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#categorytheory — Public Fediverse posts

Live and recent posts from across the Fediverse tagged #categorytheory, aggregated by home.social.

  1. 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

  2. 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

  3. 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

  4. 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.eduCyberneticsStructural ModelingSystems Science
    cc: Conceptual GraphsLaws of FormMathstodonResearch 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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 youtube.com/watch?v=FKkpVKuspm and you can see more about this seminar at sci.brooklyn.cuny.edu/~noson/S. The preprint I mention is arxiv.org/abs/2308.00677 and the talk by Joyal which was mentioned at the end can be found at youtube.com/watch?v=MxClaWFiGK.

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

  12. 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 youtube.com/watch?v=FKkpVKuspm and you can see more about this seminar at sci.brooklyn.cuny.edu/~noson/S. The preprint I mention is arxiv.org/abs/2308.00677 and the talk by Joyal which was mentioned at the end can be found at youtube.com/watch?v=MxClaWFiGK.

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

  13. 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 youtube.com/watch?v=FKkpVKuspm and you can see more about this seminar at sci.brooklyn.cuny.edu/~noson/S. The preprint I mention is arxiv.org/abs/2308.00677 and the talk by Joyal which was mentioned at the end can be found at youtube.com/watch?v=MxClaWFiGK.

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

  14. 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 youtube.com/watch?v=FKkpVKuspm and you can see more about this seminar at sci.brooklyn.cuny.edu/~noson/S. The preprint I mention is arxiv.org/abs/2308.00677 and the talk by Joyal which was mentioned at the end can be found at youtube.com/watch?v=MxClaWFiGK.

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

  15. 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 youtube.com/watch?v=FKkpVKuspm and you can see more about this seminar at sci.brooklyn.cuny.edu/~noson/S. The preprint I mention is arxiv.org/abs/2308.00677 and the talk by Joyal which was mentioned at the end can be found at youtube.com/watch?v=MxClaWFiGK.

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

  16. 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.

    In particular, I think I may want to use a horizontal categorification of clones, rather than traditional multicategories.

    More info:
    sci.brooklyn.cuny.edu/~noson/S

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

  17. 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.

    In particular, I think I may want to use a horizontal categorification of clones, rather than traditional multicategories.

    More info:
    sci.brooklyn.cuny.edu/~noson/S

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

  18. 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.

    In particular, I think I may want to use a horizontal categorification of clones, rather than traditional multicategories.

    More info:
    sci.brooklyn.cuny.edu/~noson/S

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

  19. 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.

    In particular, I think I may want to use a horizontal categorification of clones, rather than traditional multicategories.

    More info:
    sci.brooklyn.cuny.edu/~noson/S

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

  20. 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.

    In particular, I think I may want to use a horizontal categorification of clones, rather than traditional multicategories.

    More info:
    sci.brooklyn.cuny.edu/~noson/S

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

  21. I will be giving a talk on a functorial semantics for neural nets on Friday at the #Cambridge computer lab's Logic and Semantics Seminar. If you happen to be in the area at the time, consider checking it out!

    This will be my first talk on a synthesis of my paper on discrete neural nets (arxiv.org/abs/2308.00677) and the categorified invariant theory appearing in my thesis (aten.cool/documents/thesis.pdf).

    Abstract and other info: cst.cam.ac.uk/seminars/list/20

    #CategoryTheory #AppliedCategoryTheory #AI #MachineLearning #ComputerScience #logic #math #combinatorics #UniversalAlgebra

  22. I will be giving a talk on a functorial semantics for neural nets on Friday at the #Cambridge computer lab's Logic and Semantics Seminar. If you happen to be in the area at the time, consider checking it out!

    This will be my first talk on a synthesis of my paper on discrete neural nets (arxiv.org/abs/2308.00677) and the categorified invariant theory appearing in my thesis (aten.cool/documents/thesis.pdf).

    Abstract and other info: cst.cam.ac.uk/seminars/list/20

    #CategoryTheory #AppliedCategoryTheory #AI #MachineLearning #ComputerScience #logic #math #combinatorics #UniversalAlgebra

  23. I will be giving a talk on a functorial semantics for neural nets on Friday at the #Cambridge computer lab's Logic and Semantics Seminar. If you happen to be in the area at the time, consider checking it out!

    This will be my first talk on a synthesis of my paper on discrete neural nets (arxiv.org/abs/2308.00677) and the categorified invariant theory appearing in my thesis (aten.cool/documents/thesis.pdf).

    Abstract and other info: cst.cam.ac.uk/seminars/list/20

    #CategoryTheory #AppliedCategoryTheory #AI #MachineLearning #ComputerScience #logic #math #combinatorics #UniversalAlgebra

  24. Good morning UK! In two weeks I'll be visiting #Cambridge for about two weeks. Please let me know if there's anything cool I should check out while I'm there.

