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

tail, domain // ∂0ff ∈C1 // ∂1f head, codomainoo

Opposite or dual graph of C = (C0, C1, ∂0, ∂1) is Cop = (C0, C1, ∂1, ∂0)

Graph homomorphism F : D → Chas 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

1 //

OO

2

OO

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

yg

xxqqqqqqqqqqqqqhg

<<<<<<<<<<<<<<<<<

z

h++VVVVVVVVVVVVVVVVVVVVVVVVVV x

f

iiSSSSSSSSSSSSSSSSSSSS

h(gf)=(hg)f

gfoo

t

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

y

g

1y

&&MMMMMMMMMMMMM x

f

foo

z ygoo

Example: N0 = x , N1 = N , 1x = 0 , ∀m,n ∈ N , nm = m+n ; —

one object, lots of arrows [monoid of natural numbers under addition]

Equation: 3 + 5 = 4 + 4 Commuting diagram:x

4 //

3

x

4

x5// x

Example: N1 = N , ∀ m,n ∈ N , |N(m,n)| =

1 if 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 offunctions: 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.

• 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→ max0, 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 .

Ex: R2 → R2; (x1, x2) 7→ (x1, 0) in Set.

Lemma. For xr((y

shh with r s = 1y:

• 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 InvX 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 .

Global conditions:

Isomorphism: F0 and F1 are isomorphisms.

Essentially surjective: ∀ c ∈ C0 , ∃ d ∈ D0 . c ∼= F0d

Local conditions:

Full: Each restriction (4.1) is surjective.

Faithful: Each restriction (4.1) is injective.

Example: While the forgetful or underlying set functor

U : Grp→ Set; [f : (G1, ·,−1 , 1)→ (G2, ·,−1 , 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;

[f : (M1, ·, 1)→ (M2, ·, 1)] 7→ [f |M∗1 : (M∗1 , ·,−1 , 1)→ (M∗

2 , ·,−1 , 1)]

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 functorbetween 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 catgeoriescorresponds 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, anatural transformation τ : F → G is a “vector” (τx | x ∈ D0) ofcomponents τx : Fx→ Gx in C1 such that,

for all f : x→ y in D1, the rectangle of the naturality diagram

x

f

Fxτx //

Ff

Gx

Gf

y Fy τy// Gy

. . . 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 functorLA : Set → Set; [f : X → Y ] 7→ [A × X → A × Y ; (a, x) 7→ (a, fx)].Then a function α : A→ B gives a natural transformationLα : LA → LB with components LαX : A×X → B×X; (a, x) 7→ (α a, x)and naturality diagram

X

f

(a, x) LαX //_

LAf

(α a, x)_

LBf

Y (a, fx) LαY

// (α a, fx)

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 ei : V → R; ej 7→ δij.

Then V → V ∗; ei 7→ ei 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:

y

g

xfoo

gf

||xxxxxxxxxxxxxxxxxxy

f // x

z In C z

g

OO

gf or gf

;;wwwwwwwwwwwwwwwwwwIn 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 theone-object monoid category of the monoid (M, , 1M) with xy = 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 → Cis a (“covariant” or usual) functor F : D → Cop or F : Dop → C.

Thus F (1x) = 1Fx as usual, but F (g f) = Ff Fg.

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

g

Xf

oo

θ g θ g f∗ // θ g f

T

θ_

g∗

OO

θ_

g∗

OO

:

(gf)∗

==zzzzzzzzzzzzzzzzzzzzz

Z

EE

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 Chas graph maps F,G : D → C as objectsand natural transformations σ : F → G as morphisms. Composition:

x

f

Fx(τ•σ)x //

Ff

σx((QQQQQQQQQQQQQQQ Hx

Hf

Gx

Gf

τx

66mmmmmmmmmmmmmmm

Gy

τy((PPPPPPPPPPPPPPP

y Fy(τ•σ)y

//

σy

66nnnnnnnnnnnnnnnHy

. . . in D . . . in C

Constant objects and morphisms: x

f

cθ //

1c

c′

1c

y cθ

// c′

Functor category: Category D, functors F,G, . . .

Example: Linear representations of a group G are objects R : G→ Lof the functor category LG for the one-object group category G,so group homomorphisms R : G→ L(V, V )∗ = AutV = GL(V ).

The morphisms are intertwiners or equivariant maps τ : R1 → R2,

so ∀ g ∈ G , V1

R1(g)

τx // V2

R2(g)

V1 τx// V2

E.g., G = S3, V1 = R3 = Spane1, e2, e3, R1(π) : ei 7→ eπ(i).

V2 = R2 = Spane2−e1, e3−e2, R2(π) : (ei+1−ei) 7→ (eπ(i+1)−eπ(i))

τx︷ ︸︸ ︷1

3

[−2 1 1−1 −1 2

]·

R1(1 2 3)︷ ︸︸ ︷0 0 11 0 00 1 0

=

R2(1 2 3)︷ ︸︸ ︷[0 −11 −1

]·

τx︷ ︸︸ ︷1

3

[−2 1 1−1 −1 2

]

8

8. Products and coproducts

Product X × Y = (x, y) | x ∈ X , y ∈ Y of sets X, Y :

X X × YπXoo πY // Y

Zf

[[

g

CC

fug

OO

Universality property: ∀ Z ∈ Set0, “solid”↓ implies ↓“dashed”bijection Set(Z,X)× Set(Z, Y )→ Set(Z,X × Y ); (f, g) 7→ f u gwith 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 = • • :

Zf //

fug ''PPPPPPPPPPPPP X

X × YπX

77nnnnnnnnnnnnn

X × YπY

((PPPPPPPPPPPPPP

Z g//

fug77nnnnnnnnnnnnnn

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,

c

OO AAd

OO

but c× d does not.

