1. Directed Graphs Or Quivers What Is Category Theory? • Graph Theory on Steroids • Comic Book Mathematics • Abstract Nonsense • the Secret Dictionary

Home , Graph theory, Kleisli category, Quiver (mathematics)

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 = {X | X/∈ X}, have S ∈ S ⇔ S/∈ 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 ∈ C0, write C(x, y) := {f ∈ C1 | ∂0f = x, ∂1f = y}

f ∈C1 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] = {x, y}, so x ≤ z ≤ y ⇒ z = x or z = y.

Hasse diagram of (N, ≤) is 0 / 1 / 2 / 3 / ...

Hasse diagram of ({1, 2, 3, 6}, | ) is 3 / 6 O O

1 / 2

1 2

2. Categories

Category: Quiver C = (C0,C1, ∂0 : C1 → C0, ∂1 : C1 → C0) with:

• composition: ∀ x, y, z ∈ C0 , C(x, y) × C(y, z) → C(x, z); (f, g) 7→ g ◦ f

• satisfying associativity: ∀ x, y, z, t ∈ C0 , ∀ (f, g, h) ∈ C(x, y) × C(y, z) × C(z, t) , h ◦ (g ◦ f) = (h ◦ g) ◦ f

y iS qq

• identities: ∀ x, y, z ∈ C0 , ∃ 1y ∈ C(y, y) . ∀ f ∈ C(x, y) , 1y ◦ f = f and ∀ g ∈ C(y, z) , g ◦ 1y = g

f y o x MM MM 1y g MM MMM f MMM M& zo g y

Example: N0 = {x} , N1 = N , 1x = 0 , ∀ m, n ∈ 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 , ∀ m, n ∈ N , |N(m, n)| = 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: ∃ f : y → x . f ◦f = 1y and f ◦f = 1x. Ex: Bijective function in Set. • Monomorphism: ∀ gi : z → x , f ◦ g1 = f ◦ g2 ⇒ g1 = g2. Ex: Injective function in Set. • Epimorphism: ∀ 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→ max{0, n − 1} 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 ∀ x ∈ C0 , |C(x, >)| = 1. Examples: {0} in Set, upper bound in a poset, . . .

Initial object ⊥ for C has ∀ x ∈ C0 , |C(⊥, x)| = 1. Examples: Ø in Set, lower bound in a poset, Z in Ring,... Zero object 0 for C is both initial and terminal. Examples: {0} in categories of groups, vector spaces, . . .

Groupoid: Category where all morphisms are invertible. Examples: For a set X:

• Discrete category (X, {1x | x ∈ X}, ∂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) ∀ x, y ∈ 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: ∀ c ∈ C0 , ∃ d ∈ 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→ [f| ∗ :(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); 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→ [{0, 1, . . . , n−1} ,→ {0, 1, . . . , n−1, n}] 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 | x ∈ 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 {e1, . . . en} 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 ∈ C0, e.g., 2 = {0, 1} ∈ Set0 or R ∈ 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. Composition:

(τ•σ)x x F x / Hx QQQ mm6 QQQ mmm QQQ mmm σx QQQ mmm τx QQQ mmm ( Gx m

f F f Gf Hf Gy n6 PP nnn PPP nnn PPP nnσ τ PP nnn y y PPP nnn PP( y F y / Hy (τ•σ)y

. . . in D . . . in C

θ Constant objects and morphisms: x c / c0

f 1c 1c y c / c0 θ Functor category: Category D, functors F, G, . . . Example: Linear representations of a group G are objects R: G → L of the functor category LG for the one-object group category G, so group homomorphisms R: G → L(V,V )∗ = Aut V = GL(V ).

The morphisms are intertwiners or equivariant maps τ : R1 → R2, τx so ∀ g ∈ G,V1 / V2

R1(g) R2(g) V / V 1 τx 2 3 E.g., G = S3, V1 = R = Span{e1, e2, e3}, R1(π): ei 7→ eπ(i). 2 V2 = R = Span{e2 −e1, e3 −e2}, R2(π):(ei+1 −ei) 7→ (eπ(i+1) −eπ(i))

R1(1 2 3) τx z }| { R2(1 2 3) τx z }| { 0 0 1 z }| { z }| { 1 −2 1 1 0 −1 1 −2 1 1 · 1 0 0 = · −1 −1 2   1 −1 −1 −1 2 3 0 1 0 3 8

8. Products and coproducts Product X × Y = {(x, y) | x ∈ X , y ∈ Y } of sets X,Y :

πX πY XXo × Y / Y [ O C fug f g Z

Universality property: ∀ Z ∈ Set0, “solid”↓ implies ↓“dashed” bijection Set(Z,X) × Set(Z,Y ) → Set(Z,X × Y ); (f, g) 7→ f u g with f = πX ◦ (f u g) and g = πY ◦ (f u g). Thus f u g : z 7→ (fz, gz). Picture in Set2 for discrete “two spot” diagram 2 = •• :

f Z / X PPP nn7 PPP nnn PPP nnn PP nnπX fug PP' nnn X × Y

X × Y P fug nn7 PP π nnn PPP Y nnn PPP nnn PPP nnn PP( Z g / Y

Examples: Product in Set carries products in Grp, Ring, Mon, etc. Example: Product in a poset category is a greatest lower bound. a b a × b = c exists, O ¢A O ¢¢ ¢¢ ¢¢ c ¢ d but c × d does not.

ιX ιy Coproduct in C is the product in Cop: X / X + Y o Y ftg f g ) Z u Example: Coproduct in Set is the disjont union. Example: Coproduct in a poset category is a least upper bound.

ιU ιU - q Biproduct U j U ⊕ V 4 V in L is product and coproduct. πU πV 9

9. More limits and colimits k

& Pullback: Z ___ / X ×B Y / Y or with poset diagram r πY

πX g h - X / B f h • → • ← • category picture: Z P /6 X P nnn P P nnn r P nnn P nn πX P( nnn X ×B Y f

r f◦πX =g◦πY Z ______/ X × Y / B B 6 O f◦h=g◦k g X ×B Y n6 PP r n PPP πY n PPP n n PPP n PPP Z n /( Y k

∂0 ∂1 Ex: Domain of category composition is pullback of C1 −→ C0 ←− C1. Pushout is the dual of a pullback. k f $ / Equalizer: Z ___ / E / X Y so f ◦k = g◦k ⇒ ∃! r . e◦r = k r e g / e e In L, E = Ker(f − g) ,→ X. In Set, E = {x ∈ X | fx = gx} ,→ X.

k f / # Coequalizer: X Y / C ___ / Z ; k◦f = k◦g ⇒ ∃! r. r◦u = k g / u r In Set, C is quotient of Y by equiv. rel’n. gen. by {(fx, gx) | x ∈ X}. In L, u projects from Y to C = Coker(f − g) := Y/Im(f − g). Extended First Isomorphism Theorem in L is the exact sequence

f 0 / Kerf / X / Y / Cokerf / 0

g1 g2 where exact means Im g1 = Ker g2 for each / • / .

Similar in Ab, RMod, ModR, ModK (commutative unital ring K), or any abelian category A where each A(X,Y ) is an abelian group. 10

10. General limits and colimits Diagram D, category C, constant or diagonal for θ ∈ C(c, c0) is nat. tr. ∆θ : ∆c → ∆c0 with ∆: D → C;[f : x → y] 7→ c / c0 . θ

θ c / c0 Limit of graph map F : D → C is projection π : ∆ lim F → F such ←− that ∀ κ: ∆Z → F, ∃ r = lim κ ∈ C(Z, lim F ) . π ◦ ∆ lim κ = κ. ! ←− ←− ←− κx x Z / F x NN 7 NN ooo NNN oo NN ooo r=lim κNN oo πx ←− NN' ooo lim F ←− f F f lim F ←− r=lim κ q8 NN ←− qq NN πy qq NNN qqq NN qq NNN qq N& y in D in C Z q / F y κy

A.k.a “projective limit” or “inverse limit”, written as lim. Colimit of graph map F : D → C is insertion ι: F → ∆ lim F such −→ that ∀ κ: F → ∆Z, ∃ r = lim κ ∈ C(lim F,Z) . ∆ lim κ ◦ ι = κ. ! −→ −→ −→ A.k.a “inductive limit” or “direct limit”, written as colim.

