Category Theory¶
Job: unify mathematical structures via objects, morphisms, functors, and universal properties — the organizational language of modern algebra, topology, geometry, and logic.
Core¶
Category theory replaces “elements of sets” with “arrows and their composition.” Constructions characterized by universal properties are unique up to unique isomorphism and transport across disciplines. Functors and natural transformations are the morphisms of this meta-geometry; adjunctions encode free/forgetful and tensor/Hom dualities.
| Layer | Content |
|---|---|
| Categories & functors | \(\mathbf{Set}\), \(\mathbf{Grp}\), \(\mathbf{Top}\), \(\mathbf{R}\)-\(\mathbf{Mod}\); \(\operatorname{Hom}\)-sets |
| Natural transformations | morphisms of functors; functor categories \([\mathcal{C},\mathcal{D}]\) |
| Limits / colimits | products, equalizers, pullbacks; duals (pushouts, coeq.) |
| Adjunctions | free \(\dashv\) forgetful; unit/counit; Hom-tensor |
| Yoneda | \(y:C\hookrightarrow\operatorname{PSh}(C)\), \(C(-,X)\) fully faithful |
| Monoidal / enriched | \(\otimes\), closed categories; \(k\)-linear cats (pointer) |
| Abelian / triangulated | homological algebra setting |
| Higher (exit) | \((\infty,1)\)-categories, \(\infty\)-operads |
Reading map¶
| Page | Focus |
|---|---|
| Foundations | definitions through Yoneda and adjunctions; quals-usable package |
| Monads & Algebras | monads, Eilenberg–Moore/Kleisli, Barr–Beck, distributive laws |
| Monoidal & Enriched | tensor/coherence, closed/internal Hom, \(\mathcal{V}\)-enriched, \(k\)-linear |
| Accessible & Presentable | \(\kappa\)-accessible, locally presentable, combinatorial, Vopěnka |
| Categorical Logic & Topoi | first-order/internal, topos \(\Omega\), geometric morphism, synthetic |
| Topos theory | Sheaves, internal logic, geometric morphisms |
| Higher categories | \((\infty,1)\)-cats, \(\mathbb{E}_n\)-operads |
| HoTT | Univalence; synthetic homotopy |
| Homological algebra | Abelian/triangulated; derived functors |
| Derived AG | Derived rings, spectral schemes (pointer) |
Dependency order¶
- Comfortable algebra/topology examples (groups, modules, spaces) — not required formally, but needed for meaning.
- Foundations: categories \(\to\) functors \(\to\) natural transformations \(\to\) limits \(\to\) adjunctions \(\to\) Yoneda.
- Abelian categories \(\to\) homological algebra.
- Sheaves / toposes \(\to\) topos theory; then higher cats if needed.
Interfaces¶
| Area | Contact |
|---|---|
| Abstract algebra | modules, \(\operatorname{Hom}\), \(\otimes\), free functors |
| Algebraic topology | homotopy functors, homology as functor |
| Algebraic geometry | \(\mathbf{Sch}\), sheaves, fiber products as limits |
| Logic / type theory | CCC, toposes, univalent foundations |
| Representation theory | Tannakian reconstruction (pointer) |
Notation (paper-standard)¶
- \(\mathcal{C},\mathcal{D}\); \(\operatorname{Hom}_{\mathcal{C}}(X,Y)\) or \(\mathcal{C}(X,Y)\); \(1_X\)
- \(F:\mathcal{C}\to\mathcal{D}\), \(\eta:F\Rightarrow G\); \(F\dashv G\); unit \(\eta\), counit \(\varepsilon\)
- \(\lim\), \(\operatorname{colim}\), \(X\times Y\), \(X\sqcup Y\), pullback squares
- \(y(X)=\mathcal{C}(-,X)\); \(\operatorname{PSh}(C)=[C^{\mathrm{op}},\mathbf{Set}]\)
See it¶
Derived functors as universal \(\delta\)-functors. \(\operatorname{Ext}\), \(\operatorname{Tor}\), sheaf cohomology: once exactness failures of \(\operatorname{Hom}\) and \(\otimes\) are formalized, spectral sequences and long exact sequences are formal consequences — research papers assume this fluency.
Yoneda and moduli. Representable functors \(h_X\) encode “families of objects parameterized by \(X\).” Moduli problems ask for representing objects (or stacks when autos intervene). The fully faithful embedding \(C\hookrightarrow\operatorname{PSh}(C)\) is the formal reason “natural constructions are determined by Hom-functors.”
Hom-tensor adjunction. \(-\otimes_R N\dashv\operatorname{Hom}_R(N,-)\) is the prototype: free modules, extension of scalars, and sheaf \(\otimes\)/\(\underline{\mathrm{Hom}}\) are the same pattern.
Check¶
State the universal property of the product \(X\times Y\) without using elements.
Object \(P\) with \(\pi_X:P\to X\), \(\pi_Y:P\to Y\) such that for any \(Z\) with \(f:Z\to X\), \(g:Z\to Y\) there exists unique \(h:Z\to P\) with \(\pi_X h=f\), \(\pi_Y h=g\).
Why is ‘the’ free group on a set unique only up to unique iso?
Free group is left adjoint to forgetful \(\mathbf{Grp}\to\mathbf{Set}\); left adjoints unique up to unique natural iso when they exist.
What does the Yoneda lemma assert for a presheaf \(F\)?
\(\operatorname{Nat}(h_X,F)\cong F(X)\) naturally in \(X\) and \(F\); in particular \(y:C\to\operatorname{PSh}(C)\) is fully faithful.
Pitfalls¶
- Size: \(\mathbf{Set}\) is not small; “category of all categories” needs care (universes / conglomerates).
- Equality vs isomorphism: universal properties determine objects up to unique iso, not “on the nose.”
- Concrete categories (faithful forgetful to \(\mathbf{Set}\)) are special — not all categories are concrete (Freyd).
- Limits that “look like products of elements” fail in general categories; always use the UP.
Refs¶
- Mac Lane, Categories for the Working Mathematician
- Riehl, Category Theory in Context
- Borceux, Handbook of Categorical Algebra (reference)
- Leinster, Basic Category Theory (concise)
Next¶
Foundations · Topos theory · HoTT · Homological algebra · Higher categories · Algebraic geometry · Monads & Algebras · Monoidal & Enriched · Accessible & Presentable · Categorical Logic & Topoi