Coproduct in C is the product in Cop: XιX //

f ))

X + Y

ftg Y

ιyoo

guuZ

Example: Coproduct in Set is the disjont union.

Example: Coproduct in a poset category is a least upper bound.

Biproduct UιU --

U ⊕ VπU

jjπV

44 VιUqq

in L is product and coproduct.

9

9. More limits and colimits

Pullback: Z

k

&&

h --

r//___ X ×B Y πY

//

πX

Y

g

X

f// B

or with poset diagram

• → • ← • category picture: Zh //

r((PPPPPPP X

f

X ×B YπX

66nnnnnnnnnnnnnn

Zr //______

fh=gk

66X ×B YfπX=gπY // B

X ×B YπY

((PPPPPPPPPPPPPP

Zk

//

r

66nnnnnnnY

g

OO

Ex: Domain of category composition is pullback of C1∂0−→ C0

∂1←− C1.

Pushout is the dual of a pullback.

Equalizer: Z

k

$$

r//___ E e

// Xf //g// Y so fk = gk ⇒ ∃! r . er = k

In L, E = Ker(f − g)e→ X. In Set, E = x ∈ X | fx = gx e

→ X.

Coequalizer: Xf //g// Y u

//

k

##C r

//___ Z ; kf = kg ⇒ ∃! r. ru = k

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

0 // Kerf // Xf // Y // Cokerf // 0

where exact means Im g1 = Ker g2 for eachg1 // • g2 // .

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, c′) is nat.

tr. ∆θ : ∆c→ ∆c′ with ∆: D → C; [f : x→ y] 7→ cθ

// c′

cθ // c′

.

Limit of graph map F : D → C is projection π : ∆ lim←−F → F suchthat ∀ κ : ∆Z → F , ∃! r = lim←−κ ∈ C(Z, lim←−F ) . π ∆ lim←−κ = κ.

x

f

Zκx //

r=lim←−κ ''NNNNNNNNNNNNN Fx

Ff

lim←−Fπx

77ooooooooooooo

lim←−Fπy

&&NNNNNNNNNNNNN

y in D in C Z κy//

r=lim←−κ88qqqqqqqqqqqqq

Fy

A.k.a “projective limit” or “inverse limit”, written as lim.

Colimit of graph map F : D → C is insertion ι : F → ∆ lim−→F suchthat ∀ 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 categoriesx : (N,≤)→ (R ∪ ∞,≤) : n 7→ xn. Then lim−→x = limn→∞ xn.

Example: F : Nr 0, 1 → Ring;n 7→ Z/nZ. Thenr = ∆ lim←−κ : Z → lim←−F =

∏∞n=2 Z/nZ for κn : Z→ Z/nZ;x 7→ x+nZ.

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 : Pfin(V )→ L;X 7→ Span(X). Then lim−→F = V .

Theorem: Each algebra is the (directed) colimitof 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, x′), (y, y′)

)× (B × C)

((y, y′), (z, z′)

)→ (B×C)

((x, x′), (z, z′)

):((f, f ′), (g, g′)

)7→ (f ′f, g′g).

Universality: B B × CπBoo πC // C

DF

[[

G

CC

FuG

OO

— graph maps or functors.

Example: B′ B′ × C ′π′Boo

π′C // C ′

B

F

OO

B × CπBoo πC //

F×G

OO

C

G

OO

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.

Then ∀ f : b→ b′, g : c→ c′: S(b, c)Rb(g)=S(b,g)

//

Lc(f)=S(f,c)

S(f,g)

((QQQQQQQS(b, c′)

Lc′ (f)=S(f,c′)

S(b′, c)Rb′ (g)=S(b′,g)

// S(b′, c′)

Conversely, given Rb : C → D and Lc : B → Dwith ∀b ∈ B0, c ∈ C0 , Lc(b) = Rb(c)and commuting solid square,the diagonal defines a bifunctor S : B × C → D.

Example: Locally small C, B = Cop, Rb : C → Set; b 7→ C(b, c),Lc : Cop → Set; b 7→ C(b, c) (like dualizing), Rb(c) = C(b, c) = Lc(b).

For h ∈ C(b′, c), so bf // b′

h // cg // c′ , have

h f Rb(g) // g h f C(b, c)Rb(g) // C(b, c′)

h_

Lc(f)

OO

Rb′ (g)

// g h_Lc(f)

OO

C(b′, c)

Lc(f)

OO

Rb′ (g)// C(b′, c′)

Lc′ (f)

OO

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.