Example: Functor (order-preserving) between poset categories x:( , ≤) → ( ∪ {∞}, ≤): n 7→ x . Then lim x = lim x . N R n −→ n→∞ n

Example: F : N r {0, 1} → Ring; n 7→ Z/nZ. Then r = ∆ lim κ: ,→ lim F = Q∞ /n for κ : → /n ; x 7→ x+n . ←− Z ←− n=2 Z Z n Z Z Z Z

Directed diagram D: ∀ x, y ∈ D0 , ∃ z ∈ D0 . x → z ← y. Then have directed limits and directed colimits. Example: (Real) vector space V , directed poset Pfin(V ), ⊆ of finite subsets. Functor F : P (V ) → L; X 7→ Span(X). Then lim F = V . fin −→ Theorem: Each algebra is the (directed) colimit of its finitely generated subalgebras. 11

11. Product categories and bifunctors

Product B × C of quivers B,C has (B × C)0 = B0 × C0, (B × C)1 = B1 × C1, pointwise ∂i(f, g) = (∂if, ∂ig) for i = 0, 1. Product B × C of categories B,C : pointwise identities, composition: (B × C)(x, x0), (y, y0) × (B × C)(y, y0), (z, z0) → (B ×C)(x, x0), (z, z0): (f, f 0), (g, g0) 7→ (f 0 ◦f, g0 ◦g).

πB πC Universality: BBo × C / C — graph maps or functors. [ O C F uG F G D

0 0 πB πC Example: B0 o B0 × C0 / C0 O O O F F ×G G πB πC B o B × C / C

Bifunctor S to D on B and C is a functor S : B × C → D — graph, diagram or quiver bimap if B,C are just quivers.

Proposition: Given bifunctor S : B × C → D: For (b, c) ∈ (B × C)0, deﬁne Rb := S(b, ): C → D and Lc := S( , c): B → D.

Rb(g)=S(b,g) Then ∀ f : b → b0, g : c → c0: S(b, c) / S(b, c0) Q Q S(f,g) Q 0 Lc(f)=S(f,c) Q L 0 (f)=S(f,c ) Q Q c Q( 0 0 0 S(b , c) 0 / S(b , c ) Rb0 (g)=S(b ,g) Conversely, given Rb : C → D and Lc : B → D with ∀b ∈ B0, c ∈ C0 ,Lc(b) = Rb(c) and commuting solid square, the diagonal deﬁnes a bifunctor S : B × C → D.

op Example: Locally small C, B = C , Rb : C → Set; b 7→ C(b, c), op Lc : C → Set; b 7→ C(b, c) (like dualizing), Rb(c) = C(b, c) = Lc(b). f h g For h ∈ C(b0, c), so b / b0 / c / c0 , have

Rb(g) Rb(g) h ◦ f / g ◦ h ◦ f C(b, c) / C(b, c0) O O O O Lc(f) Lc(f) Lc(f) Lc0 (f) _ _ h / g ◦ h C(b0, c) / C(b0, c0) Rb0 (g) Rb0 (g) 12

12. Cartesian monoidal categories Cartesian monoidal category: category C with all ﬁnite products. Idea: Think of (C, ×, >) as like a monoid, say (N, +, 0) or (R, ·, 1). Problem of non-associativity: e.g. in Set,(x, (y, z)) 6= ((x, y), z). Fix: Bifunctor C × C → C;(X,Y ) 7→ X × Y , trifunctors C × C × C → C;(X,Y,Z) 7→ X × (Y × Z) or (X × Y ) × Z, nat. isom. α with components αX,Y,Z : X × (Y × Z) → (X × Y ) × Z which commute with projections; both sides give a product of X,Y,Z. πY ×Z πY [Typical two-stage projection πY : X × (Y × Z) −−−→ Y × Z −→ Y .] Problem of non-unitality: e.g. in Set with > = {∗}, have (∗, x) 6= x. Fix: Functors C → C; X 7→ > × X or X × > nat. isom. to identity, so components λX : > × X → X and ρX : X × > → X .

Potentially large “monoid” (C, ×, >) “up to natural isomorphisms”.

Pentagon: (W × X) × (Y × Z) i jUU αW ×X,Y,Z iii UUU αW,X,Y ×Z iiii UUUU iiii UUUU tiiii UUU ((W × X) × Y ) × ZW × (X × (Y × Z)) O αW,X,Y ×1Z 1W ×αX,Y,Z

αW,X×Y,Z (W × (X × Y )) × Z o W × ((X × Y ) × Z)

αX,>,Y Triangle: X × (> × Y ) / (X × >) × Y O OOO ooo OOO ooo 1 ×λ OO ooρ ×1 X Y OO' wooo X Y X × Y

λ> ( Digon: > × > 6 > ρ>

Coherence [CWM §VII.2]: If these three diagrams commute, then w.l.o.g. have a strict monoidal category: the nat. isoms. are identities.

• Coherence holds for Cartesian monoidal categories. 13

13. Groups in categories

Group (G, ∇: G × G → G, S : G → G, η : > → G) in Set, satisfying:

1G×∇ 1G×η G × G × G / G × G and G × G o G × > (so a monoid) O KK KK ∇ ∇×1G ∇ η×1G KK ρG KK KK K% G × G / G > × G / G ∇ λG

1G×S πG and G × G / G × G with G × G / G w; GG fM O ∆ ww GG ∇ M ∆ ww GG πG M M 1G ww GG M ww η G# M G / > / G GGo GG w; 1G GG ww GG ww ∆ GG ww ∇ G# ww G × G / G × G (so a group). S×1G

Group in a Cartesian monoidal category: interprets the diagrams.

Example: A topological group is a group in the category Top of continuous maps between topological spaces.

Example: The additive group functor CRing Ga : K 7→ (K, +, −, 0) is a group in Grp .

Example: The multiplicative group or group-of-units functor ∗ −1 CRing Gm : K 7→ (K , ·, , 1) is a group in Grp .

Example: The p-th roots of unity functor p p−1 CRing µp : K 7→ ({k ∈ K | k = 1}, ·, , 1) is a group in Grp .

CRing Example: (SL2, ∇, S, η) as a group in Grp : a b SL2 : CRing → Grp; K 7→ a, b, c, d ∈ K , ad−bc = 1 . c d

a b a0 b0 aa0 + bc0 ab0 + bd0 ∇ : , 7→ , K c d c0 d0 ca0 + dc0 cb0 + dd0 a b d −b 1 0 S : 7→ , and η : 1 7→ . K c d −c a K 0 1 14

14. Spaces, bases, adjunctions Category L of linear transformations of vector spaces over a ﬁeld K. Forgetful or underlying set functor U : L → Set. r n P o For set X, v. sp. with basis X is FX = λi ·xi | λi ∈ K , xi ∈ X , i=1 r P the space of formal linear combinations λi · xi of elements of X. i=1

At X, have unit ηX : X → UFX; x 7→ 1 · x which inserts the basis. r P At V , have counit εV : FUV → V ; λi · vi 7→ v, where i=1 v = λ1v1 + ... + λrvr, the formal combination worked out in V . r r P P For f : X → Y , lin. transf. F f : FX → FY ; λi · xi 7→ λi · f(xi). i=1 i=1 So have functors F : Set → L and U : L → Set.

∼ Nat. isom. with components ϕX,V : L(FX,V ) = Set(X,UV ) (*)

θ ηX Uθ Mutually inverse ϕX,V :[FX −→ V ] 7→ [X −→ UFX −→ UV ] (informally, restricting θ to X); note unit ηX = ϕX,F X (1FX );

−1 f F f εV and dually, ϕX,V :[X −→ UV ] 7→ [FX −→ FUV −→ V ] −1 (informally, extending f to FX); note counit εV = ϕUV,V (1UV ).

Adjunction (F, U, η, ε) with left adjoint F and right adjoint U.

−1 Thus ϕX,V : θ 7→ Uθ ◦ ηX and ϕX,V : f 7→ εV ◦ F f.

−1 Triangular identities: ∀ X ∈ Set0 , 1FX = εFX ◦F ηX [= ϕX,F X (ηX )]

and ∀ V ∈ L0 , 1UV = UεV ◦ ηUV [= ϕUV,V (εV )]. The triangular identities are necessary and suﬃcient for an adjumction.

F Other notations: F a U or L v ⊥ Set 5 U

Mnemonic: In the box (*), put the functors at the extreme edges. The left adjoint (F ) is on the left; the right adjoint (U) is on the right. 15

15. Three adjunctions with monoids • Free module functor F : Set → Mon is left adjoint to the forgetful functor U : Mon → Set.

“Tensor” notation x1 ⊗ ... ⊗ xn for n-tuple (x1, . . . , xn). For set or alphabet X, coproduct FX := P Xn, with X0 = {1} and n∈N word concatenation associative product (x1⊗...⊗xm, y1⊗...⊗yn) 7→ x1⊗...⊗xm⊗y1 ...⊗yn.

Note λX = 1X gives 1 ⊗ x = x and similarly x ⊗ 1 = x by ρX = 1X .

Unit ηX : x 7→ x (“alphabet letter makes a one-letter word”) and counit εM : FUM → M; m1 ⊗ m2 7→ m1 · m2 (“multiplication table”).

