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See: Yoneda Lemma, Topos, Logic, Limits vs Colimits, Type theory, Homotopy type theory, Category theory glossary, Eduardo Ochs, Category theory obstacles, Category theory progression, Statistics, Adjunction, Algebra of requirements, Algebra of perspectives, Category theory study group, 20200524 Exercises, 20200705 Exercises, 20200726 Exercises, 20200809 Exercises, Set theory, Polynomial functors

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The Purpose and Nature of Category Theory

I'm learning category theory and related disciplines (type theory, topos theory, homotopy type theory) to understand how to express more precisely certain concepts in my philosophy such as:

• the algebra of perspectives (composition)
• the four levels of knowledge (Yoneda lemma)
• divisions of everything (adjunction - adjoint strings)
• extending the domain (Kan extensions)
• Curry-Howard-Lambek correspondence

Category theory also has ideas that I can absorb

• notions of identity, equality, equivalence
• algebra of requirements
• duality
• internal structure vs. external relationships

Algebra of perspectives

• Gilles Fauconnier's theory of mental spaces.
• Algebra of concern. I can care what another cares about. And they can care about what another person cares about.
• The purpose of perspective is to preserve the {{Truth}}.
• Perspective is a morphism.
• There is composition: Mary's perspective takes up Anna's perspective
• Perspective is associative: Anna takes up Betty's perspective of Charles' perspective may be composed in either direction (by stepping out from Charles to Betty to Anna, or by stepping in from Anna to Betty to Charles)
• The identity morphism is the Truth and is a ZeroStructure
• What are the objects? They are what is preserved by the perspective. So they are, in some sense, truths. But in what sense? As sets of truths that define a state of mind. These states of mind are divisions of everything.
• The lost child.
• Viską aprėpus, noriu "požiūriais" ir būtent "Dievo požiūriu" bei jų "bendryste" naujai suvokti, apžvelgti ir išplėsti visus savo ankstyvesnius atradimus.
• A related idea to relative idempotence and relative commutativity that I had is that we may think of God's view as transparent and Human's view as opaque. Then we may be able to "escape a view" by "bisecting a view". That may also relate to adjoints.
• Think of perspectives (upon a diagram) as being defined by adding an object to a cone. And think of limits as stepping in (arrows go in), and colimits as stepping out (arrows come out).

The purpose of category theory

• Category theory externalizes the internal structure of mathematical objects. It reexpresses that internal structures in terms of constraints on external relationships between structures.
  • Lygmuo Kodėl viską išsako ryšiais. O tas ryšys yra tarpas, kuriuo išsakomas Kitas. Kategorijų teorijoje panašiai, tikslas yra pereiti iš narių (objektų) nagrinėjimą į ryšių (morfizmų) nagrinėjimą.
  • Internal discussion with oneself vs. external discussion with others (Vygotsky) is the distinction that category theory makes between internal structure and external relationships.
  • Category theory relates God's outer perspective (on the general, external "black box" relationships) and our inner perspective, within the system, in terms of the properties of our particular system. The question of God's necessity and nature includes the relationship between God and human's perspectives.
• Category theory models perspectives and attention shifting.
  • Category theory shines light on the big picture. Perspectives shine light on the big picture (God's) or the local picture (human's).
• Category theory expresses a duality between objects and mappings. Thus functors likewise have this dual nature, and so do natural transformations.
• A purpose of category theory is to hide implementation details, which if not hidden, not segregated are the source of the paradoxes. Subsystems and entropy also relate to the existence or not of this "wall". This wall is perhaps the wall that separates the conscious and the unconscious in the six restructurings.
  • Eugenia Cheng: Mathematics is the logical study of how logical things work. Abstraction is what we need for logical study. Category theory is the the math of math, thus the logical study of the logical study of how logical things work.
  • Category theory for me: distinguishing what observations are nontrivial - intrinsic to a subject - and what are observations are content-wise trivial or universal - not related to the subject, but simply an aspect of abstraction.