[Typical two-stage projection πY : X × (Y × Z)πY×Z−−−→ Y × Z πY−→ 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)αW×X,Y,Z

ttiiiiiiiiiiiiiiii

((W ×X)× Y )× Z W × (X × (Y × Z))

αW,X,Y×ZjjUUUUUUUUUUUUUUUU

1W×αX,Y,Z

(W × (X × Y ))× Z

αW,X,Y ×1Z

OO

W × ((X × Y )× Z)αW,X×Y,Zoo

Triangle: X × (>× Y )αX,>,Y //

1X×λY ''OOOOOOOOOOO(X ×>)× Y

ρX×1Ywwooooooooooo

X × Y

Digon: >×>λ>

((

ρ>

66 >

Coherence [CWM §VII.2]: If these three diagrams commute, thenw.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:

G×G×G 1G×∇//

∇×1G

G×G∇

G×G∇

// G

and G×G∇

%%KKKKKKKKKKK G×>1G×ηoo

ρG

>×GλG

//

η×1G

OO

G

(so a monoid)

and G×G 1G×S // G×G∇

##GGGGGGGGG

G

∆;;wwwwwwwww

∆ ##GGGGGGGGG// > η // G

G×GS×1G

// G×G∇

;;wwwwwwwww

with G×G πG //

πG

G

G G

1G

OO

1Goo

∆ffM M M M M M

(so a group).

Group in a Cartesian monoidal category: interprets the diagrams.

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

Example: The additive group functorGa : K 7→ (K,+,−, 0) is a group in GrpCRing.

Example: The multiplicative group or group-of-units functorGm : K 7→ (K∗, ·,−1 , 1) is a group in GrpCRing.

Example: The p-th roots of unity functorµp : K 7→ (k ∈ K | kp = 1, ·,p−1 , 1) is a group in GrpCRing.

Example: (SL2,∇, S, η) as a group in GrpCRing:

SL2 : CRing→ Grp;K 7→[

a bc d

] ∣∣∣∣ a, b, c, d ∈ K , ad−bc = 1

.

∇K :

([a bc d

],

[a′ b′

c′ d′

])7→[aa′ + bc′ ab′ + bd′

ca′ + dc′ cb′ + dd′

],

SK :

[a bc d

]7→[d −b−c a

], and ηK : 1 7→

[1 00 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.

For set X, v. sp. with basis X is FX = r∑i=1

λi ·xi | λi ∈ K , xi ∈ X

,

the space of formal linear combinationsr∑i=1

λi · xi of elements of X.

At X, have unit ηX : X → UFX;x 7→ 1 · x which inserts the basis.

At V , have counit εV : FUV → V ;r∑i=1

λi · vi 7→ v, where

v = λ1v1 + . . .+ λrvr, the formal combination worked out in V .

For f : X → Y , lin. transf. Ff : FX → FY ;r∑i=1

λi · xi 7→r∑i=1

λi · f(xi).

So have functors F : Set→ L and U : L → Set.

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

Mutually inverse ϕX,V : [FXθ−→ V ] 7→ [X

ηX−→ UFXUθ−→ UV ]

(informally, restricting θ to X); note unit ηX = ϕX,FX(1FX);

and dually, ϕ−1X,V : [X

f−→ UV ] 7→ [FXFf−→ FUV

εV−→ V ]

(informally, extending f to FX); note counit εV = ϕ−1UV,V (1UV ).

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

Thus ϕX,V : θ 7→ Uθ ηX and ϕ−1X,V : f 7→ εV Ff .

Triangular identities: ∀X ∈ Set0 , 1FX = εFXFηX [= ϕ−1X,FX(ηX)]

and ∀ V ∈ L0 , 1UV = UεV ηUV [= ϕUV,V (εV )].

The triangular identities are necessary and sufficient for an adjumction.

Other notations: F a U or LU

55⊥ Set

F

vv

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 adjointto the forgetful functor U : Mon→ Set.

“Tensor” notation x1 ⊗ . . .⊗ xn for n-tuple (x1, . . . , xn).

For set or alphabet X, coproduct FX :=∑n∈N

Xn, with X0 = 1 and

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(FG,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.

Galois correspondence: Mut. inverse(S(B),≤

) R // (R(A),≤

)Soo .

Polarity is a relation α ⊆ I × J . Gives Galois connection

S : (2I ,⊆)→ (2J ,⊇);X 7→ y ∈ J | ∀x ∈ X , xα y

R : (2J ,⊇)→ (2I ,⊆);Y 7→ x ∈ I | ∀y ∈ Y , x α y

Note ∀ X ⊆ I , ∀ Y ⊆ J ,

SX ⊇ Y ⇔ ∀ x ∈ X , ∀ y ∈ Y , xα 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 〈u,v〉.Polarity (u,v) | 〈u,v〉 = 0 ⊆ V × V .Closure of a subset is its orthogonal complement.

Alg. geometry: On Cn×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 IC[X1, . . . , Xn]

is its radical√I = f | ∃ 0 < n ∈ N . fn ∈ I.

Example: Radical of 〈X21 〉 in C[X1, . . . , Xn] is 〈X1〉.

17

17. Slice categories and comma categories

For b ∈ C0, slice category (C ↓ b) or (1C ↓ b) of C-objects over b

has object class ∂−11 (b), morphisms c

p<<<<<<<<f // c′

p′

b

(commuting),

composition c

p======== f//

f ′f''

c′

p′

f ′// c′′

p′′

b

, terminal object 1b : b→ 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)(a′, g′) = (a+ ga′, gg′).

