Alphabeta Math

Category Theory

42 pages in 3 parts

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. This collection develops that language and the theorems that make it worth having. Functors and natural transformations turn constructions and comparisons between them into objects of study. The Yoneda lemma says an object is determined by the arrows into it, which is what allows a universal property to serve as a definition. Limits and colimits specify an object by the arrows to or from a diagram, so products, coproducts, equalisers, pullbacks and their duals become one construction with one uniqueness argument. An adjunction pairs two functors through a natural bijection of hom-sets, equivalently through a unit and a counit satisfying the triangle identities, and left adjoints preserve colimits while right adjoints preserve limits. Reflective subcategories are the case where an inclusion has a left adjoint, the adjoint functor theorems say when an adjoint exists at all, and every adjunction induces a monad and a comonad whose algebras measure how much of it the monad remembers.

Homological algebra is the declared consumer. Its chain complexes, long exact sequences, derived functors, Ext, Tor and spectral sequences are scaffolded to take their additive and abelian categories, their exactness criteria and their diagram lemmas from the continuation of this track, and to quote the Yoneda lemma and the pointwise computation of limits in functor categories. Algebraic geometry reaches the same way, through commutative algebra.

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 Colimits46 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, 5 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.