Category Theory Study Materials

Introductory materials

• Emily Riehl. Category Theory in Context
• Steve Awodey. Category Theory Book.
• Category Theory textbook by Steve Adowey
• Videos: Category Theory, video lectures by Steve Adowey
• Lawvere, W., Schanuel, S.: Conceptual Mathematics: A first introduction to categories. Cambridge (1997)
• John Baez: Category Theory
• John Baez's Category Theory Course
• John Baez lecture notes. Winter 2016.
• Good exposition of adjoint functors and other topics.
• Abstract and Concrete Categories. The Joy of Cats. Jiri Adamek, Horst Herrlich, George E. Strecker.
• Make category theory intuitive
• Bartosz Milewski: Natural Transformations
• Tai-Danae Bradley: What is Category Theory Anyways?
• Eduardo Ochs
• Videos: Catster and at YouTube
• Videos: Partha Pratim Ghosh
• Videos: The Beginner's Introduction with examples from music, Martin Codrington
• Video: MathProofsable
• AlgTop2 video lecture
• David Spivak: "Categories = polynomial comonads", a simple demonstration

Applied Category Theory

• Community: Applied Category Theory web at Functorial wiki
• Seven Sketches in Compositionality: An Invitation to Applied Category Theory Fong, B., Spivak, D.I.
• Videos: 7 Sketches On Compositionality
• Tai-Danae Bradley: What is Applied Category Theory?
• Applied Category Theory Course, Discussion and Textbook
• Videos: Applied Category Theory 2019: Spivak, Fong

Programmers

• Joseph Goguen. A Categorical Manifesto.
• Category Theory for Programmers, Bartosz Milewski.
• Videos: Bartosz Milewski for programmers
• Programmers go bananas

Music

• The Topos of Music: Geometric Logic of Concepts, Theory, and Performance. Guerino Mazzola.

Physics

• A Pre-history of n-categorical Physics by John Baez
• Category theory for scientists
• Physics, Topology, Logic and Computation: A Rosetta Stone. 2009.
• John Baez. Spans in Quantum Theory.

Categorification

• Jeff Hicks Categorification
• Categorification

Categorical logic and topos

• Categorical Logic lecture notes by Steve Awodey
• Videos: Toposes in Como 2018

Homotopy type theory

• Videos: Univalent Foundations of Mathematics

Higher category theory

• Eugenia Cheng. Higher Dimensional Category Theory: The Architecture of Mathematics
• Videos: Eugenia Cheng: Periodic table of n-categories
• Video: N-category theory lectures by Tom Leinster
• Groth, M., A Short Course on ∞-categories
• The theory of quasi-categories II
• Lurie. Higher Algebra. 2014.
• Categories Superieures et Theorie des Topos par Denis-Charles Cisinki

Philosophy

• Ralf Kroemer, Tool and Object: A History and Philosophy of Category Theory Birkhauser (2007)
• From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory (Logic, Epistemology, and the Unity of Science) Jean-Pierre Marquis
• Categories for the Working Philosopher, Discussion about the book project
• http://plato.stanford.edu/entries/category-theory/

Simplexes and simplicial sets

• Emily Riehl, A leisurely introduction to simplicial sets
• What's special about the simplex category?

Network theory

• Network theory (wiki) and Network theory (blog) by John Baez

Lie theory

• Borel-Weil theorem The Borel-Weil-Bott theorem characterizes representations of suitable Lie groups GG as space of holomorphic sections of complex line bundles over flag varieties G/BG/B, for BB a Borel subgroup.

Geometric Representation Theory

• Introductory Book for Geometric Representation Theory
• Videos: Geometric Representation Theory
• Videos: Geometric Representation Theory Seminar 2007 John Baez and James Dolan
• Video: Theory X and Life

Geometry and toposes

• Reviving the Philosophy of Geometry by David Corfield

Category theory concepts

Internal structure and external relationships

• Categorification (making math explicit) vs. Decategorification (making math implicit). Algebraic combinatorics is the concrete flip-side of the abstractness of category theory. But algebraic combinatorics comes with implicit interpretation whereas category theory comes with explicit notation.