For b ∈ C0 and T : E → C,comma category (T ↓ b) of objects T -over b

has morphisms Te

p@@@@@@@@Tf // Te′

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).

Proof. XηX

xxxxxxxxp

!!CCCCCCCC

UFXUϕ−1

X,Ap

//_______ UA

and FX

p!!CCCCCCCCFϕX,Ap //_______ FUA

εAvvvvvvvvvv

A

Cor: Given A

U

66⊥ X

F

vv, unit and counit uniquely determined.

18

18. The Yoneda Lemma

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

Then forK : A→ Set, have SetA(A(A1, ), K

) ∼= KA1; τ 7→ τA1(1A1)

Proof. • Injectivity:

A1

h

A(A1, A1)τA1 //

A(A1,h)

=L(h)

KA1

Kh

1A1_

// τA1(1A1)_

A2 A(A1, A2) τA2

// KA2 h // τA2(h) = Kh(τA1(1A1)

)

In A In Set

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

A2

f

A(A1, A2)ρA2 //

A(A1,f)

=L(f)

KA2

Kf

h_

// Kh(x)_

A3 A(A1, A3) ρA3

// KA3 f h //

Kf(Kh(x)

)=

K(f h)(x)

In A In Set

Corollary: Full, faithful (covariant) Yoneda embedding

∃ : D → D = SetDop

; [f : x→ y] 7→[D( , f) : D( , x)→ D( , y)

]Category D 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 meansthe 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 → GrpThen 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:

Lemma 1: τ : S → T is

epi

mono

in SetA iff each τA′′

epi

mono

in A.

Proof. Sτ //

τ

p-o

T

1T

T1T

// T

⇔ ∀ A′′ ∈ A0, SA′′τA′′ //

τA′′

p-o

TA′′

TA′′ TA′′

Lemma 2: For f : A′ → A, natural transformation R(f) or

A(A, f) : A(A, )→ A(A′, ) is

monoepi

iff f is

epi

split mono

.

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

Proof. A(A,A′)R(f)−−−→ A(A′, A′) epi ⇒ ∃ r ∈ A(A,A′) . r f = 1A′ .

Conv., f oprop = 1A′ ⇒ ∀ A′′ , ∃A′′(f op) ∃A′′(rop) = ∃A′′(1A′)⇒ R(f)A′′ R(r)A′′ = 1A(A′,A′′) ⇒ R(f)A′′ surj., epi; so R(f) epi.

∀ A′′ , A(A,A′′)R(f)−−−→ A(A′, A′′) mono⇔ h1f = h2f ⇒ h1 = h2 .

Theorem: In AU

44⊥ XF

tt,

U is . . . iff εA . . .full has retractfaithful is epifull, faithful is iso

Proof. Natural transformation α : A(A, )→ A(FUA, ) with

component αA′ : A(A,A′)UA,A′−−−→ X(UA,UA′)

ϕ−1UA,A′−−−−→ A(FUA,A′).

Under Yoneda Lemma, α 7→ αA(1A) = εA. Then by Lemma 2:εA split mono ⇔ α epi ⇔ ∀A′ , αA′ surj. ⇔ ∀A′ , UA,A′ surj;εA epi ⇔ α mono ⇔ ∀A′ , αA′ mono ⇔ ∀A′ , UA,A′ 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 : FUq ∼= q, isomorphic counit of an adjunction.Note (P,≤) is a poset — antireflexive!

Adjoint equivalence: AU

44⊥ XF

ttwith unit, counit iso.

Equivalence: AU

55 XF

uuwith 1X → UF , FU → 1A iso.

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 ⇒∀ f ∈ A(FX,A) , ∃! ϕX,Af ∈ X(X,UA) . FϕX,Af = ε−1

A f , . . .[Complete the adjunction, dual to the construction for linear algebra.]

(c)⇒(a): Need F full and faithful.

F faithful: X1

g

ηX1 // UFX1

UFg

X2 ηX2

// UFX2

and U faithful: FUA1

k

εA1 // A1

FUk

FUA2

εA2 // A2

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

Have X1

f

ηX1 // UFX1

Uh

X2 ηX2

// UFX2

for f = η−1X2 Uh ηX1 and X1

f

ηX1 // UFX1

UFf

X2 ηX2

// UFX2

so Uh = UFf . Then U faithful gives h = Ff.

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.

• Morita equivalence: Ring R, ring Rnn of n× n-matrices over R.

ModRU

22⊥ ModRnn

Frr

with U : M →n︷ ︸︸ ︷

M ⊕ · · · ⊕M ,

F : [Rnn → End(N)] 7→ [R→ Rn

n → End(N)].

Concrete category:Category of sets with structure (algebraic, topological,. . . )and structure-preserving functions (homomorphisms, continuous,. . . ).

• Duality: Equivalence AU

33⊥ Xop

Ftt

of concrete categories.

Dualizing object: Set T with structure T ∈ A or T ∈ X,where: ∀ A ∈ A0 , A(A, T ) ≤ Set(A, T ) = TA ∈ X

and: ∀ X ∈ X0 , X(X,T ) ≤ Set(X,T ) = TX ∈ A.

Then U = A( , T ) and F = X( , T ).

