Chapter 1. Categories, Functors, and Natural Transformations (pp. 1–48)
FC03 — Riehl, Category Theory in Context.
This chapter contains 27 definitions, 2 theorems, 2 propositions, 18 lemmas, 4 corollaries, 26 examples, 9 constructions, 9 remarks, 6 conventions, 1 criterion and 3 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition: Category
Category: objects and arrows with specified domains/codomains, identity arrows, and associative unital composition.
Remark: Objects are recoverable from identity arrows
Objects are recoverable from identity arrows; equivalently a category may be specified by its arrows, partial composition, and identities.
Example: Concrete categories
Concrete categories: Set, Top and pointed variants, Group/Ring/Field, Mod_R/Vect_k/Ab, Graph/DirGraph, Man, Meas, Poset, Ch_R, and Model_T, with their standard structure-preserving maps.
Example: Abstract examples
Abstract examples: Mat_R; one-object categories BG of monoids/groups; preorders/posets and ordinal categories; discrete categories; Htpy; and Measure.
Definition: A category is small when its collection of arrows is a set.
A category is small when its collection of arrows is a set.
Convention/notation
For a small category C, write ob C and mor C for its sets of objects and morphisms; domain, codomain, and identity are set maps.
Definition: A category is locally small when each collection of morphisms between a…
A category is locally small when each collection of morphisms between a fixed pair of objects is a set.
Convention/notation
For locally small C, the hom-set from X to Y is C(X,Y), also written Hom(X,Y).
Definition: An isomorphism f:X→Y has a two-sided inverse
An isomorphism f:X→Y has a two-sided inverse; write X≅Y.
Definition/notation
An endomorphism has equal domain and codomain; an automorphism is an invertible endomorphism.
Example: Isomorphisms specialize to bijections in Set, bijective algebraic…
Isomorphisms specialize to bijections in Set, bijective algebraic homomorphisms, homeomorphisms in Top, homotopy equivalences in Htpy, and only identities in posets.
Definition: A groupoid is a category in which every morphism is an isomorphism.
A groupoid is a category in which every morphism is an isomorphism.
Example: A group is a one-object groupoid
A group is a one-object groupoid; the fundamental groupoid Π₁(X) has points of X as objects and endpoint-preserving homotopy classes of paths as morphisms.
Definition: A subcategory has selected objects and morphisms closed under identities and…
A subcategory has selected objects and morphisms closed under identities and composition; e.g. CRing ⊂ Ring ⊂ Rng.
Lemma: Every category has a maximal subgroupoid
Every category has a maximal subgroupoid: all objects and exactly the isomorphisms.
Definition: The opposite category C^op has the same objects and reversed arrows, with…
The opposite category C^op has the same objects and reversed arrows, with reversed composition.
Construction/comparison
Passing to C^op is involutive information-wise and exhibits the syntactic self-duality of the category axioms.
Example: Opposites of MatR reverse matrix handedness
Opposites of Mat_R reverse matrix handedness; opposite preorders reverse order; (BG)op≅B(Gop), with G^op governing right actions.
Convention/principle
Duality principle: a theorem universally quantified over categories yields a dual theorem by reversing every arrow.
Lemma: For f:x→y, the following are equivalent
For f:x→y, the following are equivalent: f is an isomorphism; postcomposition C(c,x)→C(c,y) is bijective for every c; precomposition C(y,c)→C(x,c) is bijective for every c.
Remark: In a locally small category, isomorphisms are detected representably by…
In a locally small category, isomorphisms are detected representably by isomorphisms of all postcomposition (equivalently precomposition) hom-set maps.
Definition: A monomorphism is left-cancellable and an epimorphism is right-cancellable
A monomorphism is left-cancellable and an epimorphism is right-cancellable; the notions are dual.
Comparison
Equivalently, f is monic iff every postcomposition map on hom-sets is injective; f is epic iff every precomposition map is injective.
Example: In Set, monomorphisms are exactly injections and epimorphisms exactly…
In Set, monomorphisms are exactly injections and epimorphisms exactly surjections.
Example: If rs=1, then s is a section/split mono, r a retraction/split epi, and x is…
If rs=1, then s is a section/split mono, r a retraction/split epi, and x is a retract of y.
Example: Every isomorphism is monic and epic, but the converse fails
Every isomorphism is monic and epic, but the converse fails: Z→Q is monic and epic in Ring without being an isomorphism.
Lemma: Composites of monos are mono and monicity of gf implies monicity of f
Composites of monos are mono and monicity of gf implies monicity of f; dually composites of epis are epi and epicity of gf implies epicity of g.
Definition: A functor F:C→D assigns objects and morphisms preserving domains/codomains,…
A functor F:C→D assigns objects and morphisms preserving domains/codomains, identities, and composition.