Partial category

• Ruling a black box (100 trillion neurons) with culture (100 thousand concepts). Modeling and controlling the brain. Inferring objects: defining "objects" from the relationships of their possibilities. The brain is such a partial category (partially defined category) as opposed to a total category. The partial category may not have all of its objects well defined, and in particular, it may lack identity morphisms. Also, the brain may not actually work perfectly, just extremely well.

Basic kinds of categories

• Awodey writes in his book about the preorders and the monoids as describing two different pedagogically useful extreme cases that emphasize arrow and object outlooks on categories.

Threesome

• I am thinking that categories should be considered on three levels:
  • Objects (of being - what is)
  • Arrows (of doing)
  • Equations (of reflecting) that relate arrows (or objects), especially in composition.

The composition of the arrows always seems to me underexplained. And that is where the different levels of equivalence become relevant. Also to be considered is whether an object should be thought of as an arrow to itself.

Contradiction

• Natural contradiction inherent in the approach of defining categories by starting with a category of categories and then taking a category to be its object.

Math discovery

• Category theory concepts such as adjoints (least upper bounds, greatest lower bounds) and limits-colimits are actually concepts of analysis.

Identity

• For each object {$x$}, the identity morphism {${id}_x$} must be unique. Because consider {${id1}_x \circ {id2}_x$}. The identity is whichever disappears.
• Identity morphism (and what it gets map to) is the root of the tree (from which walks proceed).

Morphism

• A morphism can be factored into an epic map and an injective map. You can't lose any information.

Relevance

• Kernel: f(ker f)=0 in Y gives what is irrelevant because it is internal. The Cokernel Y/Im f is what's left over when you map, what is irrelevant because it is external.

Duality

• There are always dual categories {$C$} and {$C^{op}$}.
• Duality is based on morphisms and their directionality.

Loop

• A loop can be understood as a circle or as a complicated closed curve. It is a loop to itself, but in another context it is a complicated curve, perhaps in the complex numbers, perhaps in a multi-dimensional space.

Definiteness and types

• Why I feel strange that Set is not definite: "every function should have a definite class as domain and a definite class as range". Riehl quotes Eilenberg and Maclane: ". . . the whole concept of a category is essentially an auxiliary one; our basic concepts are essentially those of a functor and of a natural transformation . . . . The idea of a category is required only by the precept that every function should have a definite class as domain and a definite class as range, for the categories are provided as the domains and ranges of functors. Thus one could drop the category concept altogether and adopt an even more intuitive standpoint, in which a functor such as “Hom” is not defined over the category of “all” groups, but for each particular pair of groups which may be given. [EM45]"

Functors

• A functor is defined by what it does on a composition triangle of morphisms, and what it does on the identity morphism: {$F(a \overset{f}{\rightarrow} b \overset{g}{\rightarrow} c) \Rightarrow F(a) \overset{F(f)}{\rightarrow} F(b) \overset{F(g)}{\rightarrow} F(c)$}
• A functor is an interpretation that takes us from a syntax category to a semantics category.
• Milewski: A functor embeds one category in another.
• Milewski: A functor may collapse multiple objects/functions into one, but it never breaks connections.
• John Baez: "Every sufficiently good analogy is yearning to become a functor."
• Set = programs, actions, etc. as possible outputs. Working with a category: action in a system. Functor takes you outside your category: output. Outputs: objects become relevant.

