Categories, functors, natural transformations and their universal constructions. Modern foundational language linking algebra, topology, logic and computer science (type theory, programming-language semantics, databases).
category-theory
Category
A collection of objects and morphisms (arrows) between them, with associative composition and identity arrows. Examples: Set, Grp, Top,…
Object
A 'thing' in a category. Internal structure is invisible; everything is accessed through morphisms. Dually, an arrow into or out of an…
Morphism (arrow) f: A → B
A structured 'map' between two objects. Must compose and have identities. In Set these are functions; in Grp, group homomorphisms; in Top,…
Isomorphism
A morphism f: A → B with a two-sided inverse. Captures 'the right notion of sameness' in every category: bijection in Set, homeomorphism in…
Functor F: C → D
A structure-preserving map between categories: sends objects to objects, morphisms to morphisms, preserves composition and identities.…
Natural transformation η: F ⇒ G
A family of morphisms η_A: F(A) → G(A) commuting with every morphism of C. Morphisms between functors — arrows between arrows.
Universal property
A characterisation of an object by the arrows in or out of it that satisfy a terminality condition. Determines the object uniquely up to…
Limit and colimit
Universal cone / cocone over a diagram. Products, pullbacks and equalisers are limits; coproducts, pushouts and coequalisers are colimits.
Adjunction F ⊣ G
A pair of functors with a natural bijection Hom_D(F A, B) ≅ Hom_C(A, G B). Free/forgetful pairs are the paradigm; every adjunction gives a…
Yoneda lemma
For any functor F: C → Set and object A, natural transformations Hom_C(A, −) ⇒ F are in bijection with F(A). 'An object is determined by…
Monad
An endofunctor T : 𝒞 → 𝒞 with natural transformations η : Id ⇒ T (unit) and μ : T² ⇒ T (multiplication) satisfying associativity and unit…
Kan extension
The left (Lan) or right (Ran) Kan extension of F : 𝒞 → 𝓔 along K : 𝒞 → 𝒟 is the universal functor 𝒟 → 𝓔 with a natural transformation…
Pullback and pushout
Pullback (limit of a cospan A→C←B) and pushout (colimit of a span A←C→B); specialise to fibre products, amalgamated free products, fibre…
Topos
A category behaving like Set: finite limits, exponentials, and a subobject classifier Ω. Elementary toposes host intuitionistic…
Sheaf
A presheaf ℱ : 𝒪(X)^op → Set satisfying the gluing axiom: compatible local sections on an open cover patch to a unique global section. The…
Derived category D(𝒜)
Localisation of the homotopy category of chain complexes over an abelian category 𝒜 at quasi-isomorphisms. Houses derived functors and…
Model category
Quillen's framework: a category with three distinguished classes (weak equivalences, fibrations, cofibrations) satisfying lifting and…
∞-category (quasi-category)
A simplicial set whose inner horns admit fillers (Boardman–Vogt/Joyal). Models (∞,1)-categories; Lurie's Higher Topos Theory develops the…
Adjoint functors
F ⊣ G iff Hom(F A, B) ≅ Hom(A, G B) natural in A, B. Universal constructions (free/forgetful, product/diagonal, tensor/hom). Right adjoints…
Monoidal category
Category with ⊗ bifunctor + unit I + associator/unitor coherence. String diagrams. Braided, symmetric, ribbon variants. Basis of TQFT,…
Topos
Category 'like Set': finite limits + power objects + subobject classifier. Elementary (Lawvere-Tierney) vs Grothendieck. Internal logic =…
Enriched category
Hom-sets replaced by objects of monoidal V. Ab-enriched = additive; sSet-enriched = simplicial; Cat-enriched = 2-categories.
2-category & bicategory
Objects, 1-morphisms, 2-morphisms with vertical/horizontal composition. Strict (assoc on nose) vs weak (bicategory, Benabou). Example: Cat.
Grothendieck topology / site
Family of covering sieves on each object satisfying stability, local-character, base-change. Generalizes open covers; basis for sheaves.
Descent theory
Glue local data to global object over a cover. Faithfully-flat descent (Grothendieck); effective for quasi-coherent sheaves; stackification.
Quillen adjunction / equivalence
Left adjoint preserves cofibrations + trivial cofibs. Derived adjunction LF ⊣ RG on Ho. Equivalence when induces equivalence of homotopy…
Simplicial set
Functor Δ^op → Set. Kan complexes = fibrant; nerve of small category is sSet. Model for ∞-groupoids (homotopy types).
Sheaf topos Sh(C,J)
Sheaves on a site form Grothendieck topos. Elementary: Cartesian closed + subobject classifier + finite limits/colimits.
Abelian category
Additive with kernels, cokernels, mono = ker of coker, epi = coker of ker. Freyd-Mitchell: small abelian embeds in R-Mod. Derived functors…
Triangulated category
Additive with shift [1] and distinguished triangles X→Y→Z→X[1] satisfying axioms (rotation, octahedral). Verdier; D(A), stable ∞-cat.
Ends and coends
End ∫_c F(c,c) of a functor F: C^op × C → D is the universal dinatural wedge (equalizer of the two natural maps). Coend ∫^c is the dual…
Day convolution
Given a symmetric monoidal base (C, ⊗), the presheaf category [C^op, V] carries a symmetric monoidal structure via the Day coend formula.…
Grothendieck construction
Equivalence between pseudofunctors C → Cat and Grothendieck fibrations over C. The total category ∫F has objects (c, x) with x ∈ F(c) and…
Beck-Chevalley condition
For a commutative square of functors that is a pullback in Cat, the mate of its identity 2-cell under adjunction is an isomorphism. In Set:…
Distributive law of monads
A natural transformation λ: TS ⇒ ST satisfying compatibility with the units and multiplications of S and T. Yields a monad structure on ST…
Coequalizer
Colimit of a parallel pair f, g : A ⇒ B: the universal object Q with q: B → Q such that qf = qg. In Set it is B/~ where ~ is the…
Kleisli category
For a monad T on C, the Kleisli category Kl(T) has the same objects as C and morphisms A → B are C-morphisms A → TB, composed via the…
Cartesian closed category
Category with finite products and exponentials: the functor (−) × B has a right adjoint (−)^B. Internal hom enables currying. Set, Cat,…