• Group of Units functor U : Mon → Grp;(M, ·, 1) 7→ (M ∗, ·,−1 , 1) is right adjoint to Forgetful F : Grp → Mon;(G, ·,−1 , 1) 7→ (G, ·, 1). ∼ Natural isomorphism ϕG,M : Mon(F G, M) = Grp(G, UM); θ 7→ θ , since g · g−1 = 1 ⇒ θ(g) · θ(g)−1 = 1 and dually, so θ(g) ∈ M ∗. ∗ ∗ Unit ηG : G → G ; g 7→ g (note G = G) and ∗ counit εM : M ,→ M; u 7→ u (embedding group of units into monoid).

• M-sets for a monoid M — categoriﬁcation of a monoid M — e.g., permutation representations for M a group.

Functor L: M → Set; ∗ 7→ X, m 7→ [Lm : X → X; x 7→ mx], can also be written as (X,M), a set X with “scalars” from M, or as the monoid homomorphism L: M → Set(X,X); m 7→ Lm. Category SetM of M-sets. Forgetful functor U : SetM → Set; L 7→ L(∗) or (X,M) 7→ X. Free M-set functor F : Set → SetM ; X 7→ (M × X,M) with m(n, x) = (mn, x). Free algebra functor is left adjoint to the underlying set functor:

Unit ηX : X → M × X; x 7→ (1, x) (embedding generators into the free algebra) and counit ε(X,M) :(M × X,M) → (X,M); (m, x) 7→ mx (action in the M-set). 16

16. Poset adjunctions and Galois correspondences Poset categories (A, ≤), (B, ≤), functors R: A → B, S : B → A. Galois connection: adjunction A(Sb, a) ∼= B(b, Ra), so Sb ≤ a ⇔ b ≤ Ra. Unit: ∀ b ∈ B , b ≤ RSb. Counit: ∀ a ∈ A , SRa ≤ a. Thus ∀ a ∈ A , Ra ≤ RSRa and ∀ b ∈ B , SRSb ≤ Sb (plug in). Also ∀ b ∈ B , Sb ≤ SRSb and ∀ a ∈ A , RSRa ≤ Ra (use S,R). Closed elements: In S(B) ⊆ A or R(A) ⊆ B.

Closure of a ∈ A is SRa = dom εa, and of b ∈ B is RSb = cod ηb.

R / Galois correspondence: Mut. inverse S(B), ≤ o R(A), ≤ . S

Polarity is a relation α ⊆ I × J. Gives Galois connection S : (2I , ⊆) → (2J , ⊇); X 7→ {y ∈ J | ∀x ∈ X , x α y} R: (2J , ⊇) → (2I , ⊆); Y 7→ {x ∈ I | ∀y ∈ Y , x α y}

Note ∀ X ⊆ I, ∀ Y ⊆ J,

SX ⊇ Y ⇔ ∀ x ∈ X, ∀ y ∈ Y , x α y ⇔ X ⊆ RY .

Galois theory: Group permutation representation or G-set (X,G). Fixed point relation {(x, g) ∈ X × G | gx = x}. Right adjoint R: 2G → 2X is the ﬁxed point functor. Left adjoint S : 2X → 2G is the (pointwise) stabilizer functor.

Polar geometry: Vector space V with quadratic form hu, vi. Polarity {(u, v) | hu, vi = 0} ⊆ V × V . Closure of a subset is its orthogonal complement.

n Alg. geometry: On C ×C[X1,...,Xn], polarity {(x, f) | f(x) = 0}. Closed subsets of Cn are algebraic sets or varieties. Closed subsets of C[X1,...,Xn] are radical ideals. Hilbert’s Nullstellensatz:√ The closure of an ideal I¡C[X1,...,Xn] is its radical I = {f | ∃ 0 < n ∈ N . f n ∈ I}. 2 Example: Radical of hX1 i in C[X1,...,Xn] is hX1i. 17

17. Slice categories and comma categories

For b ∈ C0, slice category (C ↓ b) or (1C ↓ b) of C-objects over b f has object class ∂−1(b), morphisms c / c0 (commuting), 1 < << ¡¡ << ¡ p < ¡¡ 0 << ¡¡ p b Ð¡ f 0◦f 0 ' 00 composition c / c / c , terminal object 1b : b → b. = f f 0 == == p0 p = 00 == p b

Dually, slice category (b ↓ C) or (b ↓ 1C ) of C-objects under b. Examples: Down-sets, and up-sets (or principal ﬁlters), in posets. Example: For a group G and G-module A in AbG, the split extension p: A × G → G;(a, g) 7→ g in (Grp ↓ G). Here (a, g)(a0, g0) = (a + ga0, gg0).

For b ∈ C0 and T : E → C, comma category (T ↓ b) of objects T -over b T f has morphisms T e / T e0 . @ @@ }} @@ } p @ }} 0 @@ }} p b ~}

Dually, for b ∈ C0 and S : D → C, comma category (b ↓ S) of objects S-under b.

Proposition: For adjunction (F : X → A,U : A → X, η, ε), unit ηX : X → UFX is an initial object of (X ↓ U) and counit εA : FUA → A is a terminal object of (F ↓ A).

F ϕX,Ap Proof. X and FX ______/ FUA C C ηX x CC p C vv xx C CC vv xx CC CC vv xx CC p C vv εA {x ! C! {vv UFX ______/ UA A −1 UϕX,Ap

F v Cor: Given A ⊥ 6 X , unit and counit uniquely determined. U 18

18. The Yoneda Lemma Yoneda Lemma: Let A be locally small. For object A1 of A, and f : A2 → A3 in A1, remember A(A1, f): A(A1,A2) → A(A1,A3); h 7→ f ◦ h post-composes with f.

A ∼ Then for K : A → Set, have Set A(A1, ),K = KA1; τ 7→ τA1 (1A1 ) Proof. • Injectivity: τ A1 A1 A(A1,A1) / KA1 1A1 / τA1 (1A1 ) _ _ A(A1,h) h =L (h) Kh ◦ A2 A(A1,A2) / KA2 h / τA2 (h) = Kh τA1 (1A1 ) τA2

In A In Set

• Surjectivity, ρ: A(A1,A2) → K, ρA2 : h 7→ Kh(x) for x ∈ KA1 nat:

ρA2 A2 A(A1,A2) / KA2 h / Kh(x) _ _

f A(A1,f) Kf =L◦(f) Kf Kh(x) = A3 A(A1,A3) ρ / KA3 f ◦ h A3 / K(f ◦ h)(x)

In A In Set

Corollary: Full, faithful (covariant) Yoneda embedding

Dop ∃: D,→ Db = Set ;[f : x → y] 7→ D( , f): D( , x) → D( , y)

Category Db of (set-valued) pre-sheaves over D. Note: “∃” is Katakana for “Yo”.

Example: Poset category (P, ≤). For element x, slice category D( , x) is (ess.) the down-set ↓ x of x. Then for f : x ≤ y, natural transformation D( , f) is the inclusion ↓ x ,→↓ y. 19

19. Reflective subcategories and counit properties Reflective subcategory A of B means the inclusion K : A ,→ B is full (not required in CWM), and has a left adjoint L: B → A, called the localization or reflector. Example: K : Ab ,→ Grp Then L: G 7→ G/[G, G], the largest abelian quotient of G. Reflective adjunction: A(LB, A) ∼= B(B,A)

Unit: ηB : B → LB; counit: εA : LA → A is an isomorphism. So, when are counits of adjunctions isomorphisms? Need lemmata: epi A epi Lemma 1: τ : S → T is in Set iﬀ each τ 00 in A. mono A mono

τ τ 00 00 A00 00 Proof. S / T ⇔ ∀ A ∈ A0,SA / TA

τ p-o 1T τA00 p-o T / T TA00 TA00 1T 0 Lemma 2: For f : A → A, natural transformation R◦(f) or mono epi A(A, f): A(A, ) → A(A0, ) is iﬀ f is . epi split mono

[Note R◦(f) 7→ R◦(f)(1A) = f under the Yoneda Lemma.]

0 R◦(f) 0 0 0 Proof. A(A, A ) −−−→ A(A ,A ) epi ⇒ ∃ r ∈ A(A, A ) . r ◦ f = 1A0 . op op 00 00 op 00 op 00 Conv., f r = 1A0 ⇒ ∀ A , ∃A (f ) ◦ ∃A (r ) = ∃A (1A0 ) ⇒ R◦(f)A00 ◦ R◦(r)A00 = 1A(A0,A00) ⇒ R◦(f)A00 surj., epi; so R◦(f) epi.

00 00 R◦(f) 0 00 ∀ A , A(A, A ) −−−→ A(A ,A ) mono ⇔ h1 ◦f = h2 ◦f ⇒ h1 = h2 .

