Category Theory: Foundations

Job: categories, functors, natural transformations, limits, adjunctions, and Yoneda — compact toolkit for reading modern algebra and topology.

Core

Categories

A category \(\mathcal{C}\) consists of objects \(\operatorname{ob}\mathcal{C}\), morphisms \(\mathcal{C}(A,B)\) (locally small: these are sets), associative composition, and identities \(\mathrm{id}_A\).

Example Objects Morphisms
\(\mathbf{Set}\) sets functions
\(\mathbf{Grp}\), \(\mathbf{Ab}\), \(\mathbf{Ring}\) groups, etc. homs
\(\mathbf{Top}\), \(\mathbf{hTop}\) spaces continuous / homotopy classes
\(R\text{-}\mathbf{Mod}\) left \(R\)-modules \(R\)-linear maps
\(\mathcal{C}^{\mathrm{op}}\) same objects reversed arrows

Iso: \(f\) with inverse. Mono / epi: left / right cancellable (in \(\mathbf{Set}\): injective / surjective). Initial \(0\): unique arrow \(0\to X\) all \(X\); terminal \(1\): unique \(X\to 1\). Zero object: both (e.g. \(0\) in \(\mathbf{Ab}\)).

Small: \(\operatorname{ob}\) is a set. Large: \(\mathbf{Set}\), \(\mathbf{Grp}\). Locally small: Hom-sets.

Functors and natural transformations

Functor \(F:\mathcal{C}\to\mathcal{D}\): map on objects and arrows preserving \(\mathrm{id}\) and \(\circ\). Contravariant: \(F:\mathcal{C}^{\mathrm{op}}\to\mathcal{D}\).

Natural transformation \(\alpha:F\Rightarrow G\): arrows \(\alpha_X:FX\to GX\) with \(Gf\circ\alpha_X=\alpha_Y\circ Ff\) for all \(f:X\to Y\). Natural iso: each \(\alpha_X\) iso.

Functor category \([\mathcal{C},\mathcal{D}]\): objects functors, morphisms natural transformations.

Limits and colimits

A limit of \(F:J\to\mathcal{C}\) is a universal cone: object \(L\) with \(\pi_j:L\to Fj\) commuting with \(F\) on arrows in \(J\), terminal among such. Colimit: dual (universal cocone).

Limit Colimit
product \(\prod X_i\) coproduct \(\coprod X_i\)
equalizer coequalizer
pullback pushout
inverse limit \(\varprojlim\) direct limit \(\varinjlim\)
terminal object initial object

\(\mathbf{Set}\), \(R\text{-}\mathbf{Mod}\), \(\mathbf{Top}\) are complete and cocomplete (all small limits/colimits exist). Finite limits: products + equalizers.

Adjunctions

\(F:\mathcal{C}\to\mathcal{D}\) left adjoint to \(G:\mathcal{D}\to\mathcal{C}\) (\(F\dashv G\)) if natural bijection $$ \mathcal{D}(FX,Y)\cong\mathcal{C}(X,GY) $$ or equiv. unit \(\eta:\mathrm{id}_{\mathcal{C}}\Rightarrow GF\) and counit \(\varepsilon:FG\Rightarrow\mathrm{id}_{\mathcal{D}}\) satisfying triangle identities.

Left \(F\) Right \(G\) Example
Free group Forget \(\mathbf{Grp}\to\mathbf{Set}\) words mod relations
Free \(R\)-module Forget to \(\mathbf{Ab}\)/\(\mathbf{Set}\)
\(-\otimes_R N\) \(\operatorname{Hom}_R(N,-)\) tensor-hom adjunction
Sheafification Forget sheaves \(\to\) presheaves
Suspension Loop space (homotopy) \(\Sigma\dashv \Omega\)

Left adjoints preserve colimits; right adjoints preserve limits. Reflective subcategory: inclusion has left adjoint (e.g. abelianization \(\mathbf{Grp}\to\mathbf{Ab}\)).

Yoneda lemma

Theorem (Yoneda). For \(F:\mathcal{C}^{\mathrm{op}}\to\mathbf{Set}\) and object \(A\), $$ \operatorname{Nat}\bigl(\mathcal{C}(-,A),F\bigr)\cong F(A), $$ naturally. In particular \(y:A\mapsto\mathcal{C}(-,A)\) is fully faithful (Yoneda embedding \(\mathcal{C}\hookrightarrow[\mathcal{C}^{\mathrm{op}},\mathbf{Set}]\)).

Moral: \(A\) is determined by the functor of maps into \(A\); “generalized elements” \(X\to A\).

In abelian categories (e.g. \(R\text{-}\mathbf{Mod}\)): kernels, cokernels, exact sequences. Additive functors; left/right exactness. Derived functors via resolutions — see modules.

See it

Universal property of localization. \(S^{-1}R\) is initial among rings receiving a map from \(R\) that inverts \(S\) — explains why localization is unique and functorial without inspecting fractions.

Eilenberg–Steenrod. Homology theories characterized by axioms (functoriality, homotopy, excision, dimension) — uniqueness on CW complexes; category theory organizes “what a homology theory is.”

Check

Prove that left adjoints preserve initial objects (hence colimits in general: special case).

\(\mathcal{D}(F0,Y)\cong\mathcal{C}(0,GY)\) is a singleton for all \(Y\) \(\Rightarrow\) \(F0\) initial in \(\mathcal{D}\).

Is the forgetful \(U:\mathbf{Top}\to\mathbf{Set}\) monadic? (One-line pointer.)

Yes: algebras for the ultrafilter monad / compact Hausdorff is the cleaner monadic forgetful; general \(\mathbf{Top}\) is topological over \(\mathbf{Set}\) but monadicity fails for all spaces. (Quals: know free-forgetful adjunctions; monads optional.)

Pitfalls

  • Covariance of \(\operatorname{Hom}(A,-)\) vs contravariance of \(\operatorname{Hom}(-,B)\) — mix-ups break diagram chases.
  • Limits in \(\mathbf{Top}\): underlying set is the \(\mathbf{Set}\)-limit, but topology is initial/final as appropriate — not always obvious.
  • “Canonical” maps should mean natural or uniquely determined by UP — not merely “standard choice.”
  • Equivalence of categories (fully faithful + essentially surjective) \(\neq\) isomorphism of categories (inverse on the nose).

Refs

  • Mac Lane, Categories for the Working Mathematician
  • Riehl, Category Theory in Context
  • Leinster, Basic Category Theory

Next

Category theory overview · Modules · Group theory · Algebraic topology · Abstract algebra · Monads & Algebras · Monoidal & Enriched · Accessible & Presentable · Categorical Logic & Topoi