文章 发现 ms@ms.lt +370 607 27 665 My work is in the Public Domain for all to share freely. 读物 书 影片 维基百科 Software Upload 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 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? 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 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)? 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? M-category 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 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. 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 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 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 Philosophy Simplexes and simplicial sets 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 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. 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. 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 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. 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
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