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...
  3. N

    MHB Functors | 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 Type and Category 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

    It goes without saying that theoretical physics has over the years become overrun with countless distinct - yet sometimes curiously very similar - theories, in some cases even dozens of directly competing theories. Within the foundations things can get far worse once we start to run into...
  9. Math Amateur

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

    I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.5 Isomorphisms ... I need some further help in order to fully understand some aspects of Definition 1.4, Page 12 ... ... The start of Section 1.5, including Definition 1.4 ... reads as...
  10. Math Amateur

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

    I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.5 Isomorphisms ... I need some further help in order to fully understand some further aspects of Definition 1.3, Page 12, including some remarks Awodey makes after the text of the definition ... ...
  11. Math Amateur

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

    I am reading Tom Leinster's book: "Basic Category Theory" and am focused on Chapter 1: Introduction where Leinster explains the basic idea of universal properties ... I need help in order to fully understand the proof of Lemma 0.7 ... Lemma 0.7 and its proof read as follows:In the above proof...
  12. pairofstrings

    I System to represent objects in Mathematics

    Hi. Usually, Computer Programmers use Flow Charts, Algorithms, or UML diagrams to build a great software or system. In the same manner, in Mathematics, what do Mathematicians use to build a great system that they want to build. Category Theory is at the highest level of abstraction; then...
  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

    Insights Why Higher Category Theory in Physics? - Comments

    Urs Schreiber submitted a new PF Insights post Why Higher Category Theory in Physics? Continue reading the Original PF Insights Post.
  15. SrVishi

    Algebra Categories for the Working Mathematician

    Hello. I am about to start learning category theory. I keep hearing mixed opinions on the book Categories for the Working Mathematician, by Sanders MacLane (I am aware he is one of the founders of the theory). Some say it's a "must read", and others have called it "outdated." What would seem...
  16. N

    Trying To Learn Category Theory

    I'm trying to learn Category Theory; this isn't homework or anything. I've attached a problem from the text "Basic Homological Algebra" by Osborne and I show my attempt at a solution. My solution doesn't seem exactly correct and I state why in the attachment as well. Can someone take a look...
  17. Geofleur

    Something like category theory but for physics

    I was impressed how in R. Geroch's book, Mathematical Physics, category theory is used to unify so many different branches of mathematics. Is there a single framework that, in a similar way, unifies many or all branches of physics? If so, what are some good resources for learning it? So far...
  18. H

    Exploring the Equivalence of Category Theory and Versatility

    I am a final year math major thinking of taking Category Theory (CT). The problem is people keep telling me its useless. Why? I have a biology degree and decided to get a second degree in mathematics. My motivation for studying mathematics is related partly to an idea I stumbled across in high...
  19. N

    Understanding the Coproduct in Grp as a Universal Object

    Homework Statement Coproducts exist in Grp. This starts on page 71. of his Algebra. Homework Equations [/B] Allow me to present the proof in it's entirety, modified only where it's convenient or necessary for TeXing it. I've underlined areas where I have issues and bold bracketed off my...
  20. N

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

    A couple of notes first: 1. \hom_{A}(-,N) is the left-exact functor I'm referring to; Lang gives an exercise in the section preceeding to show this. 2. This might be my own idiosyncrasy but I write TFDC to mean 'The following diagram commutes' 3. Titles are short, so I know that the hom-functor...
  21. Math Amateur

    MHB Category Theory - Initial and Final Objects

    I am reading Paolo Aluffi's book, Algebra: Chapter 0. I am currently focused on Chapter I, Section 5: Universal Properties. I need some help Example 5.3 and Exercise 5.2 in this section. QUESTION 1Example 5.3 reads as follows: Can someone clearly explain why \emptyset is initial ... We...
  22. Mandelbroth

    Category Theory: Inverse Limit in Sets

    I think this looks like a homework problem, so I'll just put it here. Homework Statement Demonstrate that, for any index category ##\mathscr{J}## and any diagram ##\mathcal{F}:\mathscr{J}\to\mathbf{Sets}##, $$\varprojlim_{\mathscr{J}}A_j=\left\{a\in \prod_{j\in \operatorname{obj}(...
  23. Math Amateur

    MHB Category Theory - Definition of Hom(X,X) in Awodey

    I am reading Steve Awodey's book, "Category Theory" (Second Edition). In Chapter 1 within a small section on monoids, Awodey defines Hom_{Sets} (X,X) as follows: " ... ... for any set X, the set of functions from X to X, written as Hom_{Sets} (X,X) is a monoid under the operation of...
  24. Math Amateur

    MHB Category Theory - Isomorphisms of Posets

    I am beginning to read "Category Theory: Second Edition" by Steve Awodey. On page 12 (see attachment) he defines isomorphisms as follows: ------------------------------------------------------------------------------- Definition 1.3. In any category, C , an arrow f \ : \ A \to B is called an...
  25. Math Amateur

    MHB Category Theory - is it just "abstract nonsense"

    In my reading of various texts on abstract algebra on a number of topics, including recently, tensor products of modules, that a number of authors tend to make use of the concepts of category theory. Do members of MHB believe it is worth investing time on category theory? Will it bring a better...
  26. alyafey22

    MHB Exploring Category Theory: Discover the Beauty of Mathematics

    Hey guys , what is the best book to read about category theory ?
  27. verty

    Category Theory - what is its current state?

    When I was very young, I read a book from the 60's about classical mathematics. I was far too young to understand anything in it of course, but I remember it mentioned that category theory was the new thing that was threatening to become a new foundation for mathematics. So I was wondering...
  28. S

    Getting Started With Category Theory: What You Need to Know

    I am interested in learning category theory, just for extracurricular knowledge so I watched a 30 minute YouTube video but I had no idea what the guy was talking about :( what do I need to know before learning Cat. theory?
  29. S

    Algebra text with category theory

    I'm looking for a textbook that covers all of the standard undergrad algebra topics but from a more modern perspective. For example, most books reprove the isomorphism theorems for groups, rings, modules, instead of showing that all of these structures are universal algebras. Ideally the text...
  30. A

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

    I've attached the problem statement and the solution in a pdf file, please check it and see if I've done it correctly. I'm new to abstract reasoning, I've only had a one semester introduction to group-theory and parts of ring theory based on baby Herstein, so I need others to check my proofs to...
  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

    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

    Category Theory and 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.