Category Theory
The mathematics of structure itself — objects and arrows, functors and naturality, universal properties and limits, the Yoneda lemma, and adjunctions.
Categories, Functors & Naturality
- Categories & Examples — Objects, arrows, composition, identities — and why a single monoid or ordered set already is a category.
- Isomorphisms, Monos & Epis — Injective and surjective, rebuilt out of arrows — and the categories where they come apart.
- Functors — Maps between categories: objects and arrows carried across, composites intact, sometimes with the arrows turned around.
- Natural Transformations — Maps between functors: one arrow per object, all of them fitting into commuting squares — the precise meaning of a construction with no arbitrary choices.
- Duality & Opposite Categories — Reverse every arrow and every theorem comes with a second one, proved for free.
- Equivalence of Categories — Isomorphism of categories demands a bijection on objects and almost never holds; equivalence asks only for full, faithful and essentially surjective — and…
Universal Properties & Limits
- Initial & Terminal Objects — The first universal property: an object fixed not by what it contains but by there being exactly one arrow out of it, or into it.
- Products & Coproducts — A cartesian product, a disjoint union and a greatest common divisor, all pinned down by the same demand about arrows.
- Equalizers & Coequalizers — Cut a domain down to where two maps agree, or squash a codomain until they must — and discover that kernels and quotient groups were these all along.
- Pullbacks & Pushouts — Intersections, preimages and matching pairs are one construction; turn the arrows around and it becomes gluing.
- Limits & Colimits — Terminal objects, products, equalizers and pullbacks are one construction read over four different shapes.
Yoneda & Adjunctions
- Representable Functors — Some functors only look like they build something new: they are the arrows out of one fixed object, which is what makes forgetting structure the same as…
- The Yoneda Lemma — A natural transformation out of a hom-functor is determined by where it sends one identity arrow, which is why an object is exactly what it looks like from…
- Adjoint Functors — A homomorphism out of a free group is just a choice of images for the generators — one bijection of hom-sets, natural in both variables, and the same pattern…
- Units, Counits & Triangle Identities — Feed the identity arrows into the adjunction bijection and two natural transformations fall out; the triangle identities say that a generator, read as a…
- Adjoints Preserve Limits — A right adjoint cannot lose a limit — which turns "does this functor have an adjoint?" into a check you can finish on one line.
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