An introduction to Yoneda structures - Irif

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Ideal completion. Every partial order A generates a free complete ∨-lattice P A. A −→ P A whose elements are the d
An introduction to Yoneda structures Paul-André Melliès CNRS, Université Paris Denis Diderot

Groupe de travail Catégories supérieures, polygraphes et homotopie Paris

21 May 2010

1

Bibliography

Ross Street and Bob Walters Yoneda structures on 2-categories Journal of Algebra 50:350-379, 1978

Mark Weber Yoneda structures from 2-toposes Applied Categorical Structures 15:259-323, 2007

2

Covariant and contravariant presheaves

A few opening words on Isbell conjugacy

3

Ideal completion Every partial order A generates a free complete A

−→

W

-lattice P A

PA

whose elements are the downward closed subsets of A, with ϕ

≤ PA

PA

ψ

⇐⇒

=

ϕ



ψ.

Aop ⇒ {0, 1}

4

Free colimit completions of categories Every small category A generates a free cocomplete category P A A

PA

−→

whose elements are the presheafs over A, with ϕ

−→P A

ψ

PA

⇐⇒

=

ϕ

natural

−→

ψ.

Aop ⇒ Set

5

Contravariant presheaves ϕ(Y ) y ϕ(X)

ϕ(f )

x

Y

C

f

X

Replaces downward closed sets 6

Filter completion Every partial order A generates a free complete A

−→

V

-lattice Q A

QA

whose elements are the upward closed subsets of A, with ϕ

≤ QA

QA

ψ

⇐⇒

=

ϕ



ψ.

( A ⇒ {0, 1} )op

7

Free limit completions of categories Every small category A generates a free complete category Q A A

−→

QA

whose elements are the covariant presheafs over A, with ϕ

−→Q A

QA

=

ψ

ϕ

⇐⇒

( P Aop )op

=

natural

←−

ψ.

( A ⇒ Set )op

8

Covariant presheaves ϕ(Y ) y ϕ(X)

ϕ(f )

x

Y

C

f

X

Replaces upward closed sets 9

Related to the Dedekind-MacNeille completion A Galois connection L

PA _





QA

L(ϕ)

=

{ y | ∀x ∈ ϕ, x ≤A y }

R(ψ)

=

{ x | ∀y ∈ ψ, x ≤A y }

R

ϕ ⊆ R(ψ)

⇐⇒

∀x ∈ ϕ, y ∈ ψ, x ≤A y

⇐⇒

L(ϕ) ⊇ ψ

The completion keeps the pairs (ϕ, ψ) such that ψ = L(ϕ) and ϕ = R(ψ)

10

The Isbell conjugacy An adjunction L

PA _





QA

L(ϕ) : Y 7→ P A(ϕ, Y) =

R

R(ψ) : X 7→ Q A(X, ψ) =

R

x∈A

ϕ(X) ⇒ hom(X, Y)

Y∈A

ψ(Y) ⇒ hom(X, Y)

R

P A(ϕ, R(ψ)) 

R X,Y∈A

ϕ(X) × ψ(Y) ⇒ hom(X, Y)  Q A(L(ϕ), ψ)

11

Yoneda structures

An axiomatic approach by Street and Walters

12

General idea Suppose a universe U1 inside a larger universe U2. –

Set is the category of sets in the universe U1,



CAT is the 2-category of categories in the universe U2.

In particular, the category Set is an object of CAT .

13

Admissible functors An object A of CAT is called admissible when its homsets A (A, A0) are all in the category Set. More generally, an arrow in CAT F

:

A −→ B

is called admissible when the homsets B (FA, B) are all in the category Set.

14

Yoneda structures A Yoneda structure in a 2-category K is defined as – a class of admissible arrows such that the composite arrow A

F /

G

B

/

C

is admissible whenever the arrow B

G /

C

is admissible.

15

Admissible objects An object A is called admissible when the identity arrow idA

:

A

−→

A

is admissible.