U is . . . iﬀ εA ... F full has retract Theorem: In A t ⊥ X , 4 faithful is epi U full, faithful is iso Proof. Natural transformation α: A(A, ) → A(F UA, ) with −1 U ϕ 0 A,A0 0 UA,A0 0 component αA0 : A(A, A ) −−−→ X(UA, UA ) −−−−→ A(F UA, A ). Under Yoneda Lemma, α 7→ αA(1A) = εA. Then by Lemma 2: 0 0 εA split mono ⇔ α epi ⇔ ∀A , αA0 surj. ⇔ ∀A ,UA,A0 surj; 0 0 εA epi ⇔ α mono ⇔ ∀A , αA0 mono ⇔ ∀A ,UA,A0 inj;. 20

20. Category equivalence Equivalence: Full, faithful, essentially surjective functor F : X → A. ∼ Recall essentially surjective: ∀ A ∈ A0 , ∃ X ∈ X . εA : FX = A. Preorder: Set (Q, ≤) with reflexive transitive relation ≤ on set Q, or a small category with ∀ x, y ∈ Q, |Q(x, y)| ≤ 1. Define α on Q by x α y ⇔ x ≤ y and y ≤ x , an equivalence relation. Set P of equivalence class representatives: ∀ q ∈ Q, ∃ p ∈ P . p ∼= q . Inclusion functor F :(P, ≤) ,→ (Q, ≤) is an equivalence. “Election” functor U :(Q, ≤) → (P, ≤) chooses representatives. ∼ Then ∀ q ∈ Q , εq : F Uq = q, isomorphic counit of an adjunction. Note (P, ≤) is a poset — antireflexive! F t Adjoint equivalence: A ⊥ 4 X with unit, counit iso. U F u Equivalence: A 5 X with 1X → UF , FU → 1A iso. U

Theorem: Functor F : X → A. TFAE: (a) F is an equivalence; (b) F is part of an adjoint equivalence of categories; (c) F is part of an equivalence of categories. ∼ (a)⇒(b): ∀ A ∈ A0 , ∃ UA ∈ X . εA : FUA = A. Full, faithful F ⇒ −1 ∀ f ∈ A(FX,A) , ∃! ϕX,Af ∈ X(X,UA) . F ϕX,Af = εA ◦ f,... [Complete the adjunction, dual to the construction for linear algebra.] (c)⇒(a): Need F full and faithful.

ηX1 εA1 F faithful: X1 / UFX1 and U faithful: FUA1 / A1

g UF g k F Uk

εA2 X2 / UFX2 FUA2 / A2 ηX2

F full: For h ∈ A(FX1,FX2), want h = F f for f ∈ X(X1,X2).

ηX1 ηX1 Have X / UFX for f = η−1 ◦ Uh ◦ η and X / UFX 1 1 X2 X1 1 1

f Uh f UF f X2 / UFX2 X2 / UFX2 ηX2 ηX2 so Uh = UF f. Then U faithful gives h = F f. Corollary: Essentially surjective K : A ,→ B gives a reﬂection. 21

21. Typical equivalences • Skeleton S of C: unique representative for each isomorphism class. Like poset (P, ≤) induced in preorder (Q, ≤), essentially surjective K : S,→ C has reﬂection L: C → S. Ex: {i ∈ N | i < n} n ∈ N as object set of skeleton of FinSet.

n • Morita equivalence: Ring R, ring Rn of n × n-matrices over R. F n r z }| { Mod ⊥ Mod n with U : M → M ⊕ · · · ⊕ M , R 2 Rn U n n F :[Rn → End(N)] 7→ [R → Rn → End(N)]. Concrete category: Category of sets with structure (algebraic, topological,. . . ) and structure-preserving functions (homomorphisms, continuous,. . . ).

F t op • Duality: Equivalence A ⊥ 3 X of concrete categories. U Dualizing object: Set T with structure T ∈ A or T ∈ X, A where: ∀ A ∈ A0 , A(A, T ) ≤ Set(A, T ) = T ∈ X X and: ∀ X ∈ X0 , X(X,T ) ≤ Set(X,T ) = T ∈ A. Then U = A( ,T ) and F = X( ,T ).

Example: Category Lfin of fin.-dim. vector spaces over a field K. −1 ∗∗ Then A = X = Lfin, T = K and εV : V → V ; v 7→ [f 7→ f(v)].

Example: Fourier transforms, Pontryagin duality. Then A = Ab, X = CAb (compact abelian groups), and T = (R/Z, +, 0) “1-dimensional torus” or (S1, ·, 1) “circle group”. Ab := UA = Ab(A, T ), the group of characters χ: A → T . −1 εA : A → FUA; a 7→ [χ 7→ χ(a)].

Example: Category A of ﬁnite Boolean algebras, X = FinSet, dualizing object T = 2 := {0, 1}, so power set FX (char. fns.). ηX : X → UFX; x 7→ [χ 7→ χ(x)].

Note: Can extend from a category Af.g. of ﬁnitely generated algebras to a category A of all algebras: treat as colimits of f.g. algebras, which will dualize to limits of Xf.g.-objects. 22

22. Preservation, reflection and creation Diagram D : J → A, functor G: A → B. J

D A / B G

G preserves J-limits if it “pushes limits forward”: π Diagram D : J → A has a limit lim D −→j D ←− j Gπ implies GD : J → B has a limit G(lim D) −−→j GD . ←− j G reflects J-limits if it “pulls limits back”: Gπj Diagram GD : J → B has a limit of the image form GL −−→ GDj π implies D : J → A already had a limit lim D = L −→j D . ←− j G creates J-limits if it both preserves and reflects, and if lim GD exists, then it exists in the image form. ←− Corresponding definitions for colimits.

Example: U : Grp → Set preserves, reﬂects limits, directed colimits. [Consider “pointwise” structure on the underlying sets.] Doesn’t preserve or reﬂect general colimits.

Example: U : Top → Set preserves, but doesn’t reﬂect, limits:

πY E / Y in Top

πX p-b g X / B f means E = {(x, y) ∈ X × Y | f(x) = g(y)} has the subspace topology.

Example: Full and faithful K : Ab ,→ Grp preserves limits, but not colimits.

Theorem: Full and faithful G: A → B reﬂects limits and colimits.

Example: In Ab, coproduct C2 + C3 or C2 ⊕ C3 is C6. In Grp, coproduct C2 +C3 or C2 ∗C3 is the modular group PSL2(Z). Doesn’t violate K : Ab ,→ Grp reﬂecting colimits: PSL2(Z) 6= KC6. 23

23. Preservation and adjunction Diagram D : J → A, adjoint functors F,U: J

D F v A ⊥ 6 X U

πj κj Suppose limit lim D −→ D exists: A / D ←− j NN 7 j NN ppp NNN pp NN ppp r=lim κ NN pp πj ←− NN& ppp lim D ←− Thus AJ (∆A, D) ∼ A(A, lim D). = ←− κj = πj ◦ r 7→ r

Theorem: Right adjoints preserve limits. Proof. UD h a s l i m i t U lim D: ←− XJ (∆X,UD) ∼ AJ ∆(FX),D ∼ A(FX, lim D) ∼ X(X,U lim D) . = = ←− = ←− Corollary: Left adjoints preserve colimits.

Example: In L(FX,V ) ∼= Set(X,UV ), have U(V1 ⊕ V2) = V1 × V2 and F (X1 + X2) = FX1 ⊕ FX2.

Example: Multiplicity of the Euler ϕ-function or totient function ∗ ϕ(n) = |{r | 1 ≤ r ≤ n and gcd(r, n) = 1}| = (Z/n, ×, 1) . Recall group of U nits functor U : Mon → Grp;(M, ·, 1) 7→ (M ∗, ·,−1 , 1) is right adjoint to F orgetful F : Grp → Mon;(G, ·,−1 , 1) 7→ (G, ·, 1). ∼ For coprime m, n, have (Z/mn, ×, 1) = (Z/m, ×, 1) × (Z/n, ×, 1).

∗ ∗ Then ϕ(mn) = (Z/mn, ×, 1) = (Z/m, ×, 1) × (Z/n, ×, 1) ∗ ∗ ∗ ∗ = (Z/m, ×, 1) ×(Z/n, ×, 1) = (Z/m, ×, 1) × (Z/n, ×, 1) = ϕ(m)ϕ(n).

