概论
数学
发现
Andrius Kulikauskas
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用中文
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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
范畴论
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 CurryHowardLambek 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?
Category
 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 twolevel 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 qanalogues 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)?
Composition
 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?
Bijections
 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?
Elements
 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.
Loops
 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?
Cardinality
 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?
Duality
 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.
Functors
 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?
Equality
 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?
Mcategory
 How to deal with selfidentity or nonidentity f of an Mcategory? with copies of an Mcategory? 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)
 CurryHowardLambek 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.
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 contentwise trivial or universal  not related to the subject, but simply an aspect of abstraction.
Category Theory Study Materials
Introductory materials
Applied Category Theory
Programmers
Music
Physics
Categorification
Categorical logic and topos
Homotopy type theory
Higher category theory
Philosophy
Simplexes and simplicial sets
Network theory
Lie theory
 BorelWeil theorem The BorelWeilBott 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 flipside 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 limitscolimits 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 multidimensional 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 superwidget 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 Theorem.
Mcategory, 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
 Willemsstyle types of behaviors (control theory)
 Hamiltonian systems
 Lagrangian systems
 inside  variables (output) as components of states (variables depend on each other) postcomposition, elements
 negation?  states
 outside  variables (input) as components of states, precomposition, whole
 network to sequence, tree, network
 limit and colimit
Substitution  make inner collapse
Brendan Fong, David I Spivak
 nyccategorytheory.slack.com
 https://www.youtube.com/watch?v=5I7v9mvOC2E 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.
