more at ease in a place which was foreign to me, an American mathematician, in multiple senses.

(1/2)

#ComputerScience #CategoryTheory #AppliedCategoryTheory #queer

#queer #appliedcategorytheory #categorytheory #computerscience #cambridge

Charlotte Aten @[email protected] · 2023-12-01 · 20:35 UTC

As I prepare to head back home from #Cambridge, I'm reflecting on how much I've learned and on the hospitality shown to me by Jamie Vicary and his graduate students, @jonmsterling, and the computer lab as a whole. My first trip to the UK has been the experience of a lifetime and I look forward to seeing new friends I've made here again soon.

I was surprised to find that one person did a great deal to make me feel welcome despite the fact that I never met her. I noticed the pictured pride flag on my first day in the computer lab, and I assumed everyone signed it as part of some event. I don't talk much about being queer on Mastodon, but seeing this definitely helped me feel much more at ease in a place which was foreign to me, an American mathematician, in multiple senses.

(1/2)

#ComputerScience #CategoryTheory #AppliedCategoryTheory #queer

#cambridge #computerscience #categorytheory #appliedcategorytheory #queer

Charlotte Aten @[email protected] · 2023-12-01 · 20:35 UTC

As I prepare to head back home from #Cambridge, I'm reflecting on how much I've learned and on the hospitality shown to me by Jamie Vicary and his graduate students, @jonmsterling, and the computer lab as a whole. My first trip to the UK has been the experience of a lifetime and I look forward to seeing new friends I've made here again soon.

I was surprised to find that one person did a great deal to make me feel welcome despite the fact that I never met her. I noticed the pictured pride flag on my first day in the computer lab, and I assumed everyone signed it as part of some event. I don't talk much about being queer on Mastodon, but seeing this definitely helped me feel much more at ease in a place which was foreign to me, an American mathematician, in multiple senses.

(1/2)

#ComputerScience #CategoryTheory #AppliedCategoryTheory #queer

#queer #appliedcategorytheory #categorytheory #computerscience #cambridge

Charlotte Aten @[email protected] · 2023-11-30 · 11:55 UTC

My talk from yesterday on a categorical semantics for neural nets for the New York Category Theory Seminar has already been posted on YouTube! You can find it at https://www.youtube.com/watch?v=FKkpVKuspmA and you can see more about this seminar at https://www.sci.brooklyn.cuny.edu/~noson/Seminar/. The preprint I mention is https://arxiv.org/abs/2308.00677 and the talk by Joyal which was mentioned at the end can be found at https://www.youtube.com/watch?v=MxClaWFiGKw.

#CategoryTheory #AppliedCategoryTheory #AI #MachineLearning #ComputerScience #UniversalAlgebra #NeuralNets

#neuralnets #universalalgebra #computerscience #machinelearning #ai #appliedcategorytheory

Charlotte Aten @[email protected] · 2023-11-30 · 11:55 UTC

My talk from yesterday on a categorical semantics for neural nets for the New York Category Theory Seminar has already been posted on YouTube! You can find it at https://www.youtube.com/watch?v=FKkpVKuspmA and you can see more about this seminar at https://www.sci.brooklyn.cuny.edu/~noson/Seminar/. The preprint I mention is https://arxiv.org/abs/2308.00677 and the talk by Joyal which was mentioned at the end can be found at https://www.youtube.com/watch?v=MxClaWFiGKw.

#CategoryTheory #AppliedCategoryTheory #AI #MachineLearning #ComputerScience #UniversalAlgebra #NeuralNets

#neuralnets #universalalgebra #computerscience #machinelearning #ai #appliedcategorytheory

Charlotte Aten @[email protected] · 2023-11-30 · 11:55 UTC

My talk from yesterday on a categorical semantics for neural nets for the New York Category Theory Seminar has already been posted on YouTube! You can find it at https://www.youtube.com/watch?v=FKkpVKuspmA and you can see more about this seminar at https://www.sci.brooklyn.cuny.edu/~noson/Seminar/. The preprint I mention is https://arxiv.org/abs/2308.00677 and the talk by Joyal which was mentioned at the end can be found at https://www.youtube.com/watch?v=MxClaWFiGKw.

#CategoryTheory #AppliedCategoryTheory #AI #MachineLearning #ComputerScience #UniversalAlgebra #NeuralNets

#neuralnets #universalalgebra #computerscience #machinelearning #ai #appliedcategorytheory

Charlotte Aten @[email protected] · 2023-11-30 · 11:55 UTC

My talk from yesterday on a categorical semantics for neural nets for the New York Category Theory Seminar has already been posted on YouTube! You can find it at https://www.youtube.com/watch?v=FKkpVKuspmA and you can see more about this seminar at https://www.sci.brooklyn.cuny.edu/~noson/Seminar/. The preprint I mention is https://arxiv.org/abs/2308.00677 and the talk by Joyal which was mentioned at the end can be found at https://www.youtube.com/watch?v=MxClaWFiGKw.

#CategoryTheory #AppliedCategoryTheory #AI #MachineLearning #ComputerScience #UniversalAlgebra #NeuralNets

#categorytheory #appliedcategorytheory #ai #machinelearning #computerscience #universalalgebra

Charlotte Aten @[email protected] · 2023-11-30 · 11:55 UTC

My talk from yesterday on a categorical semantics for neural nets for the New York Category Theory Seminar has already been posted on YouTube! You can find it at https://www.youtube.com/watch?v=FKkpVKuspmA and you can see more about this seminar at https://www.sci.brooklyn.cuny.edu/~noson/Seminar/. The preprint I mention is https://arxiv.org/abs/2308.00677 and the talk by Joyal which was mentioned at the end can be found at https://www.youtube.com/watch?v=MxClaWFiGKw.

#CategoryTheory #AppliedCategoryTheory #AI #MachineLearning #ComputerScience #UniversalAlgebra #NeuralNets

#neuralnets #universalalgebra #computerscience #machinelearning #ai #appliedcategorytheory

Charlotte Aten @[email protected] · 2023-11-29 · 21:50 UTC

In a couple hours I'll be giving a talk online for the New York Category Theory Seminar! This is going to be another iteration of "A categorical semantics for neural networks", which I first discussed last week at the Cambridge Logic and Semantics Seminar. I got some really helpful questions/feedback last week, so I'll have some new thoughts to share today.