Example: Category Lfin of fin.-dim. vector spaces over a field K.Then A = X = Lfin, T = K and ε−1

V : 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”.

A := UA = Ab(A, T ), the group of characters χ : A→ T .ε−1A : 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 algebrasto 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

AG// B

G preserves J-limits if it “pushes limits forward”:

Diagram D : J → A has a limit[

lim←−Dπj−→ Dj

]implies GD : J → B has a limit

[G(lim←−D)

Gπj−−→ GDj

].

G reflects J-limits if it “pulls limits back”:

Diagram GD : J → B has a limit of the image form[GL

Gπj−−→ GDj

]implies D : J → A already had a limit

[lim←−D = L

πj−→ Dj

].

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:

EπY //

πX

p-b

Y

g

Xf// B

in Top

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

A

U

66⊥ X

F

vv

Suppose limit[

lim←−Dπj−→ Dj

]exists: A

κj //

r=lim←−κ &&NNNNNNNNNNNNN Dj

lim←−Dπj

77ppppppppppppp

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 functionϕ(n) = |r | 1 ≤ r ≤ n and gcd(r, n) = 1| =

∣∣(Z/n,×, 1)∗∣∣.

Recall 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).

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).

Example: Equivalence AU

55 XF

uu

implies A

U

66⊥ X

F

vvand A

U

66> X

F

vv,

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.By (∗), have y( z =

∑x | x · y ≤ z.

Example: Boolean algebra with implication p( q = p→ q = (¬p)∨qNegation (pseudocomplement) ¬x := x( 0 in any Heyting algebra.

Example: 0 ≤ 12≤ 1, where 1

2( 0 = maxx | x · 1

2≤ 0 = 0.

Then ¬¬12

= ¬0 = 1 6= 12; “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 | XrS 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 L(X,L(Y, Z)

)⊆ Set

(X,Set(Y, Z)

) ∼= Set(X × Y, Z), so

L(X,L(Y, Z)

)tracks the bilinear maps X × Y → Z.

In all three cases, ∼= is a natural isomorphism of sets, so on theleft 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→ Cand 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:

(X+X ′)⊗Y = (X⊗Y )+(X ′⊗Y ) or(∑

Xi

)⊗Y =

∑(Xi⊗Y )

Also [Y, ] a right adjoint ⇒ preserves products ⇒ “exponentiation”:

[Y, Z1 × Z2] = [Y, Z1]× [Y, Z2] or [Y,∏Zi] =

∏[Y, Zi]

Compare C(Y,∏Zi) ∼=

∏C(Y, Zi)

Arithmetic: (l + l′) ·m = l ·m+ l′ ·m and (n1 · n2)m = nm1 · nm2in 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:

C(w, x)⊗ C(x, y)⊗ C(y, z)⊗1 //

1⊗

C(w, y)⊗ C(y, z)

C(w, x)⊗ C(x, z) // C(w, z) and

C(x, y)

SSSSSSSSSSSSSSSSS

SSSSSSSSSSSSSSSSS

jx⊗1 //

1⊗jy

C(x, x)⊗ C(x, y)

[recall B = I ⊗B, etc.]

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” theenriched 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 ofthe constant diagram S → b is the copower or multiple

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).

Example: For V ∈ L, have S · V =

|S| copies︷ ︸︸ ︷V ⊕ . . . ⊕ V for S finite.

For arbitrary S, have power V S = 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.

For x, y ∈ BC0 := C0, define BC(x, y) := C(x, y) · I =∑

f∈C(x,y)

I.

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) =∑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.

Then have composition∑

C(x,y)×C(y,z) I //___∑

C(x,z) I

I

ι(f,g)

OO

ιgf

66nnnnnnnnnnnnnn

.

Example: For a category C, and Boolean algebra 2,the free 2-enriched 2C is the preorderobtained 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 =((

(Xre)× (Y rd))∪(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 (x0−→ y) = (x→ 0→ y).

Kernel: Ker fker f−−→ x is the equalizer of x

f**

0

44 y .

Cokernel: ycoker f−−−−→ Coker f is the coequalizer of x

f**

0

44 y .

Lemma: Ker fker f−−→ x is mono; and dually y

coker f−−−−→ Coker f is epi.

Proof. ∀ z r,r′−−→ Ker f , (ker f) r = (ker f) r′ =: κx

⇒ Ker fker f // x

f**

0

44 y

z

r,r′

OO κx

<<zzzzzzzzzz⇒ r = r′ .

Example: For f ∈ (> ↓ Set)(Xd, Ye), have Ker f =(f−1e

)d→ Xd.

Object c of C with zero, (co)kernels, preorders(∂−1

1 c,∣∣ ) and

(∂−1

0 c,∣∣op )

.

Ker gker g // c adjunction c

coker f //

g

Coker f

zzttttttttttttt

(ker g)∣∣ f ⇔ d

f

OOddHHHHHHHHHHHH⇔ g f = 0⇔ b ⇔ (coker f)

∣∣ gSo: ker g = ker coker ker g and coker f = coker ker coker f

29

29. Factorization of morphisms, abelian categories

First Isom. Thm. for sets: Xf //

e !!DDDDDDDDD Y so f = m e ,

f(X)

m

OO

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),

have kernel Ker fker f−−→ A and cokernel B

coker f−−−−→ Coker f

(A3) Every monomorphism is a kernel; every epimorphism is a cokernel.