F u Example: Equivalence A 5 X U F F v v implies A ⊥ 6 X and A > 6 X , U U so F and U preserve limits and colimits. 24

24. Heyting algebras and topologies Preorder (P, ≤) with all finite products, sufficiently including: The empty product (terminal object) > with ∀ x ∈ P , x ≤ >; The (comm., assoc.) meet or g.l.b with ∀ x, y ∈ P , x ← x · y → y. For each fixed a in P , functor S(a):(P, ≤) → (P, ≤); x 7→ (x · a). Suppose each S(a) has a right adjoint R(a): z 7→ (a ( z):

∀ x, y, z ∈ P, x · y ≤ z ⇔ x ≤ y ( z (∗) Example: Propositions, “and” is product; “deduce q from p” is p → q. Then p ( q would be proposition “p implies q”. Bounded lattice: poset, finite products, coproducts, 0 = ⊥, 1 = >. Complete lattice: poset with all products and coproducts. Heyting algebra is a bounded lattice with the adjunctions (∗). Prop: Heyting algebras are distributive: S(a) preserves coproducts. Prop: Complete Heyting algebras are completely distributive. P By (∗), have y ( z = {x | x · y ≤ z}. Example: Boolean algebra with implication p ( q = p → q = (¬p)∨q Negation (pseudocomplement) ¬x := x ( 0 in any Heyting algebra. 1 1 1 Example: {0 ≤ 2 ≤ 1}, where 2 ( 0 = max{x | x · 2 ≤ 0} = 0. 1 1 Then ¬¬ 2 = ¬0 = 1 6= 2 ; “Law of the excluded middle” does not hold. Regular elements x = ¬¬x in Heyting algebra form Boolean algebra. Topology: In any topological space (X, O), the subset O of 2X comprising the open sets forms a complete Heyting algebra. Unions in 2X , but infinite intersections differ, take interior. ◦ Here P ( Q = [(X r P ) ∪ Q] • Indiscrete topology O = {Ø,X} • Discrete topology O = 2X • Alexandrov topology of poset (P, ≤) is the set of all downsets. • Cofinite topology of set X has O = {Ø}∪{S ⊆ X | X rS finite} • For monoid M and an M-set X, take O as the set of M-subsets. If M is a group, get a Boolean algebra. 25

25. Currying ∼ Heyting algebra: ∀ x, y, z ∈ P0 ,P (x · y, z) = P (x, y ( z) ∼ In particular, ∀ y, z ∈ P0 ,P (y, z) = P (1, y ( z). ∼ Currying: ∀ X,Y,Z ∈ Set0 , Set(X×Y,Z) = Set X, Set(Y,Z) ∼ In particular, ∀ Y,Z ∈ Set0 , Set(Y,Z) = Set >, Set(Y,Z) .

∼ Tensor product: ∀ X,Y,Z ∈ L0 , L(X ⊗ Y,Z) = L X, L(Y,Z) ∼ In particular, ∀ Y,Z ∈ L0 , L(Y,Z) = L K, L(Y,Z) , “linear spaces” as modules over commutative ring K., e.g., Z for Ab. Note LX, L(Y,Z) ⊆ SetX, Set(Y,Z) ∼= Set(X × Y,Z), so LX, L(Y,Z) tracks the bilinear maps X × Y → Z.

In all three cases, ∼= is a natural isomorphism of sets, so on the left hand side of the lower ∼= is a hom-set of the locally small category.

Strict symmetric monoidal category (C, ⊗,I): X ⊗ Y = Y ⊗ X, X ⊗ (Y ⊗ Z) = (X ⊗ Y ) ⊗ Z, and I ⊗ X = X = X ⊗ I.

E.g: Heyting algebra (P, ·, 1), Cartesian (Set, ×, >), linear (L, ⊗,K).

Closed monoidal category: Adjunction C(X ⊗Y,Z) ∼= C(X, [Y,Z]) with internal hom-object [Y,Z], set isom. C(Y,Z) ∼= C(I, [Y,Z]).

Bifunctors: monoidal product ⊗: C × C → C and internal hom [ , ]: Cop × C → C.

Heyting algebras: monoid product → adjunction → internal hom. Linear spaces: internal hom → adjunction → monoid product.

Note ⊗ Y a left adjoint ⇒ preserves coproducts ⇒ distributivity: 0 0 P P (X +X )⊗Y = (X ⊗Y )+(X ⊗Y ) or Xi ⊗Y = (Xi ⊗Y )

Also [Y, ] a right adjoint ⇒ preserves products ⇒ “exponentiation”: Q Q [Y,Z1 × Z2] = [Y,Z1] × [Y,Z2] or [Y, Zi] = [Y,Zi] Q ∼ Q Compare C(Y, Zi) = C(Y,Zi)

0 0 m m m Arithmetic: (l + l ) · m = l · m + l · m and (n1 · n2) = n1 · n2 in the skeleton (N, ·, 1) of (FinSet, ×, >). 26

26. Enriched categories Bicomplete category: All limits and colimits. Base category: bicomplete symmetric monoidal category (B, ⊗,I), e.g., (Set, ×, >), (L, ⊗,K), poset [0, ∞], ≥ , +, 0 with x+∞ = ∞.

B-enriched category: quiver C with ∀ x, y ∈ C0 ,C(x, y) ∈ B0 and: • composition: ∀ x, y, z ∈ C0 , ◦ ∈ B C(x, y) ⊗ C(y, z),C(x, z)

• identities: ∀ x ∈ C0 , jx ∈ B(I,C(x, x)) with commuting: ◦⊗1 C(w, x) ⊗ C(x, y) ⊗ C(y, z) / C(w, y) ⊗ C(y, z)

1⊗◦ ◦ C(w, x) ⊗ C(x, z) ◦ / C(w, z) and

jx⊗1 C(x, y) / C(x, x) ⊗ C(x, y) [recall B = I ⊗ B, etc.] S SSSSS SSSSS 1⊗jy SSSS ◦ SSSSS SSSS C(x, y) ⊗ C(y, y) ◦ / C(x, y) ... for w, x, y, z ∈ C0 .

Locally small category is enriched over (Set, ×, >).

Pre-additive category is enriched over (Ab, ⊗, Z).

Linear category L is enriched over (L, ⊗,K).

Closed monoidal category is enriched over itself.

Preorder is enriched over the Boolean algebra 2 = {⊥ < >}, ∧, >.

Directed metric spaces are enriched over ([0, ∞], +, 0). Thus d(x, y) ∈ [0, ∞] for x, y ∈ C0, and composition means d(x.y) + d(y, z) ≥ d(x, z).

If the symmetric monoidal (B, ⊗,I) is closed, can “impoverish” the enriched category C to Co with Co(x, y) = B(I,C(x, y)) for x, y ∈ C0. 27

27. Copowers and free enriched categories

For a set S and an object b of a cocomplete category B, the colimit of the constant diagram S → {b} is the copower or multiple P S · b = s∈S b, with insertions ιs : b → S · b for s ∈ S.

Example: For X ∈ Set, have ιs : X → S × X = S · X; x 7→ (s, x).

|S| copies z }| { Example: For V ∈ L, have S · V = V ⊕ ... ⊕ V for S ﬁnite. S ∼ For arbitrary S, have power V = Set(S,UV ) = L(FS,V ) ∈ L0, and copower S · V = f : S → V ∞ > |{s ∈ S | f(s) 6= 0}| , a subobject of V S, proper if S is inﬁnite.

Category C, bicomplete closed symmetric monoidal base category (B, ⊗,I). Free B-enriched category BC on C: left adjoint to impoverishment.

Object class BC0 = C0. P For x, y ∈ BC0 := C0, deﬁne BC(x, y) := C(x, y) · I = I. f∈C(x,y)

For x ∈ C0, deﬁne jx = ι1x : I → C(x, x) · I.

For x, y, z ∈ C0, distributivity and unitality give BC(x, y)⊗BC(y, z) = P P P P P f∈C(x,y) I⊗ g∈C(y,z) I = f∈C(x,y) g∈C(y,z) I⊗I = (f,g)∈C(x,y)×C(y,z) I. P ◦ P Then have composition C(x,y)×C(y,z) I ___ / C(x,z) I . O 6 ι nnn (f,g) nnn nnn ιg◦f nnn I nn

Example: For a category C, and Boolean algebra 2, the free 2-enriched 2C is the preorder obtained by “forgetting arrow labels” of C.

Group rings: For linear (L, ⊗,K), and a one-object group G = G1 on G0 = {∗}, the group ring over K is the one-object free L-category LG, with morphism set G · K.

Standard Hopf algebra notation: ηG = j∗ = ι1∗ : K → G · K. 28

28. Pointed sets, kernels, and cokernels

Pointed set Xe has chosen element e, so e: > → X with image {e}.

Category of pointed sets is the slice category (> ↓ Set).

Internal hom [Xe,Yd] = [X,Y ]{d} with constant d: X → Y ; x 7→ d.