Adjunction

Natural transformations

• Given functors F and G, both from C to D, a natural transformation {$eta$} maps every particular object X in C to a particular morphism {$eta_X$} from {$F(x)$} to {$G(x)$}. In this sense, the object is why (as a generalization of how) and the morphism is whether (as a generalization of what). Why and whether hold beyond circumstances (the functor), whereas how and what make sense within circumstances (the functor).
• Natural transformations don't depend on the structure internal to the objects, but only on their external relationships, as expressed by the category.
• The components of natural transformations depend only on the objects. If you know these components, then the morphisms carry over automatically.
• Natural transformations say that the trivially existing bijection (between FA and GA, FB and GB) is actually a morphism in the category D.
• Morphisms only diminish internal information (though they can enrich the context). A natural transformation is the diminishment of such diminishment. A natural transformation relates parallel worlds: thus the world of the identity map is related to an object in the parallel world. And this happens by way of the relationship, the paralellism, between an object and its identity.
• The desired natural transformation has components which are indexed by the object x which they send to be evaluated upon. Whereas in the other direction, given the natural transformation, we
• Natural transformations are important (meaningful?) because they separate two levels of knowledge. They are organized around indices from the functor's input category and describe relations in the output category.

Existence and universality

• Universal mapping property relates unique existence (all distinct) and universality (all objects are covered).

Limits and colimits

• Goguen: The colimit of a diagram of widgets creates a super-widget from the system of widgets.
• Goguen: The behavior of a system is given by a limit construction.
• In product, the information from A and B is stored externally in A x B. In coproduct, the information from A and B is stored internally in A union B (A+B). Note: multiplication is external, and addition is internal.

Kleisli categories

• Goguen: View an arbitrary adjunction as a kind of a theory. Many different problems of unification (of solving systems of equations) are finding equalizers in Kleisli categories. Kleisli categories provide an abstract notion of "computation".

Sheaves

• Categorical models for psychological consciousness. Sheaf theory - consciousness.

Axiom of forgetfullness

Higher category theory

• Riehl and Dominic Verity. Model independent higher category theory.
• Higher order category theory can express a perspective on a perspective on a perspective. Arrow from P to the arrow from P to P. But I expect that to be the maximum of structural complexity, and that it comes from the foursome, as in the Yoneda Lemma and the Yates Index Set Theorem.

M-category, a category with two classes of morphisms: tight and loose.

Attach:quivermatrix.jpg Δ

Notes

Composition

• Video: David Jaz Myers. Paradigms of Composition.
• Parameter setting: Composing by setting parameters to variables of state
  • System of differential equations
  • Moore Machine (deterministic automata)
  • Markov decision process
• Port plugging: Compose by plugging in exposed ports
  • Circuits
  • Population flow graphs
  • Labeled transition systems
• Variable sharing: Compose by sharing variables
  • Willems-style types of behaviors (control theory)
  • Hamiltonian systems
  • Lagrangian systems
• inside - variables (output) as components of states (variables depend on each other) post-composition, elements
• negation? - states
• outside - variables (input) as components of states, pre-composition, whole
• network to sequence, tree, network
• limit and colimit

Substitution - make inner collapse

• Applied Category Theory
• https://q.uiver.app/ Make category diagrams
• Yukio-Pegio Gunjia, Hideki Higashi. The origin of universality: making and invalidating a free category.
• Programming with Categories - Lecture 7
• Functors of themselves are weakly losing information. But the functors may put information in a category where that information takes on new meaning thus increases information. Because the second category leverages the first category.
• Wenbo https://www.youtube.com/playlist?list=PLbgaMIhjbmEnaH_LTkxLI7FMa2HsnawM_
• Bartosz Milewski - Category theory for programmers, Part 3 section 9, section 14
• https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/
• In category theory, there is the following asymmetry: external relationships include identity morphisms, which is to say, explicit equality, explicit self-identity. Whereas internal structure does not make self-identity explicit but rather a structure is what it is. For example, a set is not an element of itself.
• How and why is it that limits express external relationships but colimits express internal structure?
• https://golem.ph.utexas.edu/category/2020/01/profunctor_optics_the_categori.html
• https://www.appliedcategorytheory.org/adjoint-school-act-2019/ for Wenbo
• https://arxiv.org/abs/1803.05316 Seven Sketches in Compositionality: An Invitation to Applied Category Theory

