Andrius Kulikauskas

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Introduction E9F5FC

Questions FFFFC0


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


Understand these key concepts

  • Understand the Yoneda lemma and what it says about knowledge and its four levels.
  • Understand how limits and colimits relate external relationships and internal structure.
  • Understand the Curry-Howard-Lambek correspondence and its implications, how it relates logic, execution and structure.
  • Understand the unity behind various concepts and constructions in category theory, perhaps through Kan extensions, and what that says about structures, perspectives, composition, identity and duality.
  • Understand the role of concepts from probability and statistics: nondeterminism, determinism, randomness, entropy, choice, possibility, actuality, sampling.

Philosophical connections

  • How does categorification relate to internalization, as with the representations of the sixsome?
  • In what sense are sequences, hierarchies, networks external relations as in category theory?


  • Can a category be simply considered as an algebra of paths? Which is to say, rather than think in terms of objects and arrows, simply think in terms of paths and the conditions on them: identity paths and composition of paths. Relate these paths to a matrix and to symmetric functions on the eigenvalues of a matrix.

The nature of category theory

  • In category theory, what is the relationship between structure preservation of the objects, internally, and their external relationships? What do I mean by internal structure and external relationships?
  • Think about how (structure preserving) morphisms relate the entire structure, whereas maps deal with particular elements. There is a two-level nature (parts and wholes). How are the two levels related?
  • Is it possible to categorify everything, that is, to understand all inner properties of a system in terms of external relationships?
  • In what sense are q-analogues the opposite of categorification?
  • Is there a category of universal properties?
  • What can graph theory (for example, random graphs, or random order) say about category theory?
  • How do Hopf algebras with multiplication and comultiplication relate internal structure and external relationships?
  • How do we get different kinds of categories (different kinds of "truths") so that we could have functors (from one species of category to another)?


  • In what sense is composition (and the underlying equivalence classes of path) a Turing machine? Compare with the word problem for group generators.
  • Learn more about branches of mathematics where category theory is important: algebraic topology, homology, homological algebra, algebraic geometry.

Algebra of requirements

  • What is the algebra of requirements (choices) in the unfolding of category theory in terms of various concepts such as cones, limits, adjunctions?
  • How are those requirements variously met, for example, in the case of adjunctions?


  • What is the role of bijections in category theory (natural transformations, Yoneda lemma).

Hidden assumptions

  • Is it possible to show that category theory presumes the Axiom of Choice?

How distinctions are made

  • Does category theory distinguish between automorphisms and isomorphisms?


  • In most every category, can we (arbitrarily) define (uniquely) distinguished "generic objects" or "canonical objects", which are the generic equivalents for all objects that are equivalent to each other? For example, in the category of sets, the generic set of size one.


  • Does a loop in string theory not have an object that it goes from and to? Is it an objectless relation? How does that relate to category theory? How does category theory talk about loops?


  • In category theory, how do we distinguish cardinalities? How can we distinguish countable and second countable?
  • What is the significance of a category which has one morphism from one object to another? What is the significance of its cardinality? Note that all objects are equivalent. On what basis can we distinguish such a category if it has one object, two objects, three objects, or four objects?
  • In the category of Sets, is there any way to distinguish between the integers and the reals? Are all infinities the same?
  • In the category Set, how can you distinguish between a countable and uncountable set?
  • What can combinatorics tell us about the category of finite sets? and categories in general?


  • In what sense and on what basis is the category of sets breaking duality by having an initial object but no terminal object? What is the basis for the empty function, what is its significance, what does it mean to have it or not to have it? What is the significance of the empty set? In what sense is the empty function describing a "do nothing" action? The empty set is the object which only has the do nothing action to itself. An initial object is a distinguished object, and likewise a terminal object is a distinguished object.


  • Understand functors in terms of information.
  • How do functors collapse information yet also place it in context?
  • How can information be understood as an inverse of what a functor does?
  • Compare functors (as analogies) with metaphors, blends.
  • Can a functor or a function add information? Investigate: A function can add context in the codomain.
  • Why are the notions of function, and functor, and exact sequence asymmetric? Do they build out from zero?

Limits and colimits

  • How to relate category theory concepts such as various limits and colimits with concerns and perspectives, and with God's perspective?
  • Understand category theory concepts in terms of examples from mathematics.

Functional programming

  • In functional programming with monoids and monads, can we think of each function as taking us from a question type to an answer type? In general, in category theory, can we think of each morphism as taking us from a question to an answer?
  • monad = black box?


  • At what level is equality defined in defining a category? Equality is needed for the properties of identity and associativity. But is it the same identity as the identity for other morphisms?


  • How to deal with self-identity or non-identity f of an M-category? with copies of an M-category? Perhaps by embedding it in a bigger system?

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

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

Applied Category Theory





Categorical logic and topos

Homotopy type theory

Higher category theory


Simplexes and simplicial sets

Network theory

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

Geometry and toposes

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.


  • 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.


  • 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.


  • 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).


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


  • 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.


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


  • 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]"


  • 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.


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".


  • 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



  • 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

Brendan Fong, David I Spivak

  • 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

  • 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
  • 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

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.
  • 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

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

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.

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