    I'll be speaking at this seminar (talks.cam.ac.uk/show/index/637) about a functorial semantics for neural nets. I still need to send in an abstract, so my talk isn't listed yet. This topic is a synthesis of my preprint «Discrete neural nets and polymorphic learning» (arxiv.org/abs/2308.00677) as well as some of the results from my thesis (aten.cool/documents/thesis.pdf).

    I am planning on making a trip to #Nottingham at some point, as well.

    #AI #MachineLearning #AppliedCategoryTheory #Combinatorics #CategoryTheory

  25. Good morning UK! In two weeks I'll be visiting #Cambridge for about two weeks. Please let me know if there's anything cool I should check out while I'm there.

    I'll be speaking at this seminar (talks.cam.ac.uk/show/index/637) about a functorial semantics for neural nets. I still need to send in an abstract, so my talk isn't listed yet. This topic is a synthesis of my preprint «Discrete neural nets and polymorphic learning» (arxiv.org/abs/2308.00677) as well as some of the results from my thesis (aten.cool/documents/thesis.pdf).

    I am planning on making a trip to #Nottingham at some point, as well.

    #AI #MachineLearning #AppliedCategoryTheory #Combinatorics #CategoryTheory

  26. Good morning UK! In two weeks I'll be visiting #Cambridge for about two weeks. Please let me know if there's anything cool I should check out while I'm there.

    I'll be speaking at this seminar (talks.cam.ac.uk/show/index/637) about a functorial semantics for neural nets. I still need to send in an abstract, so my talk isn't listed yet. This topic is a synthesis of my preprint «Discrete neural nets and polymorphic learning» (arxiv.org/abs/2308.00677) as well as some of the results from my thesis (aten.cool/documents/thesis.pdf).

    I am planning on making a trip to #Nottingham at some point, as well.

    #AI #MachineLearning #AppliedCategoryTheory #Combinatorics #CategoryTheory

  27. Good morning UK! In two weeks I'll be visiting #Cambridge for about two weeks. Please let me know if there's anything cool I should check out while I'm there.

    I'll be speaking at this seminar (talks.cam.ac.uk/show/index/637) about a functorial semantics for neural nets. I still need to send in an abstract, so my talk isn't listed yet. This topic is a synthesis of my preprint «Discrete neural nets and polymorphic learning» (arxiv.org/abs/2308.00677) as well as some of the results from my thesis (aten.cool/documents/thesis.pdf).

    I am planning on making a trip to #Nottingham at some point, as well.

    #AI #MachineLearning #AppliedCategoryTheory #Combinatorics #CategoryTheory

  28. Good morning UK! In two weeks I'll be visiting #Cambridge for about two weeks. Please let me know if there's anything cool I should check out while I'm there.

    I'll be speaking at this seminar (talks.cam.ac.uk/show/index/637) about a functorial semantics for neural nets. I still need to send in an abstract, so my talk isn't listed yet. This topic is a synthesis of my preprint «Discrete neural nets and polymorphic learning» (arxiv.org/abs/2308.00677) as well as some of the results from my thesis (aten.cool/documents/thesis.pdf).

    I am planning on making a trip to #Nottingham at some point, as well.

    #AI #MachineLearning #AppliedCategoryTheory #Combinatorics #CategoryTheory

  29. I'm giving another talk on discrete neural nets and polymorphic learning at the #CUBoulder PALS seminar (math.colorado.edu/algebralogic) at 230pm MDT today! You can find the preprint I'm discussing at arxiv.org/abs/2308.00677. As usual, a recording will be available in case you'd like to watch later.

    #AI #CategoryTheory #algebra #AbstractAlgebra #Combinatorics #ComputerScience #AppliedCategoryTheory

  30. I'm giving another talk on discrete neural nets and polymorphic learning at the #CUBoulder PALS seminar (math.colorado.edu/algebralogic) at 230pm MDT today! You can find the preprint I'm discussing at arxiv.org/abs/2308.00677. As usual, a recording will be available in case you'd like to watch later.

    #AI #CategoryTheory #algebra #AbstractAlgebra #Combinatorics #ComputerScience #AppliedCategoryTheory

  31. I'm giving another talk on discrete neural nets and polymorphic learning at the #CUBoulder PALS seminar (math.colorado.edu/algebralogic) at 230pm MDT today! You can find the preprint I'm discussing at arxiv.org/abs/2308.00677. As usual, a recording will be available in case you'd like to watch later.

    #AI #CategoryTheory #algebra #AbstractAlgebra #Combinatorics #ComputerScience #AppliedCategoryTheory

  32. I'm giving another talk on discrete neural nets and polymorphic learning at the #CUBoulder PALS seminar (math.colorado.edu/algebralogic) at 230pm MDT today! You can find the preprint I'm discussing at arxiv.org/abs/2308.00677. As usual, a recording will be available in case you'd like to watch later.

    #AI #CategoryTheory #algebra #AbstractAlgebra #Combinatorics #ComputerScience #AppliedCategoryTheory