Image: [Im fim f−−→ B] := [Ker(coker f)

ker coker f−−−−−→ B] ,a monomorphism, smallest subobject of B that divides f .

Coimage: [Acoim f−−−→ Coim f ] := [A

coker ker f−−−−−→ Coker(ker f)] ,an epimorphism, smallest quotient of A that divides f .

Factorization [Af−→ B] = [A

q−→ Im fim 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.

Proof. Have Bcoker f−−−−→ 0 since f epi, and 1B = ker coker f .

Since f is mono, Af−→ 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: rf = 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: X X × Yoo // Y X //

f

--

X + Y[f

g

]

Yoo

gqqA

[f g]

OO

f

VV

g

HH

A

Exact: 0→ XιX−→ X + Y

[0

1

]−−→ Y → 0, 0→ X

[1 0]−−→ X × Y πY−→ Y → 0

Theorem: 0→ X + Y

[1 00 1

]−−−−→ X × Y → 0 exact,

so X+Y ∼= X×Y =: X⊕Y , biproduct.

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

[1

1

]−−→ X.

For f, g ∈ A(A,B), define Af+Lg //

∆

B

A⊕ A[f

g

];;wwwwwwwww

and Af+Rg //

[f g] ##GGGGGGGGG B

B ⊕BΣ

OO .

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).

Proof. Both sides are A∆−→ A⊕ A

[f h

g k

]−−−−→ B ⊕B Σ−→ 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.

Proof. For f : A→ B, have A⊕A

[1 f

0 1

]−−−−→ B⊕B monic and epic, so an

isomorphism with inverse B ⊕B

[1 g

0 1

]−−−−→ 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 = •yy

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 Cat(1,C(A,B)

).

2-cells: morphisms in the categories C(A,B).

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

Natural transformations τ : F → G as 2-cells AF %%

G99⇓ τ B

Horizontal 2-cell composition: Cat(A,B)×Cat(B,C)−→ Cat(A,C);

AF %%

G99⇓ τ B

F ′ %%

G′99⇓ τ ′ C 7→ A

F ′F ''

G′G77⇓ τ ′ τ C

with F ′ FaF ′τa

τ ′Fa //

(τ ′τ)a

&&MMMMMMMMMM G′ FaG′τa

F ′ Gaτ ′Ga

// G′ Ga

Identity: Cat(1,Cat(C,C)) 3 jC : •yy7→ C

1C %%

1C

99⇓ id C

Vertical 2-cell composition on Cat(A,B): Fa

σa !!DDDDDDDD(τ•σ)a // Ha

Ga

τa

<<zzzzzzzz

Entropic or interchange law: (τ ′ • σ′) (τ • σ) = (τ ′ τ) • (σ′ 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 groupswith trivial G0, and homomorphisms ρm,n : Gm ×Gn → Gm+n

satisfying Gl ×Gm ×Gn

ρl,m×1//

1×ρm,n

Gl+m ×Gn

ρl+m,n

Gl ×Gm+n ρl,m+n

// Gl+m+n

. Category G with G0 = 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,with + =

⋃(ρm,n : Gm×Gn → Gm+n

)on morphisms.

Symmetric groups Sn =〈τ1, . . . , τn−1 | τ 2

i = 1, τiτj = τjτi for |i− j| > 1, τiτi+1τi = τi+1τiτi+1〉.

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

Braid groupsBn = 〈σ1, . . . , σn−1 | σiσj = σjσi for |i− j| > 1, σiσi+1σi = σi+1σiσi+1〉

give the braid category B.

Braid relation:

i i+ 1 i+ 2t t tZZZZ

σi t t t

ZZZZ

σi+1 t t t

ZZZZ

σi t t t

i i+ 1 i+ 2

=

i i+ 1 i+ 2t t tZZZZ

σi+1t t t

ZZZZ

σit t t

ZZZZ

σi+1t t t

i i+ 1 i+ 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 givenby a structure map α : TX → X in X(TX,X) for some X ∈ X0.

Algebra (homo)morphism θ : (X,α)→ (Y, β)

is given by commuting diagram TX

α

Tθ // TY

β

Xθ// Y

in X.

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 givenby a structure map α : X → TX in X(X,TX) for some X ∈ X0.

Coalgebra (homo)morphism θ : (X,α)→ (Y, β)

is given by commuting diagram TXTθ // TY

X

α

OO

θ// Y

β

OO in X.

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

Endofunctors form a monoidal category(XX, , 1X

).

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

):

T 3 Tµ //

µT

T 2

µ

T 2

µ

>>>>>>>> TTηoo Here, µT is:

T 2µ// T T

ηT

OO

T XT // X

T 2%%

T99⇓ µ X

or

XT &&

T

88⇓ idT XT 2

%%

T99⇓ µ X = µ idT , whiskering.

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 • Fη and 1U = Uε • ηU .

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

Proposition: Unital law UFUFUεF

$$JJJJJJJJJ UFUFηoo

UF

ηUF

OO

UF

Proof. Triangular FUF

εF ""FFFFFFFFF FFηoo

F

and UFUUε

""FFFFFFFFF

U

ηU

OO

U

Proposition: µ : UFUF → UF associative.