∼ Currying: (> ↓ Set)(Xe ∧ Yd,Zc) = (> ↓ Set) Xe, [Yd,Zc] with the smash product Xe ∧ Yd = (X r {e}) × (Y r {d}) ∪ {(e, d)} . {(e,d)}

Suppose category C has a zero object 0, e.g., (> → >) in (> ↓ Set).

Zero morphism in C(x, y) is the composite (x −→0 y) = (x → 0 → y).

f ker f Kernel: Ker f −−→ x is the equalizer of x 4* y . 0 f coker f Cokernel: y −−−−→ Coker f is the coequalizer of x 4* y . 0 ker f coker f Lemma: Ker f −−→ x is mono; and dually y −−−−→ Coker f is epi.

0 r,r 0 Proof. ∀ z −−→ Ker f , (ker f) ◦ r = (ker f) ◦ r =: κx f ker f ⇒ Ker f / x 4* y O z< zz 0 0 z r,r zzκ zz x zz 0 z ⇒ r = r .

−1 Example: For f ∈ (> ↓ Set)(Xd,Ye), have Ker f = f {e} d ,→ Xd.

−1 −1 op Object c of C with zero, (co)kernels, preorders ∂1 {c}, and ∂0 {c}, .

ker g coker f Ker g / c adjunction c / Coker f dH O HH tt H tt HH f g tt HH tt HH tt HH tt (ker g) f ⇔ d ⇔ g ◦ f = 0 ⇔ b zt ⇔ (coker f) g

So: ker g = ker coker ker g and coker f = coker ker coker f 29

29. Factorization of morphisms, abelian categories

f First Isom. Thm. for sets: X / Y so f = m ◦ e , DD O DD DD m e DD D! f(X) m mono, e epi

Abelian category [Freyd]: (A0) A has a zero object; (A1) For A, B ∈ A0, product A × B and coproduct A + B exist; (A2) For A, B ∈ A0 and f ∈ A(A, B), ker f coker f have kernel Ker f −−→ A and cokernel B −−−−→ Coker f (A3) Every monomorphism is a kernel; every epimorphism is a cokernel.

im f ker coker f Image: [Im f −−→ B] := [Ker(coker f) −−−−−→ B], a monomorphism, smallest subobject of B that divides f.

coim f coker ker f Coimage: [A −−−→ Coim f] := [A −−−−−→ Coker(ker f)] , an epimorphism, smallest quotient of A that divides f.

f q im f Factorization [A −→ B] = [A −→ Im f −−→ B] with q an epimorphism. Indeed, coker q 6= 0 would mean f divided by a smaller subobject of B.

Theorem: f : A → B mono and epi ⇒ f is an isomorphism.

coker f Proof. Have B −−−−→ 0 since f epi, and 1B = ker coker f. f Since f is mono, A −→ B is also a kernel of coker f [and so A ∼= B]. Thus f has a section: f ◦s = 1B. Dually, it has a retraction: r◦f = 1A. Since r = r ◦ 1B = r ◦ f ◦ s = 1A ◦ s = s, have f invertible.

Corollary: Im f ∼= Coim f, and f = im f ◦ coim f

See https://math.stackexchange.com/questions/3268091/ coimage-and-image-in-abelian-categories 30

30. Enriching abelian categories Abelian category A (Freyd’s deﬁnition).

Matrices: XXo × Y / YX / X + Y o Y V O H f [f g] f f g g g A - A q

0 1 [1 0] Exact: 0 → X −→ιX X + Y −−→ Y → 0, 0 → X −−→ X × Y −→πY Y → 0

1 0 0 1 Theorem: 0 → X + Y −−−−→ X × Y → 0 exact, so X + Y ∼= X × Y =: X ⊕ Y , biproduct.

1 [1 1] 1 Diagonal: ∆: X −−→ X ⊕ X. Summation: Σ: X ⊕ X −−→ X.

f+Lg f+Rg For f, g ∈ A(A, B), deﬁne A / B and A / B . w; GG O ww GG ∆ ww GG Σ ww [f g] GG ww f G# A ⊕ A g B ⊕ B

Proposition: 0 +L f = f = f +L 0, 0 +R f = f = f +R 0.

Proposition: (f +L g) +R (h +L k) = (f +R h) +L (g +R k).

f h ∆ g k Σ Proof. Both sides are A −→ A ⊕ A −−−−→ B ⊕ B −→ B.

Theorem: +L = +R, commutative and associative.

Proof. Setting g = h = 0, have f +R h = f +L h =: f + h. Setting h = 0, have (f + g) + k = f + (g + k). Setting f = k = 0, have g + h = h + g.

Theorem: A(A, B) is an abelian group.

1 f 0 1 Proof. For f : A → B, have A ⊕ A −−−−→ B ⊕ B monic and epic, so an 1 g 0 1 isomorphism with inverse B ⊕ B −−−−→ A ⊕ A, then f + g = 0. 31

31. Categories and 2-categories

Category Cat of (small) categories; with Cat(D,C) = CD =: [D,C] (functor category). Cartesian closed monoidal (Cat, ×, 1), terminal category 1 = • y as the monoidal unit, and Currying Cat(A × B,C) ∼= Cat(A, [B,C]).

(Strict) 2-category: (1-)category C enriched over Cat. 0-cells: objects A, B, . . . of C. 1-cells: morphisms of C, i.e., objects in the categories C(A, B) or elements of the sets Cat1, C(A, B). 2-cells: morphisms in the categories C(A, B).

Example: 2-category Cat. Categories as 0-cells. Functors as 1-cells.

F Natural transformations τ : F → G as 2-cells A ⇓ τ% B 9 G Horizontal 2-cell composition: Cat(A, B)×Cat(B,C) −→◦ Cat(A, C);

F F 0 F 0◦F A ⇓ τ% B ⇓ τ 0 % C 7→ A ⇓ τ 0 ◦ τ' C 9 9 7 G G0 G0◦G 0 τF a with F 0 ◦ F a / G0 ◦ F a MM 0 MM (τ ◦τ)a F 0τ MM G0τ a MMM a M& 0 0 F ◦ Ga 0 / G ◦ Ga τGa

1C Identity: Cat(1, Cat(C,C)) 3 j : • y 7→ C ⇓ id % C C 9 1C

(τ•σ)a Vertical 2-cell composition on Cat(A, B): F a / Ha D < DD zz DD zz σa DD zzτa D! zz Ga Entropic or interchange law: (τ 0 • σ0) ◦ (τ • σ) = (τ 0 ◦ τ) • (σ0 ◦ σ), as bifunctorial horizontal composition respects vertical composition.

(n + 1)-category: an n-category enriched over Cat. 32

32. The braid category Monoid (M, ∇: M × M → M, η : > → M) in C with products. Monoidal category (C, ⊗, I) is a monoid in (Cat, ×, 1).

Sequence (Gn | n ∈ N) of groups with trivial G0, and homomorphisms ρm,n : Gm × Gn → Gm+n

ρl,m×1 satisfying Gl × Gm × Gn / Gl+m × Gn . Category G with G0 = N

1×ρm,n ρl+m,n Gl × Gm+n / Gl+m+n ρl,m+n and G(m, n) = Gm for m = n and Ø for m 6= n.

Monoidal category (G, +, 0) is (N, +, 0) at the object level, S with + = ρm,n : Gm ×Gn → Gm+n on morphisms.

Symmetric groups Sn = 2 hτ1, . . . , τn−1 | τi = 1, τiτj = τjτi for |i − j| > 1, τiτi+1τi = τi+1τiτi+1i.

General linear groups GL(K) with GLn(K).

Braid groups Bn = hσ1, . . . , σn−1 | σiσj = σjσi for |i − j| > 1, σiσi+1σi = σi+1σiσi+1i give the braid category B. Braid relation: i i + 1 i + 2 i i + 1 i + 2 Z Z Z Z σi tZ t t t tZ t σi+1 Z Z Z Z Z = Z σi+1 t tZ t tZ t t σi Z Z Z Z Z Z σi tZ t t t tZ t σi+1 Z Z it i t+ 1 i t+ 2 it i t+ 1 i t+ 2

First string on top layer, second in middle, third on bottom layer. • “Third Reidemeister move” in knot theory terms. • “Yang-Baxter equation” in physics. 33

33. Endofunctors, (co)algebras, monads Endofunctor category XX = [X, X] of category X.

Algebra (X, α) for an endofunctor T : X → X is given by a structure map α: TX → X in X(TX,X) for some X ∈ X0. Algebra (homo)morphism θ :(X, α) → (Y, β) T θ is given by commuting diagram TX / TY in X.