In the case of the 2-category CAT equipped with its Yoneda structure: admissible objects

=

locally small categories

16

Yoneda structures – every admissible object A induces an admissible arrow yA

:

A

−→

PA

17

Yoneda structures – every admissible arrow F

:

A

−→

B

from an admissible object A induces a diagram

P D A Y44

































y

B(F,1)

χ

A

44 44 44 44 44 44 .6 444 44 44 F 44 44 44 44 4 /

F

B

18

In the traditional case P D A Y44

































y

B(F,1)

χ

A

B(F, 1) χaF2

:

44 44 44 44 44 44 .6 444 44 44 F 44 44 44 44 4 /

B

F

:

b

λa1 . A (a1, a2)

7→

λa . B(Fa, b)

−→

λa1 . B (Fa1, Fa2) 19

In the traditional case P D A Y44

































y

B(F,1)

χ

A

B(F, 1) χ Fa1 a2

:

44 44 44 44 44 44 .6 444 44 44 F 44 44 44 44 4 /

B

F

:

b

7→

A (a1, a2)

λa . B(Fa, b) −→

B (Fa1, Fa2) 20

Axiom 1 For A and F accessible, the 2-cell

P D A Y44



































y

B(F,1)

χ

A

44 44 44 44 44 44 .6 444 44 44 F 44 44 44 44 4 /

F

B

exhibits the arrow B(F, 1) as a left extension of yA along the arrow F.

21

Axiom 1 In other words, every 2-cell

PD A

q

               

BJ

y

A

ϕ

σ /

B

F

factors uniquely as

PD A Z66

               

EM

66 66 66 66 /7 666 66 66 66 66 6

ϕ

θ

y

χF

A

q

F

/

B 22

In the traditional case A natural transformation

PD A

               

y

A

q BJ

ϕ

σ /

F

B

is the same thing as a family of functions σa1 a2

:

A (a1, a2)

−→

ϕ (Fa2) (a1)

natural in a1 contravariantly and in a2 covariantly.

23

In the traditional case A natural transformation

PD A

               

y

A

q BJ

ϕ

σ /

F

B

is the same thing as a family of functions σa1 a2

:

A (a1, a2)

−→

ϕ (Fa2) (a1)

natural in a1 contravariantly and in a2 covariantly.

24

In the traditional case A natural transformation θ in

PD A Z66

               

EM

66 66 66 66 /7 666 66 66 66 66 6

ϕ

θ

y

χF

A

q

/

F

B

is the same thing as a family of functions θa b

:

B (Fa, b)

−→

ϕ (b) (a)

natural in a contravariantly and in b covariantly.

25

In the traditional case This means that every natural transformation σa1 a2

:

A (a1, a2)

−→

ϕ (Fa2) (a1)

χFa1 a2

:

A (a1, a2)

−→

B (Fa1, Fa2)

θa1 Fa2

:

B (Fa1, Fa2)

factors as

followed by −→

ϕ (Fa2) (a1)

for a unique natural transformation θ which remains to be defined.

26

Existence Given the natural transformation σ, the natural transformation θa b

:

B (Fa, b)

−→

ϕ (Fa) (b)

Fa

−→

b

transports every morphism f

:

to the result of the action of f on the element id

σ a a ( a −→ a )



ϕ (Fa) (a)

27

Uniqueness ...

28

Axiom 2 For A and F accessible, the 2-cell

P D A Y44

































y

B(F,1)

χ

A

44 44 44 44 44 44 .6 444 44 44 F 44 44 44 44 4 /

F

B

exhibits the arrow F as an absolute left lifting of yA through B(F, 1).

29

Axiom 3(i) For A accessible, the identity 2-cell

P D A \88



y

A

88 88 88 88 88 /7 888 88 88 88 88 88 8 /

id

id y

PA

exhibits the identity arrow as a left extension of yA along yA.