Example: Functor examples include covariant power set/direct image, forgetful and…
Functor examples include covariant power set/direct image, forgetful and inclusion functors, homotopy quotients, π₁ and Π₁, cycle/boundary/homology, free groups, the derivative/Jacobian, and the Fin_* construction attached to a commutative monoid.
Theorem: Brouwer fixed point theorem
Brouwer fixed point theorem: every continuous endomorphism of D² has a fixed point.
Example: Clustering as functorial data
Clustering as functorial data: FinMetric→Cluster admits only trivial scale-invariant clusterings under the stated conditions; persistent clusters form PCluster, supporting a distinguished persistent-clustering functor.
Definition: A contravariant functor C→D is a covariant functor C^op→D; it reverses…
A contravariant functor C→D is a covariant functor C^op→D; it reverses arrows and the order of composition.
Convention/notation
For contravariant functors, arrows are drawn in the original direction in C and their images in the reversed direction; composites retain the usual C-order.
Example: Contravariant examples
Contravariant examples: inverse-image power set, vector-space dual, open/closed-subset functors, Spec on commutative rings, presheaves, and simplicial sets.
Lemma: Functors preserve isomorphisms.
Functors preserve isomorphisms.
Example: A functor BG→C is a left G-action on an object of C; BG^op→C is a right action
A functor BG→C is a left G-action on an object of C; BG^op→C is a right action. In Set, Vect_k, Top these are G-sets, representations, and G-spaces.
Corollary: A functorial G-action acts by automorphisms and (g){-1}=(g{-1}).
A functorial G-action acts by automorphisms and (g_*){-1}=(g{-1})_*.
Lemma/construction
Every functor preserves split monomorphisms and split epimorphisms.
Definition: For locally small C and c∈C, C(c,-):C→Set and C(-,c):C^op→Set are the…
For locally small C and c∈C, C(c,-):C→Set and C(-,c):C^op→Set are the covariant and contravariant functors represented by c, acting by post- and precomposition.
Terminology
A bifunctor is a functor of two variables.
Definition: The product category C×D has object and morphism pairs, with identities and…
The product category C×D has object and morphism pairs, with identities and composition componentwise.
Definition: For locally small C, the two-sided represented bifunctor C(-,-):C^op×C→Set…
For locally small C, the two-sided represented bifunctor C(-,-):C^op×C→Set sends (f,h) to g↦hgf.
Convention/notation
Cat is the locally small category of small categories and functors; CAT has locally small categories and functors as objects, with Cat↪CAT.
Definition: An isomorphism of categories is a pair of strictly inverse functors.
An isomorphism of categories is a pair of strictly inverse functors.
Example: Canonical category isomorphisms
Canonical category isomorphisms: C↦C^op; BG≅BG^op via inversion; any groupoid≅its opposite; left R-modules≅right R^op-modules; opens≅closed^op by complement; Mat_R≅Mat_R^op by transpose.
Example: For a finite Galois extension E/F with G=Aut(E/F), the fixed-field functor…
For a finite Galois extension E/F with G=Aut(E/F), the fixed-field functor yields an isomorphism O_Gop≅Field_FE (categorical fundamental theorem of Galois theory).
Construction: Set^∂ has sets as objects and partially defined functions as morphisms, with…
Set^∂ has sets as objects and partially defined functions as morphisms, with composition on the maximal compatible domain.
Construction/comparison
Adding a disjoint basepoint defines (-)*+:Set^∂→Set**; deleting the basepoint defines U:Set_*→Set^∂; U(-)_+ is strictly identity while (U-)₊ is only objectwise isomorphic to identity.
Definition: A natural transformation α:F⇒G consists of components αc:Fc→Gc satisfying…
A natural transformation α:F⇒G consists of components α_c:Fc→Gc satisfying the naturality square for every f:c→c′; it is a natural isomorphism when every component is invertible.
Example: Examples/comparisons
Examples/comparisons: V→V** by evaluation is natural; V≅V* in finite dimension is not; singleton inclusion 1_Set⇒P; natural maps between BG-actions are equivariant maps; opens and closed sets are naturally isomorphic by complement; G≅G^op naturally by inversion; only the zero natural map V→V⊗V exists.
Proposition: The splittings A≅TA⊕(A/TA) for finitely generated abelian groups cannot be…
The splittings A≅TA⊕(A/TA) for finitely generated abelian groups cannot be chosen naturally in A.
Example: Riesz representation is the natural isomorphism Σ⇒C on compact Hausdorff…
Riesz representation is the natural isomorphism Σ⇒C* on compact Hausdorff spaces, sending a signed Baire measure μ to integration g↦∫g dμ.
Example: Precomposition and postcomposition on hom-sets commute
Precomposition and postcomposition on hom-sets commute; h induces h_:C(-,y)⇒C(-,z), and f induces f:C(x,-)⇒C(w,-).