Proof. Need UFUFUFUFUεF//

UεFUF

UFUF

UεF

UFUFUεF

// UF

or FUFUFUε //

εFU

FU

ε

FU ε

// 1A

.

Naturality: FUA

εA

FUFUAεFUA //

FUεA

FUA

εA

A FUA εA// A

. . . 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 〈x1 . . . xr〉 in the alphabet.

Coalgebra ηX : X → UFX; letter x 7→ one-letter “word” or list 〈x〉.Multiplication µX = UεFX : UFUFX → UFX;

list of words or lists 7→ concatenation of list:〈〈x11 . . . x1r1〉 . . . 〈xs1 . . . xsrs〉〉 7→ 〈x11 . . . x1r1 . . . xs1 . . . xsrs〉

— removes inner brackets.

35

35. Eilenberg-Moore algebras

Does a monad (T, µ, η) on X give adjunction A

U

66⊥ X

F

vv?

• Monoid (M,m : M2 →M, e : > →M) is a monoid in (Set,×,>).

M-sets X have action a : M ×X → X with

associativity: M2 ×X 1M×a //

m×1X

M ×Xa

M ×X a

// X

, unitality: Xe×1X //

HHHHHHHHHH

HHHHHHHHHH M ×XaX

morphisms: M ×X1

a1

1M×f // M ×X2

a2

X1f

// X2

, category SetM of M -sets.

Free FX = (M2 ×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.

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

Eilenberg-Moore algebra a : TX → X in X(TX,X) for X ∈ X0:

with associativity: T 2XTa //

µX

TX

a

TX a// X

and unitality: XηX //

DDDDDDDD

DDDDDDDD TX

aX

,

morphisms: TX1

a1

Tf // TX2

a2

X1f// X2

, cat. XT of Eilenberg-Moore algebras.

Forgetful UT : XT → X; (TXa−→ X) 7→ X.

Free F T : X→ XT ;X 7→ (T 2XµX−−→ TX). Note UTF TX = TX

Unit ηTX : XηX−→ TX, counit εTa : (T 2X

Ta−→ TX)a−→ (TX

a−→ X)

Eilenberg-Moore adjunction(F T , UT , ηT , εT

), yields monad (T, µ, η).

36

36. The Kleisli category of a monad

Alternative adjunction A

U

66⊥ X

F

vvfrom monad (T, µ, η) on X.

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

(XT )0 = X0, XT (X, Y ) = X(X,TY ), identities 1X = ηX : X → TX,

composition (Yg−→ TZ) (X

f−→ TY ) = (Xf−→ TY

Tg−→ T 2ZµZ−→ TZ) .

Adjunction XT (FTX, Y ) = XT (X, Y ) = X(X,TY ) = X(X,UTY )

with left adjoint FT (Xf−→ Y ) = (X

f−→ YηY−→ TY ),

right adjoint UT (Yg−→ TZ) = (TY

Tg−→ T 2ZµZ−→ TZ), unit X

ηX−→ TX,

counit εY = TY1TY−−→ TY , and monad (UTFT , UT εFT , η) = (T, µ, η).

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

Category Rel of relations XR−→ 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 (XR−→ Y

S−→ Z) := (x, z) | ∃ t ∈ Y . x R t S z.

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

Proof. XR−→ Y gives SetP-morphism X

R−→ PX;x 7→ y | x R y.Kleisli identity is ηX : X → PX;x 7→ x = x′ | x = x′,and Kleisli composition gives the relation product:

XR

// PYPS

// P2Z µZ// PZ

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(X

coevX⊗1X−−−−−−→ X ⊗X∗ ⊗X 1X⊗evX−−−−−→ X)

= 1X

and (X∗

1X∗⊗coevX−−−−−−−→ X∗ ⊗X ⊗X∗ evX⊗1X∗−−−−−→ 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 e1, . . . , en with ei(ej) = δij,then coev : K → X ⊗X∗; 1 7→

∑nj=1 ej ⊗ ej.

Yanking: ei 7→∑n

j=1 ej ⊗ ej ⊗ ei 7→∑n

j=1 ej ⊗ ej(ei) =∑n

j=1 ejδji = ei

and ei 7→ ei ⊗∑n

j=1 ej ⊗ ej 7→∑n

j=1 ei(ej)⊗ ej =∑n

j=1 δij ej = ei.

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

Example: Relation category Rel, biproduct is the disjoint union with

X 3 x&

ιX--x ∈ X + Y 3 y$

πX

kk

πY

33 y ∈ Y

ιYqq

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, (x′, x′, x)) | x, x′ ∈ X

with ((x′, x, x), x′) | x, x′ ∈ X is (x, x) | x ∈ X.

Second is similar.

38

38. Monoidal functors

Monoidal categories (C,⊗,1), (C′,⊗,1′).

