Alphabeta Math

Category Theory

42 pages in 3 parts

Category theory packages objects, arrows, and composition into a language for mathematical structures and universal constructions. The published pages develop categories, functors, natural transformations, and the Yoneda lemma; limits and colimits; adjunctions, reflective subcategories, and monads; and additive and abelian categories, exactness, and diagram lemmas. Further topics include Kan extensions and density, ends and coends, enriched and functor categories, monoidal and closed categories, rigidity, braiding, tensor and fusion categories, and Grothendieck axioms. Universal properties organize products, equalizers, pullbacks, and their duals, while adjunctions relate functors through hom-set bijections. Examples range from algebraic and topological categories to these shared constructions.

Set Theory, Abstract Algebra, Linear Algebra, and Topology supply foundational conventions and examples used by these pages. Homological Algebra builds on the additive and abelian material, exactness, and diagram lemmas; Algebraic Geometry and Algebraic Topology use categorical constructions in their own developments. Braid Groups and Categorification and Geometric Representation Theory also use its monoidal, functorial, and universal-property methods.

Pathway

The parts run in order. Everything a page needs from this group has been read by the time you reach it, and the level on each row is how many dependency steps into the group that page sits.

  1. Part 1 · Categories, functors and Yoneda

    2 pages

    A category is objects, arrows and composition, and the point of the definition is that sets, groups, rings, vector spaces, modules, spaces and posets all satisfy it. Functors and natural transformations make constructions and comparisons themselves objects of study, and the Yoneda lemma says an object is determined by the arrows into it, which is what turns a universal property into a definition.

  2. Part 2 · Limits and adjunctions

    4 pages · after Part 1

    Limits and colimits specify an object by all arrows to or from a diagram, so products, equalisers, pullbacks and their duals become instances of one universal construction. Adjunctions express the same economy for functors, and ends, coends and weighted limits extend it to bifunctors and naturality. The added page shows how this universal language becomes additive: preadditive categories identify hom-sets with abelian groups, biproducts force the needed enrichment, matrix calculus organises maps between finite biproducts, and kernels and cokernels recover finite limits and colimits. Additive functors and idempotent completion then supply the standard structural consequences.

    • Limits and Colimits47 results

      Categories, functors, natural transformations, opposite categories, universal objects, representable functors, and the Yoneda lemma supply the language for specifying an object through all morphisms to or from it.

      11 definitions, 6 lemmas, 6 propositions, 17 theorems, 6 corollaries, 1 remarkExamples & counterexamples
    • Adjunctions Units and Counits51 results

      Categories, functors and natural transformations are published, together with whiskering and horizontal composition, the interchange law, functor categories, comma…

      6 definitions, 3 lemmas, 6 propositions, 23 theorems, 4 corollaries, 4 false statements, 5 remarksExamples & counterexamples
    • Ends Coends and Weighted Limits50 results

      Opposite and product categories, natural transformations, cones, limits, and the Yoneda machinery already provide the ordinary categorical language used here.

      8 definitions, 1 lemma, 3 propositions, 23 theorems, 7 corollaries, 6 false statements, 2 remarksExamples & counterexamples
    • Preadditive and Additive Categories and Biproducts60 results

      This page develops the thesis that additivity is not extra decoration on a category once finite biproducts exist.

      10 definitions, 7 propositions, 24 theorems, 9 corollaries, 3 counterexamples, 5 false statements, 2 remarksExamples & counterexamples

  3. Part 3 · Reflections, adjoint functors and monads

    15 pages · after Part 2

    Reflective subcategories and adjoint functor theorems produce left adjoints; monads, comonads, and Beck encode that data algebraically. Abelian categories supply kernels, cokernels, exactness, generators, and diagram lemmas, while monoidal, closed, braided, and symmetric structures organize tensor products, coherence, internal homs, and commutativity. Duality and rigidity give tensor adjunctions, dual morphisms, and pivotal, spherical, and ribbon refinements of trace and dimension. Enrichment transfers these constructions to a monoidal base through enriched Yoneda, tensors, cotensors, weighted limits, and adjunctions. Finally, finite rigid k-linear abelian multitensor categories have exact tensoring; finite semisimplicity gives multifusion categories, and a simple unit gives fusion categories. Their Grothendieck rings encode fusion rules and duality's anti-involution, distinguishing a fusion unit from a possibly split multifusion unit.