α β X / Y θ

Example: Finite subset endofunctor Pﬁn : Set → Set, bounded semilattice (commutative, idempotent monoid) (X, ·, 1), structure map α: PﬁnX → X; {x1, . . . , xr} 7→ x1 · ... · xr · 1. Coalgebra (X, α) for an endofunctor T : X → X is given by a structure map α: X → TX in X(X,TX) for some X ∈ X0. Coalgebra (homo)morphism θ :(X, α) → (Y, β) T θ is given by commuting diagram TX / TY in X. O O α β

X / Y θ

Example: Coalgebra with structure map α: X → PﬁnX represents a non-deterministic dynamical system.

X Endofunctors form a monoidal category X , ◦, 1X .

2 X Monad on X is a monoid (T, µ: T → T, η : 1X → T ) in X , ◦, 1X :

T µ T η T 3 / T 2 T 2 o T Here, µT is: O >> >> µ µT µ ηT >> >> T 2 T % T 2 / TT T X / X ⇓ µ X µ 9 T or T T 2 X ⇓ id & X ⇓ µ % X = µ ◦ id , whiskering. T 8 9 T T T

Will get various kinds of algebras from monads. 34

34. Adjunctions yield monads

Adjunction (F : X → A,U : A → X, η : 1X → UF, ε: FU → 1A).

Triangular identities 1F = εF • F η and 1U = Uε • ηU.

Trace in X: UF -coalgs. ηX : X → UFX, µ := UεF : UFUF → UF .

UF η Proposition: Unital law UFUF o UF O JJ JJ UεF ηUF JJ JJ J$ UF UF

F η Proof. Triangular FUF o F and UFU FF O FF FF FF Uε FF ηU FF εF FF FF F" F" F U U Proposition: µ: UFUF → UF associative.

UF UεF F Uε Proof. Need UFUFUF / UFUF or FUFU / FU .

UεF UF UεF εF U ε UFUF / UF FU / 1 UεF ε A

εFUA Naturality: FUA FUFUA / FUA

εA F UεA εA AFUA / A εA

. . . in A . . . in A

Theorem: Adjunction (F, U, η, ε) gives monad (UF, UεF, η) on X.

Example: Free monoid adjunction, for set or alphabet X. ∗ Then UFX or X is the set of words or lists hx1 . . . xri in the alphabet.

Coalgebra ηX : X → UFX; letter x 7→ one-letter “word” or list hxi.

Multiplication µX = UεFX : UFUFX → UFX; list of words or lists 7→ concatenation of list:

hhx11 . . . x1r1 i ... hxs1 . . . xsrs ii 7→ hx11 . . . x1r1 . . . xs1 . . . xsrs i — removes inner brackets. 35

35. Eilenberg-Moore algebras F v Does a monad (T, µ, η) on X give adjunction A ⊥ 6 X ? U • Monoid (M, m: M 2 → M, e : > → M) is a monoid in (Set, ×, >). M-sets X have action a: M × X → X with 1M ×a e×1X associativity: M 2 × X / M × X , unitality: X / M × X H HHHH HHHH m×1X a HHHH a HHHH HHH X M × X a / X

1M ×f M morphisms: M × X1 / M × X2 , category Set of M-sets.

a1 a2 X1 / X2 f Free FX = (M 2 × X −−−→m×1X M × X), adjoint U(M × X −→a X) = X.

Unit ηX : X → M × X; x 7→ (e, x), counit εa = a. Gives a model endofunctor T : X 7→ M × X on Set.

2 X • Monad (T, µ: T → T, η : 1X → T ) is a monoid in (X , ◦, 1X).

Eilenberg-Moore algebra a: TX → X in X(TX,X) for X ∈ X0: T a ηX with associativity: T 2X / TX and unitality: X / TX , DDD DDDD µX a DDD a DDDD DD TX a / X X

T f T morphisms: TX1 / TX2 , cat. X of Eilenberg-Moore algebras.

a1 a2 X1 / X2 f

Forgetful U T : XT → X;(TX −→a X) 7→ X. µ Free F T : X → XT ; X 7→ (T 2X −−→X TX). Note U T F T X = TX

T ηX T 2 T a a a Unit ηX : X −→ TX, counit εa :(T X −→ TX) −→ (TX −→ X)

Eilenberg-Moore adjunction F T ,U T , ηT , εT , yields monad (T, µ, η). 36

36. The Kleisli category of a monad

F v Alternative adjunction A ⊥ 6 X from monad (T, µ, η) on X. U

Kleisli category XT of monad (T, µ, η) on X:

(XT )0 = X0, XT (X,Y ) = X(X,TY ), identities 1X = ηX : X → TX, g f f T g µ composition (Y −→ TZ) ◦ (X −→ TY ) = (X −→ TY −→ T 2Z −→Z TZ) .

Adjunction XT (FT X,Y ) = XT (X,Y ) = X(X,TY ) = X(X,UT Y ) f f ηY with left adjoint FT (X −→ Y ) = (X −→ Y −→ TY ), g T g 2 µZ ηX right adjoint UT (Y −→ TZ) = (TY −→ T Z −→ TZ), unit X −→ TX, 1TY counit εY = TY −−→ TY , and monad (UT FT ,UT εFT , η) = (T, µ, η).

Power set monad (P, µ, η) with ηX : x 7→ {x} and set family union µX : {{x, . . . }, {y, . . . },... } 7→ {x, . . . , y, . . . , . . . } — like with lists.

Category Rel of relations X −→R Y = {(x, y) | x R y} on sets:

Have Rel0 = Set0, with 1X as the identity function or equality relation on a set X. Relation product (X −→R Y −→S Z) := {(x, z) | ∃ t ∈ Y . x R t S z}.

Theorem: Category Rel is the Kleisli category for (P, µ, η).

R R Proof. X −→ Y gives SetP -morphism X −→PX; x 7→ {y | x R y}. 0 0 Kleisli identity is ηX : X → PX; x 7→ {x} = {x | x = x }, and Kleisli composition gives the relation product:

X / PY / P2Z / PZ R PS µZ

x / {t | x R t} / {{z | t S z} | x R t} / {z | ∃ t . x R t S z}

E / {{z | e S z} | e ∈ E} 37

37. Compact closed categories

Compact closed category: Symmetric, monoidal (C, ⊗, 1) with: • Contravariant duality functor ∗ : C → C; ∗ • Evaluation natural transformation evX : X ⊗ X → 1; and ∗ • Coevaluation natural transformation coevX : 1 → X ⊗ X , yanking conditions

coevX ⊗1X ∗ 1X ⊗evX X −−−−−−→ X ⊗ X ⊗ X −−−−−→ X = 1X and ∗ 1X∗ ⊗coevX ∗ ∗ evX ⊗1X∗ ∗ X −−−−−−−→ X ⊗ X ⊗ X −−−−−→ X = 1X∗

Lemma: Internal hom [X,Y ] = X∗ ⊗ Y .

∗ Example: (Lﬁn, ⊗,K) with duality X = L(X,K). If X has basis {e1, . . . , en}, ∗ and X has dual basis {eb1,..., ebn} with ebi(ej) = δij, ∗ Pn then coev: K → X ⊗ X ; 1 7→ j=1 ej ⊗ ebj. Pn Pn Pn Yanking: ei 7→ j=1 ej ⊗ ebj ⊗ ei 7→ j=1 ej ⊗ ebj(ei) = j=1 ejδji = ei Pn Pn Pn and ebi 7→ ebi ⊗ j=1 ej ⊗ ebj 7→ j=1 ebi(ej) ⊗ ebj = j=1 δijebj = ebi.

Lemma: For Y with basis {d1, . . . , dm}, morphism ei 7→ dj corresponds to tensor ebi ⊗dj.

Example: Relation category Rel, biproduct is the disjoint union with ιX ιY & - q X 3 x x ∈ X + Y 3 y y ∈ Y k 3 $ πX πY

Monoidal category Rel, ×, {0}, compact closed with X∗ = X. Evaluation {((x, x), 0) | x ∈ X}, coevaluation {(0, (x, x)) | x ∈ X}. First yanking condition: relation product of {(x, (x0, x0, x)) | x, x0 ∈ X} with {((x0, x, x), x0) | x, x0 ∈ X} is {(x, x) | x ∈ X}. Second is similar. 38

38. Monoidal functors Monoidal categories (C, ⊗, 1), (C0, ⊗, 10).

Monoidal functor F :(C, ⊗, 1) → (C0, ⊗, 1) with natural transformations µX,Y : F (X) ⊗ F (Y ) → F (X ⊗ Y ) and C0-morphism : 10 → F (1) such that:

0 αF (X),F (Y ),F (Z) F (X) ⊗ F (Y ) ⊗ F (Z) / F (X) ⊗ F (Y ) ⊗ F (Z)

1F (X)⊗µY,Z µX,Y ⊗1F (Z) F (X) ⊗ F (Y ⊗ Z) F (X ⊗ Y ) ⊗ F (Z)

µX,Y ⊗Z µX⊗Y,Z F X ⊗ (Y ⊗ Z) / F (X ⊗ Y ) ⊗ Z, F (αX,Y,Z ) — associativity, and unitality: 0 0 ρF (X) λF (X) F (X) ⊗ 10 / F (X) , F (X) o 10 ⊗ F (X) O O 1F (X)⊗ F ρX F λX ⊗1F (X) F (X) ⊗ F (1) / F (X ⊗ 1) F (1 ⊗ X) o F (1) ⊗ F (X). µX,1 µ1,X

Example: Underlying set functor U :(L, ⊗,K) → (Set, ×, {1}).

