1. Directed graphs or quivers What is category theory? • Graph theory on steroids • Comic book mathematics • Abstract nonsense • The secret dictionary Sets and classes: For S = fX j X2 = Xg, have S 2 S , S2 = S Directed graph or quiver: C = (C0;C1;@0 : C1 ! C0;@1 : C1 ! C0) Class C0 of objects, vertices, points, . Class C1 of morphisms, (directed) edges, arrows, . For x; y 2 C0, write C(x; y) := ff 2 C1 j @0f = x; @1f = yg f 2C1 tail, domain / @0f / @1f o head, codomain op Opposite or dual graph of C = (C0;C1;@0;@1) is C = (C0;C1;@1;@0) Graph homomorphism F : D ! C has object part F0 : D0 ! C0 and morphism part F1 : D1 ! C1 with @i ◦ F1(f) = F0 ◦ @i(f) for i = 0; 1. Graph isomorphism has bijective object and morphism parts. Poset (X; ≤): set X with reflexive, antisymmetric, transitive order ≤ Hasse diagram of poset (X; ≤): x ! y if y covers x, i.e., x 6= y and [x; y] = fx; yg, so x ≤ z ≤ y ) z = x or z = y. Hasse diagram of (N; ≤) is 0 / 1 / 2 / 3 / ::: Hasse diagram of (f1; 2; 3; 6g; j ) is 3 / 6 O O 1 / 2 1 2 2. Categories Category: Quiver C = (C0;C1;@0 : C1 ! C0;@1 : C1 ! C0) with: • composition: 8 x; y; z 2 C0 ; C(x; y) × C(y; z) ! C(x; z); (f; g) 7! g ◦ f • satisfying associativity: 8 x; y; z; t 2 C0 ; 8 (f; g; h) 2 C(x; y) × C(y; z) × C(z; t) ; h ◦ (g ◦ f) = (h ◦ g) ◦ f y iS qq <SSSS g qq << SSS f qqq h◦g < SSSS qq << SSS qq g◦f < SSS xqq << SS z Vo VV < x VVVV << VVVV < VVVV << h VVVV < h◦(g◦f)=(h◦g)◦f VVVV < VVV+ t • identities: 8 x; y; z 2 C0 ; 9 1y 2 C(y; y) : 8 f 2 C(x; y) ; 1y ◦ f = f and 8 g 2 C(y; z) ; g ◦ 1y = g f y o x MM MM 1y g MM MMM f MMM M& zo g y Example: N0 = fxg ; N1 = N ; 1x = 0 ; 8 m; n 2 N ; n◦m = m+n ; | one object, lots of arrows [monoid of natural numbers under addition] 4 x / x Equation: 3 + 5 = 4 + 4 Commuting diagram: 3 4 x / x 5 ( 1 if m ≤ n; Example: N1 = N ; 8 m; n 2 N ; jN(m; n)j = 0 otherwise | lots of objects, lots of arrows [poset (N; ≤) as a category] These two examples are small categories: have a set of morphisms. Example: The category Set has the class of all sets as its object class, with Set(X; Y ) as the set of all functions from X to Y , composition of functions: g ◦ f(x) = g f(x) , usual identities 1X : X ! X; x 7! x. This example is large (not small), but locally small: just a set of arrows between each pair of objects. 3 3. Special morphisms and objects Consider morphism f : x ! y. 0 0 0 • Isom. or invertible: 9 f : y ! x : f ◦f = 1y and f ◦f = 1x. Ex: Bijective function in Set. • Monomorphism: 8 gi : z ! x ; f ◦ g1 = f ◦ g2 ) g1 = g2. Ex: Injective function in Set. • Epimorphism: 8 gi : y ! z ; g1 ◦ f = g2 ◦ f ) g1 = g2. Ex: Surjective function in Set. • Retract or split epimorphism: r : y ! x with r ◦ f = 1x. Ex: r : n 7! maxf0; n − 1g retracts successor function on N. • Section or split monomorphism: s: y ! x with f ◦ s = 1y. Ex: Successor function on N is a section of r : N ! N. • Idempotent: x = @0f = @1f and f ◦ f = f. 2 2 Ex: R ! R ;(x1; x2) 7! (x1; 0) in Set. r ( Lemma. For x h y with r ◦ s = 1y: s • r is an epimorphism. • s is a monomorphism. • s ◦ r is an idempotent (said to split). Isomorphic objects x ∼= y: Have isomorphism f : x ! y. Terminal object > for C has 8 x 2 C0 ; jC(x; >)j = 1. Examples: f0g in Set, upper bound in a poset, . Initial object ? for C has 8 x 2 C0 ; jC(?; x)j = 1. Examples: Ø in Set, lower bound in a poset, Z in Ring,... Zero object 0 for C is both initial and terminal. Examples: f0g in categories of groups, vector spaces, . Groupoid: Category where all morphisms are invertible. Examples: For a set X: • Discrete category (X; f1x j x 2 Xg;@0 : 1x 7! x; @1 : 1x 7! x). • Symmetric group X! of all bijections X ! X. • The collection Inv X of all bijections between subsets of X. 4 4. Functors Functor: A graph homomorphism F : D ! C, thus with restrictions (4.1) 8 x; y 2 D0 ;F1 : D(x; y) ! C(F0x; F0y): f 7! F1f ; respecting identities, compositions: F11x = 1F0x, F1(g ◦ f) = F1g ◦ F1f. ( Isomorphism: F0 and F1 are isomorphisms. Global conditions: ∼ Essentially surjective: 8 c 2 C0 ; 9 d 2 D0 : c = F0d ( Full: Each restriction (4.1) is surjective. Local conditions: Faithful: Each restriction (4.1) is injective. Example: While the forgetful or underlying set functor −1 −1 U : Grp ! Set;[f :(G1; ·; ; 1) ! (G2; ·; ; 1)] 7! [f : G1 ! G2] is faithful, U0 : Grp0 ! Set0 is not injective (same set, different groups). Also U not full: Some functions between groups are not homomorphic. ∗ Example: For a monoid (M; ·; 1M ), write M for the group of invertible elements or units. Then Mon ! Grp; ∗ −1 ∗ −1 [f :(M ; ·; 1) ! (M ; ·; 1)] 7! [fj ∗ :(M ; ·; ; 1) ! (M ; ·; ; 1)] 1 2 M1 1 2 is a functor between large categories, the group of units functor. Moral: Mathematical constructions are functors! Example: A monoid homomorphism f : M1 ! M2 yields a functor between the corresponding small one-object categories. Note (R; ·; 1) ! ([0; 1[; ·; 1); n 7! n2 is full, but not faithful. Example: A functor F :(P1; ≤) ! (P2; ≤) between poset catgeories corresponds to an order-preserving function: x ≤ y in P1 ) F0x ≤ F0y in P2 : Trivially faithful. Example: Inclusion of a subcategory always gives a faithful functor. Full subcategory: The inclusion functor is full. Example: Category FinSet of finite sets is full in Set. Example: Functor (N; ≤) ! FinSet;[n < n+1] 7! [f0; 1; : : : ; n−1g ,! f0; 1; : : : ; n−1; ng] is essentially surjective. 5 5. Natural transformations Given graph maps F; G: D ! C from a graph D to a category C, a natural transformation τ : F ! G is a \vector" (τx j x 2 D0) of components τx : F x ! Gx in C1 such that, for all f : x ! y in D1, the rectangle of the naturality diagram τx x F x / Gx f F f Gf y F y / Gy τy . in D . in C commutes in the category C. Natural isomorphism: Each component τx is an isomorphism in C. Example: For a set A, have a functor LA : Set ! Set;[f : X ! Y ] 7! [A × X ! A × Y ;(a; x) 7! (a; fx)]. Then a function α: A ! B gives a natural transformation α A B α L : L ! L with components LX : A×X ! B ×X;(a; x) 7! (α a; x) and naturality diagram α LX X (a; x) / (α a; x) _ _ f LAf LB f Y (a; fx) α / (α a; fx) LY Example: Category L of (linear transformations between) real vector spaces. Dual space V ∗ = L(V; R) of linear functionals on vector space V . Double dual V ∗∗ = L(V ∗; R) = L(L(V; R); R). Identity functor I : L!L. Double dual functor DD : L!L; V 7! V ∗∗. Natural transformation τ : I ! DD with \evaluation" components ∗∗ τV : V ! V ; v 7! [θ 7! θ(v)]. Gives a natural isomorphism in finite dimensions. Contrast: Given basis fe1; : : : eng of V , define ebi : V ! R; ej 7! δij. ∗ Then V ! V ; ei 7! ebi does not set up a natural isomorphism. 6 6. Duality and contravariant functors Dual or opposite Cop of a category C is built on the dual graph Cop: Same identity morphisms, but composition as shown: f f y o x y / x x O w; xx ww xx ww xx ww g xx g ww xxg◦f ww g◦f or gf xx ww xx ww xx ww z |x In C z w In Cop For Eulerian notation in C, algebraic notation would be natural in Cop. Example: The dual of a one-object monoid category (M; ·; 1M ) is the one-object monoid category of the monoid (M; ◦; 1M ) with x◦y = y ·x. Example: For a set X, the dual of the poset category of P(X); ⊆ is the poset category of P(X); ⊇ . Contravariant functor F : D ! C is a (\covariant" or usual) functor F : D ! Cop or F : Dop ! C. Thus F (1x) = 1F x as usual, but F (g ◦ f) = F f ◦ F g. Generic examples: Locally small C, e.g., Set or lin. trans. cat. L. Fix a dualizing object T 2 C0, e.g., 2 = f0; 1g 2 Set0 or R 2 L0. Functor ∗: C ! Setop; Z 7! Z∗ := C(Z; T ) with (g ◦ f)∗ = f ∗ ◦ g∗: Y o X f f ∗ θ ◦ g θ ◦ g / θ ◦ g ◦ f O O = zz zz zz zz g g∗ T g∗ zz E zz(g◦f)∗ zz zz zz _ _ :zz θ θ z Z From Set, set Z∗ is the power set 2Z of characteristic functions θ. From L, vector space Z∗ is the dual space of linear functionals θ. 7 7. Diagram categories and functor categories Diagram category CD for diagram D and category C has graph maps F; G: D ! C as objects and natural transformations σ : F ! G as morphisms.