Example: Categorification of N: cartesian product, disjoint union, and exponentiation…
Categorification of N: cartesian product, disjoint union, and exponentiation of sets satisfy natural distributive/exponential isomorphisms; cardinality on Fin_iso decategorifies them to arithmetic identities.
Lemma: Natural transformations F⇒G are in bijection with functors H:C×2→D…
Natural transformations F⇒G are in bijection with functors H:C×2→D restricting to F and G at the two endpoints.
Construction/terminology
2 is the walking/free arrow; the two-object groupoid I with one arrow in each hom-set is the walking/free isomorphism, and C×I encodes natural isomorphisms.
Definition: An equivalence C≃D consists of F:C→D, G:D→C and natural isomorphisms 1C≅GF…
An equivalence C≃D consists of F:C→D, G:D→C and natural isomorphisms 1_C≅GF and FG≅1_D.
Lemma: Equivalence of categories is an equivalence relation.
Equivalence of categories is an equivalence relation.
Example: Set and Set^∂ are equivalent via adding/removing a basepoint
Set_* and Set^∂ are equivalent via adding/removing a basepoint; one composite is identity and the other naturally isomorphic to identity.
Construction/comparison
Mat_k→Vect_kbasis→Vect_kfd and chosen quasi-inverses compare matrix, based-vector-space, and abstract finite-dimensional linear algebra.
Definition: A functor is full when each hom-map is surjective, faithful when each is…
A functor is full when each hom-map is surjective, faithful when each is injective, and essentially surjective when every target object is isomorphic to some image object.
Remark: Fullness/faithfulness are local hom-set conditions
Fullness/faithfulness are local hom-set conditions. A faithful functor injective on objects is an embedding; full+faithful is fully faithful; with object-injectivity it is a full embedding.
Theorem: Equivalences are full, faithful, and essentially surjective
Equivalences are full, faithful, and essentially surjective; assuming choice, every functor with those three properties is an equivalence.
Lemma: A morphism a→b and fixed isomorphisms a≅a′, b≅b′ determine a unique…
A morphism a→b and fixed isomorphisms a≅a′, b≅b′ determine a unique transported morphism a′→b′ making the comparison square commute.
Corollary: For every field k, Matk≃Vectk^fd.
For every field k, Mat_k≃Vect_k^fd.
Definition: A category is connected when any two objects are joined by a finite zig-zag…
A category is connected when any two objects are joined by a finite zig-zag of morphisms.
Proposition: Any connected groupoid is equivalent to the automorphism group of any chosen…
Any connected groupoid is equivalent to the automorphism group of any chosen object, viewed as a one-object category.
Corollary: For path-connected X, all fundamental groups π₁(X,x) obtained from different…
For path-connected X, all fundamental groups π₁(X,x) obtained from different basepoints are isomorphic.
Remark: The inclusion π₁(X,x)→Π₁(X) is a natural equivalence in based path-connected…
The inclusion π₁(X,x)→Π₁(X) is a natural equivalence in based path-connected spaces, while inverse equivalences require path choices and need not assemble naturally.
Definition: A category is skeletal when it has one object per isomorphism class
A category is skeletal when it has one object per isomorphism class; a skeleton sk C is a skeletal category equivalent to C, unique up to isomorphism.
Remark: A skeleton is obtained by a full subcategory on chosen representatives
A skeleton is obtained by a full subcategory on chosen representatives; inclusion is an equivalence, but skeleton choices are not strictly functorial. Equivalence between skeletal categories is isomorphism.
Example: Skeletons
Skeletons: a connected groupoid gives an automorphism group; a preorder gives a poset; Vect_k^fd gives Mat_k; Fin_iso gives objects n with automorphism groups Σ_n.
Example: The translation groupoid TG X categorifies orbit-stabilizer
The translation groupoid T_G X categorifies orbit-stabilizer: components are G-orbits, object automorphism groups are stabilizers, and |G|=|O_x||G_x| when finite.
Definition: Essentially small means equivalent to a small category (equivalently, small…
Essentially small means equivalent to a small category (equivalently, small skeleton); essentially discrete means equivalent to a discrete category.
Comparison
Local smallness, being a groupoid, taking opposites/products, and isomorphism detection are invariant under equivalence of categories.
Construction: A full and faithful F:C→D is an equivalence onto its essential image, the…
A full and faithful F:C→D is an equivalence onto its essential image, the full subcategory on objects isomorphic to some Fc.
Convention/definition
A displayed directed graph in a category is commutative when any two composable paths with common source and target have equal composites.
Definition: A monoid is M∈Set with multiplication μ:M×M→M and unit η:1→M satisfying…
A monoid is M∈Set with multiplication μ:M×M→M and unit η:1→M satisfying associativity and left/right unit diagrams.
