What is Category theory: Definition and 57 Discussions

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms). A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Informally, category theory is a general theory of functions.
Several terms used in category theory, including the term "morphism", are used differently from their uses in the rest of mathematics. In category theory, morphisms obey conditions specific to category theory itself.
Samuel Eilenberg and Saunders Mac Lane introduced the concepts of categories, functors, and natural transformations from 1942–45 in their study of algebraic topology, with the goal of understanding the processes that preserve mathematical structure.
Category theory has practical applications in programming language theory, for example the usage of monads in functional programming. It may also be used as an axiomatic foundation for mathematics, as an alternative to set theory and other proposed foundations.

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  1. John Bernier

    Towards a new theory of information kinematics

    The physical world is highly parallel; many things are happening side by side simultaneously. This is true not only at the macroscopic scales that we can see but as far as we know down to the subatomic scales. This is referred to as the principle of locality [1], which states that an object is...
  2. A Sensible Introduction to Category Theory

    A Sensible Introduction to Category Theory

    Remember when I used a video with a coconut in the thumbnail to drive a stake through the heart of mathematical structure? Today, in this introduction to the...
    • malawi_glenn
    • Media item
    • Category theory
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    • Category: Set Theory & Logic
  3. N

    MHB What are the tasks that need help with in category theory?

    Hello, does anyone can help me with these tasks: I have no idea how to solve it
  4. snypehype46

    I Functor between the category of Hilbert Space and the category of sets

    I have a question that is related to categories and physics. I was reading this paper by John Baez in which he describes a TQFT as a functor from the category nCob (n-dimensional cobordisms) to Vector spaces. https://arxiv.org/pdf/quant-ph/0404040.pdf. At the beginning of the paper @john baez...
  5. S

    I Category Theory and the Riemann Hypothesis

    YouTube has been suggesting videos about category theory of late, and I have spent some time skimming through them, without really understanding where it's all going. A question came to mind, namely: It seems reasonably conceivable that group theory could perhaps supply a vital key to the...
  6. heff001

    A Are Type Theory and Category Theory Viable Alternatives to Set Theory?

    Are Type Theory and Category Theory alternatives to Set Theory?
  7. L

    A Structure preserved by strong equivalence of metrics?

    Let ##d_1## and ##d_2## be two metrics on the same set ##X##. We say that ##d_1## and ##d_2## are equivalent if the identity map from ##(X,d_1)## to ##(X,d_2)## and its inverse are continuous. We say that they’re uniformly equivalent if the identity map and its inverse are uniformly...
  8. Auto-Didact

    A Taxonomy of Theories in Theoretical Physics

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  9. Math Amateur

    MHB Category Theory .... Groups and Isomorphisms .... Awodey, Section 1.5 .... ....

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  10. Math Amateur

    MHB Category Theory .... Isomorphisms and Inverses .... Awodey, Section 1.5 .... ....

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    MHB Universal Propeties in Category Theory .... Leinster Lemma 0.7 ....

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  12. pairofstrings

    I System to represent objects in Mathematics

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  13. Krunchyman

    I What is the equivalent of a group in category theory?

    I understand that in group theory, a group consists of a set and a binary operation for the elements in the set, and of course all the group axioms. But if we move away from set theory into category theory, is a group defined on a category?
  14. Urs Schreiber

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  15. SrVishi

    Algebra Categories for the Working Mathematician

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  16. N

    Trying To Learn Category Theory

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  17. Geofleur

    Something like category theory but for physics

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  18. H

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  19. N

    Understanding the Coproduct in Grp as a Universal Object

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  20. N

    [itex]\hom_A(-,N)[/itex]Functor Takes Coproducts to Products

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  25. Math Amateur

    MHB Category Theory - is it just "abstract nonsense"

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  27. verty

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  28. S

    Getting Started With Category Theory: What You Need to Know

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  29. S

    Algebra text with category theory

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  30. A

    Category theory: Prove that two given short Ex. Seq.s are isomorphic.

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  31. A

    What are the prerequisites for studying category theory?

    well, as always, I initially took a look at what wikipedia says. the idea of talking about general mathematical objects and arrows between them sounds pretty impressive and quite exciting to me, but just like any other math stuff, the idea looks quite simple and the examples that wikipedia gives...
  32. S

    Category theory : product and coproduct

    Hi, I was trying to understand coproduct and product as defined in category theory from the website http://en.wikiversity.org/wiki/Introduction_to_Category_Theory/Products_and_Coproducts. Before I could even think of sth difficult there are some simple things which I don't seem to...
  33. F

    Category Theory: Intro Texts & Recommended Books

    I'm looking to start studying Category Theory. What books would you recommend for these? I can't seem to find an introductory book on categories. What is the standard text for such a thing?
  34. S

    Category Theory Used in Physics.