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

F (X)⊗(F (Y )⊗ F (Z)

)1F (X)⊗µY,Z

α′F (X),F (Y ),F (Z)//

(F (X)⊗ F (Y )

)⊗ F (Z)

µX,Y ⊗1F (Z)

F (X)⊗ F (Y ⊗ Z)

µX,Y⊗Z

F (X ⊗ Y )⊗ F (Z)

µX⊗Y,Z

F(X ⊗ (Y ⊗ Z)

)F (αX,Y,Z)

// F((X ⊗ Y )⊗ Z

),

— associativity, and unitality:

F (X)⊗ 1′ρ′F (X) //

1F (X)⊗ε

F (X)

F (X)⊗ F (1) µX,1// F (X ⊗ 1)

FρX

OO, F (X) 1′ ⊗ F (X)

λ′F (X)oo

ε⊗1F (X)

F (1⊗X)

FλX

OO

F (1)⊗ F (X).µ1,Xoo

Example: Underlying set functor U : (L,⊗, K)→ (Set,×, 1).Here ε : 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.StrongStrict

monoidal functor: ε and the µX,Y are

isomorphisms.

identities.

Example: Free vector space functor F : (Set,×, 1)→ (L,⊗, K).

Here F (X × Y ) = F (X)⊗ F (Y ) and F1 = K, so strong, strict.

Braided monoidal functor: F (X)⊗ F (Y )

µX,Y

σ′F (X),F (Y )// F (Y )⊗ F (X)

µY,X

F (X ⊗ Y )

FσX,Y

// F (Y ⊗X),

Symmetric monoidal functor if (C,⊗,1), (C′,⊗,1′) 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 , π†X = ιX .

Example: Category FDHilb of finite-dimensional Hilbert spaces,with ∀ x ∈ X , ∀ y ∈ Y , 〈f(x) | y〉 = 〈x | f †(y)〉 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

1coevX−−−→ X⊗X∗ τ−→ X∗⊗X 1X∗⊗f−−−−→ X∗⊗X evX−−→ 1.

Example in FDHilb:∑j ej⊗ej 7→

∑j ej⊗ej 7→

∑j ej⊗f(ej) 7→

∑k

∑j ej⊗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 , 〈f(x) | x〉 ≥ 0.

Complete positivity of f : [X,X]→ [Y, Y ] or f : X∗⊗X → Y ∗⊗Y :

∀ Z ∈ C0 , ∀ positive g : 1→ Z∗ ⊗X∗ ⊗X ⊗ Z ,

1g // Z∗ ⊗X∗ ⊗X ⊗ Z

1Z∗⊗f⊗1Z// Z∗ ⊗ Y ∗ ⊗ Y ⊗ Z is positive.

40

40. Subobjects and subobject classifiers

For object X of category C, define Presub(X) as the full subcategoryof (C ↓ X) whose defining morphisms s : S → X are monomorphisms.

Lemma: Presub(X) is a preorder: X

S

j1**

j2

44

s

??T

t

__????????

⇒ j1 = 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

Sj //

1S

77

s

??S ′

s′

OO

j′ // S

s

__????????

⇒ j′ j = 1S, similarly j j′ = 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!]:

∀ X ∈ C0 , ∀ (Ss−→ X) ∈ SubC(X) ,∃! χ . S //

s

p-b

>true

X χ//___ Ω

Example: > true−−→ false, true in Set with S = χ−1true ⊆ 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 ∈ C0.

Proof. S ′ //

p-b

S //

p-b

>true

Yf// X χ

// Ω

Remark: S ′ 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, adinatural transformation δ : F ⇒ G is a “vector” (δx | x ∈ D0) ofcomponents δx : F (x, x)→ G(x, x) in C1 such that,

for all f : x→ y in D1, the hexagon of the dinaturality diagram

x x

f

F (x, x)δx // G(x, x)

G(1x,f)

%%KKKKKKKKK

F (y, x)

F (f,1x)99sssssssss

F (1y ,f) %%KKKKKKKKKG(x, y)

y

f

OO

y F (y, y)δy

// G(y, y)G(f,1y)

99sssssssss

in Dop in D in C

commutes in the category C.

Example: δnx : C(x, x)→ C(x, x); k 7→ kn — Church numeral n.

h0

11

--

h f 7→ (h f)n

f h 7→ (f h)n

%**

44

f (h f)n = (f h)n f

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

PX ×X 3X // Ω

>>>>>>>>

>>>>>>>>

PY ×X

Pf×1X77ppppppppppp

P1Y ×f ''NNNNNNNNNNN Ω

PY × Y 3Y// Ω

commute.

So dinatural 3 : P × 1C ⇒ ∆Ω for P : Cop → 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.

Examples: Presheaf categories D = SetDop

for small D.

Grothendieck topos: E with reflective full K : E → D, for some D,

where the left adjoint L : D → E preserves finite limits.

Example: Sheaves — the presheaves F ∈ E0 ⊆ D,where D = (O,⊆) for a topological space (X,O), satisfying:

For each open cover U =⋃i∈I Ui of each U ∈ O, require equalizer

FUiF (Ui∩Uj⊆Ui) // F (Ui ∩ Uj)

FUe //___∏

k∈I FUk

p //___

q//___

πi

OO

πj

∏k,l∈I F (Uk ∩ Ul) .

πi,j

OO

πi,j

FUj

F (Ui∩Uj⊆Uj)// F (Ui ∩ 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 .

(c) Monic > true−−→ Ω, and ∀ monic Ss−→ X , ∃! χ . S //

s

p-b

>true

X χ//___ Ω

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

∃ unique X

r

such that X × Yr×1Y

ρ // Ω

PY PY × Y 3Y// Ω

Lawvere: First-order theory of topoi as a foundation for mathematics.