Definition: A topological monoid is the same multiplication/unit diagram internal to…
A topological monoid is the same multiplication/unit diagram internal to Top, so multiplication is continuous.
Construction/comparison
A unital ring is an abelian group R with μ:R⊗_Z R→R and η:Z→R satisfying the same associative/unit diagrams.
Construction/comparison
A k-algebra is V∈Vect_k with μ:V⊗_kV→V and η:k→V satisfying the same diagrams.
Definition: A diagram in C is a functor F:J→C with small indexing category J.
A diagram in C is a functor F:J→C with small indexing category J.
Lemma: Functors preserve commutative diagrams.
Functors preserve commutative diagrams.
Example: The indexing category 2×2 encodes a commutative square
The indexing category 2×2 encodes a commutative square; its diagonal is the common composite of the two edge paths.
Remark: An indexing category may be presented by a directed graph plus specified…
An indexing category may be presented by a directed graph plus specified composition/commutativity relations; a functor is determined by images of generators satisfying those relations.
Example: The ordinal 4 indexes three composable arrows together with all composites
The ordinal 4 indexes three composable arrows together with all composites; its triangular faces commute.
Example: 2×3 indexes a commutative rectangle
2×3 indexes a commutative rectangle: the two adjacent squares and outer rectangle share the induced composition relations.
Lemma: Path substitution
Path substitution: replacing a contiguous composable subpath by any equal composite leaves the total composite unchanged.
Criterion
For simple acyclic diagrams presented as posets, commutativity is checked on minimal subdiagrams/relations.
Lemma: A commuting triangle remains commuting after transporting across an isomorphism
A commuting triangle remains commuting after transporting across an isomorphism; equivalently one can cancel/insert an invertible edge.
Lemma: In a commutative square of isomorphisms, the inverse arrows form a…
In a commutative square of isomorphisms, the inverse arrows form a commutative square.
Definition: Initial i has a unique arrow i→c to every object
Initial i has a unique arrow i→c to every object; terminal t has a unique arrow c→t from every object.
Example: Initial/terminal examples in Set, Top, Set, ModR, Group, Ring/Rng, Field,…
Initial/terminal examples in Set, Top, Set_*, Mod_R, Group, Ring/Rng, Field, Cat, and preorders.
Lemma: Any two parallel composites agree if their common source is initial or their…
Any two parallel composites agree if their common source is initial or their common target is terminal.
Definition: A concrete category is a category C equipped with a faithful functor U:C→Set.
A concrete category is a category C equipped with a faithful functor U:C→Set.
Example: The familiar concrete categories admit faithful underlying-set functors
The familiar concrete categories admit faithful underlying-set functors; for Graph the combined vertex-plus-edge functor V⊔E is faithful whereas V or E alone is not.
Lemma: A faithful functor reflects commutative diagrams
A faithful functor reflects commutative diagrams; hence commutativity in a concrete category can be checked on underlying elements.
Remark: Cancellation in pasted diagrams requires cancellable outer edges
Cancellation in pasted diagrams requires cancellable outer edges: a right outer map must be mono and a left outer map epi to cancel them from a composite equality.
Lemma: If an outer rectangle commutes, then the missing inner square commutes when…
If an outer rectangle commutes, then the missing inner square commutes when the other inner square does and the common outer edge is respectively monic (right) or epic (left).
Construction: For fixed C,D, functors C→D are objects and natural transformations are…
For fixed C,D, functors C→D are objects and natural transformations are morphisms of a functor category D^C; identity components are identities.
Lemma: Vertical composition of α:F⇒G and β:G⇒H is componentwise
Vertical composition of α:F⇒G and β:G⇒H is componentwise: (β·α)_c=β_cα_c.
Corollary: Functors C→D and natural transformations form the category D^C under…
Functors C→D and natural transformations form the category D^C under vertical composition.
Remark: Functor-category size
Functor-category size: D^C is locally small when C is small and D locally small; in particular (-)^(-) gives Cat^op×Cat→Cat and Cat^op×CAT→CAT in the stated regimes.
Lemma: Horizontal composition βα:HF⇒KG has component β{Gc}∘Hαc = Kαc∘β{Fc}.
Horizontal composition β*α:HF⇒KG has component β_{Gc}∘Hα_c = Kα_c∘β_{Fc}.
Remark: Whiskering composes a natural transformation on either side with a functor,…
Whiskering composes a natural transformation on either side with a functor, producing Hα,Kα,βF,βG as used in horizontal composition.
Lemma: Middle-four interchange
Middle-four interchange: vertical-then-horizontal composition of a rectangular array of natural transformations equals horizontal-then-vertical composition.
Definition: A 2-category has objects, 1-morphisms, and 2-morphisms
A 2-category has objects, 1-morphisms, and 2-morphisms; 1-morphisms compose categorically, 2-morphisms compose vertically and horizontally, and middle-four interchange holds.