Brendan Fong, David I Spivak

• nyc-category-theory.slack.com
• Brendan Fong: Partition Logic (Ellerman)
• A morphism weakens information itself but also can strengthen the context for the information.
• Preorders are not partial orders because there may be two different objects that are less than or equal to each other but not the same. Examples of preorders: Fractions of integers (2/4 and 1/2 are distinct.) Decimal sequences (where 0.999... and 1.000... are distinct.)
• A natural transformation is like a homotopy.
• Functor relates observer, perspective, mental spaces.
• Think about a topological space, its open sets and continuous functions, as a category, where the continuous functions are understood contravariantly - the inverse image of an open set is an open set. What intuition does this capture? How is this understood in category theory? How does it relate to sheaves?
• How do prime ideals and factoring relate to universal properties?
• A functor is a restructuring from the smaller category (the conscious) into part of a larger category (the unconscious) so as to restructure it.
• The Category of small categories includes the empty category (with no object and no morphism) as its initial object (by the empty functor) and includes the trivial category 1 (with one object and one morphism) as its terminal object.
• Bertalan Pecsi. On Morita Equivalence of Categories. Profunctors, adjunctions.

Port graph, can consider in terms of

• relations (no restriction)
• functions (every input has a single output)
• bijections (function in both directions)

This can be considered in terms of levels of intelligence.

Category theory

• https://www.preposterousuniverse.com/podcast/2021/05/10/146-emily-riehl-on-topology-categories-and-the-future-of-mathematics/
• The levels of sophistication of the notion of confidence in statistics may be modeled by higher dimensional category theory.
• Garunkštis - analytic number theory - is interested in category theory.
• MIT. Applied Category Theory
• Enriched preorder category.
• Preorder is a boolean enriched category.
• Example of self-enriched category is Bool where we have the category with two objects and one morphism between them, and we map that morphism to 0 (false) or 1 (true) with 0<=1.
• HomSet is an enrichment of a category with a preorder of nonempty homsets.
• Diachronic category is preorder, synchronic category (linear time) is monoid defined on one element (all time, all actions at once).
• An object is a point in time. In the diachronic point of view, we move by morphisms from object to object. In the synchronic point of view we keep returning to the same point in time, so all morphisms act in parallel.
• Enriched category is the "graded category" that I was seeking for in statistics, etc.
• Residuated Boolean algebra, residuated semilattice
• A Categorical Theory of Patches
• https://www.math3ma.com/blog/limits-and-colimits-part-2
• Distinction between extrinsic symmetry (relative to an outside observer) and intrinsic symmetry (relative to itself).
• Arrows, structures, and functors_ the categorical imperative-Academic Press (1975)
• Logical Foundation of Cognition - Lawvere - 4. Tools for the Advancement of the Logic of Closed Categories

Signal flow graphs are expressed on four levels:

• Diagrammatic language (homotopy), can deform a graph without changing
• lambda-expressions make variables explicit
• point-free, combinator style
• matrices for efficient computation
• Tool and Object: A History and Philosophy of Category Theory. Ralf Krömer.
• How is symmetry related to groups? The group action on a set can be thought of as a functor from the group as a one object category to a set, with each element mapped to a set automorphism, which must be surjective because it is invertible.
• Group as category with one object. Can think of the object as the whole group G. Then the morphisms act on the whole group and yield the whole group. And this also expresses the relationship between a set (the whole) and its elements (the morphisms).

Mailing list

• Categories List

There are three places categories store information, and three special cases to study

• A) Store in the morphisms. Study the category with a single object and with multiple morphisms. Like a monoid or a group.
• B) Store in the relations between the objects by way of the morphisms. Limit one morphism per relationships. Preorder.
• C) Store in the objects. Have many objects but not morphisms except for identity morphisms. This is a set.

Note that we can map C into A, express the Set as a Homset.