Here : {1} → K; 1 7→ 1, and µX,Y : UX × UY → U(X ⊗ Y ) is the ! usual quotient by relations (k1x1 + k2x2, y) = k1(x1, y) + k2(x2, y), etc.

Strong isomorphisms. monoidal functor: and the µ are Strict X,Y identities.

Example: Free vector space functor F :(Set, ×, {1}) → (L, ⊗,K). Here F (X × Y ) = F (X) ⊗ F (Y ) and F {1} = K, so strong, strict.

0 σF (X),F (Y ) Braided monoidal functor: F (X) ⊗ F (Y ) / F (Y ) ⊗ F (X)

µX,Y µY,X F (X ⊗ Y ) / F (Y ⊗ X), F σX,Y

Symmetric monoidal functor if (C, ⊗, 1), (C0, ⊗, 10) symmetric. Example: ∗ : C → Cop in a compact closed category (C, ⊗, 1). 39

39. Dagger categories

Dagger category C has a contravariant functor †: C → C, with: † † †† X = X for X ∈ C0, adjoint f : Y → X of f ∈ C(X,Y ), f = f.

Example: One-object linear categories C, H with x† = x. Morphism f in dagger category C is: • Hermitian or self-adjoint if f † = f; • unitary if invertible and f † = f −1. Dagger monoidal category: † is a strict monoidal functor.

† Dagger compact closed category: ∀ X ∈ C0 , coevX = (evX ) .

†∗ ∗† Lemma: ∀ f ∈ C1 , f = f . (Necessary, not suﬃcient, for DCCC.)

† Biproduct dagger compact closed category: ∀ X ∈ C0 , πX = ιX . Example: Category FDHilb of ﬁnite-dimensional Hilbert spaces, with ∀ x ∈ X, ∀ y ∈ Y, hf(x) | yi = hx | f †(y)i for f : X → Y . Example: Rel with R∗ = R† as the converse relation. Information theory: A bit in a BDCCC is 2 := 1 ⊕ 1.

Examples: {0, 1} in Rel, or qubit C ⊕ C = C2 in FDHilb. Extract information from [1, 1], e.g., false = Ø and true = 11 in Rel, or scalar 1 7→ c in FDHilb.

Trace of f ∈ [X,X] = X∗ ⊗ X is 1 ∗ ⊗f 1 −−−→coevX X ⊗X∗ −→τ X∗ ⊗X −−−−→X X∗ ⊗X −−→evX 1. Example in FDHilb: P P P P P P j ej ⊗ebj 7→ j ebj ⊗ej 7→ j ebj ⊗f(ej) 7→ k j ebj ⊗fjkek 7→ j fjj

Positive endomorphism f : X → X if ∃ g : X → Y . f = g† ◦ g. Examples: In Rel, x R y ⇒ y R x and x R x. In FDHilb, ∀ x ∈ X, hf(x) | xi ≥ 0.

Complete positivity of f :[X,X] → [Y,Y ] or f : X∗ ⊗ X → Y ∗ ⊗ Y : ∗ ∗ ∀ Z ∈ C0 , ∀ positive g : 1 → Z ⊗ X ⊗ X ⊗ Z, g 1Z∗ ⊗f⊗1Z 1 / Z∗ ⊗ X∗ ⊗ X ⊗ Z / Z∗ ⊗ Y ∗ ⊗ Y ⊗ Z is positive. 40

40. Subobjects and subobject classifiers For object X of category C, deﬁne Presub(X) as the full subcategory of (C ↓ X) whose deﬁning morphisms s: S → X are monomorphisms.

Lemma: Presub(X) is a preorder: X ? _?? s ?? t j ?? 1 ?? * S 4 T ⇒ j1 = j2. j2

Skeleton poset SubC(X) or just Sub(X) consists of [2nd-order concept] subobjects of X: Equivalence classes of monomorphisms s: S → X. Note X ? O _? s ?? s 0 ? s ?? 0 ?? j 0 j 0 0 S / S 7/ S ⇒ j ◦ j = 1S, similarly j ◦ j = 1S0 .

1S Well-powered category C: Each object has a set of subobjects. Now assume C has ﬁnite limits, so terminal > giving elements > → X of objects X.

Subobject classiﬁer > −−→true Ω in C [makes subobjects ﬁrst-order!]: s ∀ X ∈ C0 , ∀ (S −→ X) ∈ SubC(X) , ∃! χ . S / > s p-b true X χ ___ / Ω

Example: > −−→{true false, true} in Set with S = χ−1{true} ⊆ X. Also works in category SetG of G-sets, for any group G, with trivial action of G on Ω. Proposition: If C is well-powered and locally small, ∼ SubC = C( , Ω) = ∃Ω ∈ Cb 0. Proof. S0 / S / >

p-b p-b true Y / X / Ω f χ

Remark: S0 is the inverse image of S under f : Y → X. 41

41. Dinatural transformations and power objects Given graph maps F,G: Dop × D → C for graph D and category C, a dinatural transformation δ : F ⇒ G is a “vector” (δx | x ∈ D0) of components δx : F (x, x) → G(x, x) in C1 such that, for all f : x → y in D1, the hexagon of the dinaturality diagram

δx x x F (x, x) / G(x, x) O s9 KK F (f,1x) ss KKG(1x,f) ss KK ss KK ss K% f f F (y, x) G(x, y) K 9 KK ss KK ss KK ss F (1y,f) KK ss G(f,1y) % s y y F (y, y) / G(y, y) δy

in Dop in D in C commutes in the category C.

n n Example: δx : C(x, x) → C(x, x); k 7→ k — Church numeral n. n % 1 h ◦ f 7→ (h ◦ f) * h 0 f ◦ (h ◦ f)n = (f ◦ h)n ◦ f n 4 - f ◦ h 7→ (f ◦ h)

Example: For C with (finite limits and) a subobject classifier Ω, ∼ ∼ SubC(X × Y ) = C(X × Y, Ω) = C(X, PY ) defines the power object PY of an object Y . ∼ In C(PY × Y, Ω) = C(PY, PY ), suppose 3Y 7→ 1PY .

In Set, have 3Y as characteristic function of {(S, y) ∈ PY ×Y | S 3 y}. For f : X → Y , deﬁne Pf : PY → PX as the unique morphism making 3X PX × X / Ω commute. p7 >>>> Pf×1X pp >>> ppp >>> pp >>>> ppp >> PY × X Ω N NN NNN NN P1Y ×f NN N' PY × Y / Ω 3Y op So dinatural 3: P × 1C ⇒ ∆Ω for P : C → C. 42

42. Elementary and Grothendieck topoi Elementary topos: Category C with ﬁnite limits and a power object. Properties: Finite colimits, subobject classiﬁer, Cartesian closed. op Examples: Presheaf categories Db = SetD for small D.

Grothendieck topos: E with reﬂective full K : E ,→ Db, for some D, where the left adjoint L: Db → E preserves ﬁnite limits.

Example: Sheaves — the presheaves F ∈ E0 ⊆ Db, where D = (O, ⊆) for a topological space (X, O), satisfying: S For each open cover U = i∈I Ui of each U ∈ O, require equalizer

F (Ui∩Uj ⊆Ui) FUi / F (Ui ∩ Uj) O O πi πi,j p e Q ___ / Q FU ___ / k∈I FUk k,l∈I F (Uk ∩ Ul) . q ___ / πj πi,j FUj / F (Ui ∩ Uj) F (Ui∩Uj ⊆Uj )

Typically, F (V ⊆ U): FU → FV ; f 7→ f|V (restriction of functions).

Equalizer condition means match of FUi and FUj on F (Ui ∩ Uj).

Elementary deﬁnition: A topos is a category C with the following. (a) A terminal object >. (b) Pullback of each X → B ← Y . true s (c) Monic > −−→ Ω, and ∀ monic S −→ X, ∃! χ . S / > s p-b true X χ ___ / Ω

(d) ∀ Y, ∃ (3Y : PY × Y → Ω) . ∀ (ρ: X × Y → Ω) , ρ ∃ unique X such that X × Y / Ω

r r×1Y PY PY × Y / Ω 3Y Lawvere: First-order theory of topoi as a foundation for mathematics.