Category theory
Mathematics done in terms of objects and the maps between them, where the maps carry the content and the internal structure of an object is not consulted. A category is objects, morphisms, composition and identities; a functor maps one category to another preserving that; a natural transformation maps one functor to another. Nothing in the definitions mentions elements, which is the point.
One source: bergman-invitation-general-algebra, where it arrives in Chapter 7 as a tool for universal-algebra rather than as the subject. Roughly 180 pages of that book are category theory.
What the source builds
Chapter 7 defines categories, works through examples, handles the size problem with universes, then does functors, contravariance, properties of morphisms (mono, epi and their relatives), special objects, morphisms of functors (natural transformations), properties of functor categories, and closes with a sketch of enriched categories.
Chapter 8 is the payoff, and its title is the book's: universal constructions.
- Initial and terminal objects — the simplest universal property.
- Representable functors and Yoneda's Lemma.
- Adjoint functors, the organising idea; free-algebra constructions become left adjoints.
- Direct and inverse limits, then limits and colimits in general, with two sections on which functors respect which and one on how limits and colimits interact.
- Existence theorems, and morphisms and contravariance for adjunctions.
- An unusual concrete interlude: the p-adic numbers, developed as an inverse limit.
Chapter 10 ends at Freyd's criterion for the existence of adjoints, which Bergman names "the climax of the course" and credits to Freyd's paper as the source of the chapter's ideas.
A tool, not a foundation
This source treats category theory instrumentally. It is introduced 200 pages in, after the reader has already met free groups, coproducts, tensor products and abelianization by hand — so the abstractions arrive as names for patterns the reader has already worked through, not as first principles. Bergman's own reading advice reinforces it: the hard material "begins with Chapter 8", and §§7.9–7.11 (natural transformations onward) are on the difficult list.
That ordering is a pedagogical claim, and it belongs beside the spoke's others. Where judson-abstract-algebra puts applications early because students meeting rigorous proof need motivation, Bergman puts a hundred pages of worked special cases before the general theory for the same reason, one level up: "I often first approach general results through very particular cases."
What the spoke does not have
No dedicated category-theory text — this page rests entirely on chapters inside a general-algebra book, and inherits that book's selection. Bergman himself notes that one important area of the subject is left out and hands it to another reference. Monoidal categories, topoi and the homotopy-theoretic side are all absent here. Treat this as the categorical apparatus a working algebraist needs, not as a survey of the field.
Related
universal-algebra · bergman-invitation-general-algebra · abstract-algebra · george-bergman · mathematics