Category theory

• Maclane. Categories for the Working Mathematician.
• Mind wants to go from external relationships (soft wiring) to internal structure (hard wiring). Category theorists are trying to go in the opposite direction but they are out of their minds.
• Distinguishing amounts (colimits) and units (limits).
• Thomas Krantz. The Significance of Category Theory for Philosophy. He responded to my comment.
• https://en.m.wikipedia.org/wiki/Reflective_subcategory
• Those things are which show themselves to be. Colimit (inner structure - unconscious) shows itself by limit (external relationships - conscious)
• Set functions are not symmetric. But are list functions symmetric? Note that in the category Set the initial object is different from the terminal objects and so there is no zero object. But in the category of Vector spaces there is a zero object, and products equal coproducts.
• If products equal coproducts, then do all limits equal colimits?
• If a category has all products and all equalizers, then it has all limits! Math3ma

Internal structures - external relationships

• internal coproduct {$\textrm{Map}(\coprod_{k\geq 0}A_k,B)\cong\prod_{k\geq0}\textrm{Map}(A_k,B)$} external product of relations

Beauty - expressing everything in terms of external relations - is the key idea of category theory.

Relate Ellerman's heteromorphism and comma category.

San Francisco Meet Up interests: Dependently typed programming languages. Language aspects of category theory. Functional programming. Topos, lambda calculus. Is type theory advantageous? Modeling infinitesimals.

Dan Shiebler. Kan Extensions in Data Science and Machine Learning

Category theory

• Owen Lynch, Brandon Shapiro, David Spivak. All Concepts are Cat♯
• Awodey. Continuity and Logical Completeness: An Application of Sheaf Theory and Topoi
• Awodey, Reck. Completeness and Categoricity.
• What is the relationship between universal properties as proved by the function extensionality principle, and universal properties as given by Kan extensions?

Tai-Danae Bradley distinguishes between limits (stack) and colimits (glue).

• Colimits glue, limits cut out solutions to equations
• Vector space {$\mathbb{R}^X$} and (co)presheaf category {$\textrm{Set}^C$}.

Natural Transformations playlist

Kategorija yra pasaulis. Pasaulis nusako morfizmų nusakytą objektą, tai ką morfizmai išsaugoja tuose objektuose.

Consider the category where the objects are groups but the maps are set functions. Then is it different from the category of sets? It is practically the same. So it is not the objects which characterize the category but the morphisms. In general, what depends on the morphisms and what on the objects?

John Baez, Michael Shulman. Lectures on n-Categories and Cohomology.

• (-1)-categories are hom(x,y) sets where x and y are parallel 0-morphisms in a 0-category, which is to say, a set. But the only 0-morphisms in a set are the identity morphisms. Thus hom(x,y) is either an identity morphism (when x=y) or the empty set (otherwise). These are the two possible (-1)-categories.
• (-2)-categories are hom(x,y) sets where x and y are -1-morphisms in a -1-category. But there is only one non-empty (-1)-category and it has only one morphism. Thus there is only one (-2)-category and it consists of this unique morphism. This category expresses necessary equality when there is only one choice. That is reminiscent of the choice from a single choice which is modeled by {$F_1$}, the field with one element.

Tai-Danae Bradley, John Terilla, Yiannis Vlassopolous. An Enriched Category Theory of Language: From Syntax to Semantics.

• Categories for AI talk: Category Theory Inspired by LLMs - by Tai-Danae Bradley

Have all finite limits is equivalent to

• Having terminal objects
• Having a product for any pair of objects
• Having an equalizer for any pair of parallel arrows

These are the building blocks for limits

The notion of a singleton, and more broadly, membership in a set can be reworked in the language of category theory. Similarly, rework all of the Zermelo-Frankel axioms of set theory in the language of category theory.

Relate the three-cycle (taking a stand, following through, reflecting) to Kan extension as defined by filtering objects and morphisms, acting on them, and then taking the colimit or limit on that filter.

Theo Buhler. Exact categories

In Cartesian categories you can copy and delete information. (John Baez - Rosetta Stone) How does that relate to Turing machine?

Imagination acts through external relationships but there is also internal structure, beyond the imagination.

• Spivak and Kent: Ologs

Partial knowledge

• forgetful functors
• fibrations, lenses

David Spivak: "Cultivating Strategies". Finding the Right Abstractions.