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  1. More Images

    Category theory

    Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in almost all areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. Many areas of computer science also rely on category theory, such as functional programming and semantics. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the source and the target of the morphism. Metaphorically, a morphism is an arrow that maps its source to its target. Morphisms can be composed if the target of the first morphism equals the source of the second one. Wikipedia

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  2. plato.stanford.edu

    Category theory has come to occupy a central position in contemporary mathematics and theoretical computer science, and is also applied to mathematical physics. Roughly, it is a general mathematical theory of structures and of systems of structures. As category theory is still evolving, its functions are correspondingly developing, expanding ...
  3. Category theory takes a bird's eye view of mathematics. From high in the sky, details become invisible, but we can spot patterns that were impossible to de-tect from ground level. How is the lowest common multiple of two numbers like the direct sum of two vector spaces? What do discrete topological spaces,
    Author:Tom LeinsterPublished:2014
  4. We will now give a very short introduction to Category theory, highlighting its relevance to the topics in representation theory we have discussed. For a serious acquaintance with category theory, the reader should use the classical book [McL]. Definition 6.1. A category C is the following data: (i) a class of objects Ob(C);
  5. math.jhu.edu

    category theory is mathematical analogy. Specifically, category theory provides a mathe-matical language that can be deployed to describe phenomena in any mathematical context. Perhaps surprisingly given this level of generality, these concepts are neither meaningless and nor in many cases so clearly visible prior to their advent.
  6. web.auburn.edu

    A \category" is an abstraction based on this idea of objects and morphisms. When one studies groups, rings, topological spaces, and so forth, one usually focuses on elements of these objects. Category theory shifts the focus away from the elements of the objects and toward the morphisms between the objects.
  7. web.math.ucsb.edu

    %PDF-1.5 %ÐÔÅØ 4 0 obj /S /GoTo /D (section.0) >> endobj 7 0 obj (Introduction) endobj 8 0 obj /S /GoTo /D (section.1) >> endobj 11 0 obj (What is a Category?) endobj 12 0 obj /S /GoTo /D (subsection.1.1) >> endobj 15 0 obj (Basic Definitions and First Examples) endobj 16 0 obj /S /GoTo /D (subsection.1.2) >> endobj 19 0 obj (Thinking in Diagrams) endobj 20 0 obj /S /GoTo /D (subsection.1. ...
  8. Category theory is a relatively new branch of mathematics that has transformed much of pure math research. The technical advance is that category theory provides a framework in which to organize formal systems and by which to translate between them, allowing one to transfer knowledge from one field to another. But this same organizational framework also has many compelling examples outside of ...
  9. en.wikipedia.org

    The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows (also called morphisms, although this term also has a specific, non category-theoretical sense), where these collections satisfy certain ...
  10. pi.math.cornell.edu

    Category theory has been around for about half a century now, invented in the 1940's by Eilenberg and MacLane. Eilenberg was an algebraic topologist and MacLane was an algebraist. They realized that they were doing the same calcu-lations in different areas of mathematics, which led them to develop category theory.

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