    I just have a quick, but odd question. Next term I will be taking a Category Theory class with a visiting professor from Moscow. The professor is known to have done most of his work in Physics. Does anyone have any experience using results from Category Theory applied to Physics? It just...
  35. Fredrik

    Model theory, category theory and universal algebra

    If someone is willing to briefly explain what these three branches of mathematics are about, I'd appreciate it. I don't even understand if they are three completely different things or if they're overlapping a lot. I understand the definition of a category, and Landau just made me aware of the...
  36. R

    Category theory - right adjoint problem

    I'm stuck with the attached practice problems. does anyone with knowledge of category theory please have solutions?
  37. R

    Looking for Category Theory Paper

    Is there any free (and legal, obviously) place where I can acquire a copy of Eilenberg and MacLane's paper "The General Theory of Natural Equivalences" on the internet? I'm currently studying category theory and am interested in finding a copy of this paper. Thanks!
  38. F

    Simple category theory isomorphism

    Hey all, Okay, let me give this a wack. I want to show that A \times 1 is isomorphic to A. I'm aware that this is trivial, even for a category theory style. However, sticking to the defs and conventions is tricky if you aren't aware of the subtleties, which is why I'm posting this. So here...
  39. M

    Help w/ Proof in Category Theory

    I don't understand this proof, specifically the part in red, I don't understand. Please help me understand this step in the proof. Thanks! Let Tors be the category whose objects are torsion abelian groups; if A and B are torsion abelian groups, we define \text{Mor}_{\text{\textbf{Tors}}}(A...
  40. T

    Abstract Math: Beyond Category Theory

    Category theory is considered extremely abstract. What are some other branches of mathematics which are considered as abstract or even more abstract then category theory?
  41. M

    Where to Find Introductory Books on Category Theory?

    Does anyone know of a good introductory book on category theory?
  42. E

    Category theory and Computer Science

    There is a similar thread below, but I think it is more appropriate to make a new thread because it asks totally different questions. I am new to category theory and thinking about its possible application to computer science. Deductive systems based on first order logic (sound and complete)...
  43. T

    Set theory and category theory

    They seem to be different fields but both try to underpin maths. There has been suggestions that set theory is problematic, where some paradoxes cannot be resolved. But how about Category theory? Any problems or paradoxes? Is it more promising then set theory?
  44. S

    Category theory question

    Homework Statement Show that there are only two possible categories with one object and two morphisms. Homework Equations None The Attempt at a Solution My thinking here is as follows : Let's say that the object in our category is A, and that the two morphisms are f and 1, where...
  45. M

    Books on Character theory and category theory

    As the title says i would like to get recommended some books on Character theory and category theory. I like my books to be really matematically and to not skip corners, but I don't like books that think that I'm a master mind and just write 'clear' or 'easy' instead of proving the theorem...
  46. N

    Dashed arrow and solid arrow in category theory

    In category theory,what is the difference between dashed arrow and solid arrow?I am just curious why there is not any textbook which I could find mention about it formally .heh...
  47. K

    How Does Category Theory Enhance Understanding of Physics?

    I am currently working my way through classical Yang-Mills theory with the help of John Baez's book on gauge fields and some others. I have recently just began to notice the new, well new to myself, research on higher gauge theory. This looks very interesting but I feel that my background in...
  48. Shaun Culver

    Exploring the Link between Math Structures with Category Theory

    Does 'category theory' link different mathematical structures? Can you give a superficial example of how this has been done (eg. Name two mathematical structures and, with waving of hands, explain how they have been linked)? Could you direct me to a link where this has been done elegantly in...
  49. L

    Category Theory: ?What is up with these Diagrams?

    Category Theory: ?What is up with these Diagrams? So I found a basic category theory book online and was trying to learn some of the basics. Many of the proofs are done in diagram form (and it seems to very greatly reduce their lengths). However, no where in the book does the author prove...
  50. S

    Learn Category Theory: Best Introductory Books & Articles w/ Physics Examples

    Essentially I'd just like to learn more about it in my own spare time. Is there any particularly good introductory books or articles? I'd particularly like anything that uses examples from physics, but it isn't essential.
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