30

Axiom 3(ii) For A, B, F, G accessible, the 2-cell

P D A



;C

y

A

F∗ o



























y

χB(1,F)

F

P D B Z44

44 44 44 44 44 /7 44 44 44 G 44 44 44 4

C(G,1)

/

χ

B

G

/

C

exhibits the arrow F∗ ◦ C(G, 1) as a left extension of yA along G ◦ F.

31

The inverse arrow Given an arrow between admissible objects A and B F

A

−→

B

PB

−→

PA

:

the arrow F∗

:

is defined as follows:

P D A \88

88 88 88 88 88 ∗ /7 888 88 88 B (1,F) 88 88 88 8 / /

F = P B ( B (1,F) , 1 )

y

χ

A

F

B

y

PB

where B (1, F) denotes the composite arrow y ◦ F. 32

The inverse arrow The 2-dimensional cell

PC A ]::

::  ::   ::   :: ∗   ::  7 / ::   ::   B(1,F) ::   ::   ::   / /

F = P B ( B (1,F) , 1 )

y

χ

A

F

B

y

PB

factors as a pair of Kan extensions:

PD O A

       .6    F      /

y

B(F,1)

PO A \99 B(F,1)

χ

A

F

B

B

99 99 99 99 9 ∗ .6 999 99 99 99 99 9

F

y

/

PB 33

Existential image Given an arrow between admissible objects A and B F

:

A

−→

B

PA

−→

PB

the arrow ∃F

:

is defined as follows: ∃F

P D A



y

A

/

P D B



























y

F

/

B

34

Universal image Given an arrow between admissible objects A and B F

:

A

−→

B

PA

−→

PB

the arrow ∀F

:

is defined as follows: ∀F

P D A Z55



PB

D

55

55

55 B(F,1)

55

55

/7 55

55

55

55

55

55



5

/

χ

y

A

/

KS

χF F

y

B

35

Monads with arity

An idea by Mark Weber

36

Category with arity A fully faithful and dense functor i0

:

Θ0

−→

A

where Θ0 is a small category.

37

Category with arity This induces a fully faithful functor A(i0, 1)

:

A

−→

P Θ0

which transports every object A of the category A into the presheaf

A(i0, A)

:

Θ0

−→

Set

p

7→

A(i0p, A)

38

Monads with arity A monad T on a category with arity (A, i0) such that: (1) the natural transformation exhibits the functor T AX22

E 0

T◦i

id

Θ0

i0

22 22 22 22 22 08 22 22 22 22 22 22 2

T

/

A

as a left kan extension of the functor T ◦ i0 along the functor i0,

39

Monads with arity (2) this left kan extension is preserved by the inclusion functor to P Θ0.

P O Θ0 A(i0 ,1)

AZ66

C         0         

T◦i

Θ0

66 66 66 66 66 .6 666 66 66 66 66 66 6 /

T

id

i0

A 40

Equivalently... The natural transformation

P O Θ0 A(i0 ,1)

AZ66

C         0         

T◦i

Θ0

66 66 66 66 66 .6 666 66 66 66 66 66 6 /

T

id

i0

A

exhibits A(i0, 1) ◦ T as a left kan extension of A(i0, 1) ◦ T ◦ i0 along i0. 41

Equivalently... For every object A, the canonical morphism

Z p∈Θ 0 A(i0n, Ti0p) × A(i0p, A)

−→

A(i0n, TA)

is an isomorphism.

42

Unique factorization up to zig-zag Every morphism i0 n

−→

TA

Ti0p

−→

in the category A decomposes as i0n

e

−→

Tf

TA

for a pair of morphisms e : i0n → Ti0p and f : i0p → A.

43

Unique factorization up to zig-zag The factorization should be unique up to zig-zag of

Ti0pD

{= {{ { 1{{{{{ {{ {{ { {{

DD DD DD DD 1 DD DD DD DD !

e

i0n C

Tf

CC CC CC CC CC 2 CCCC C!

e

Ti0 u



TA

= zz z zz zz z zz zz z 2 zz zz

Tf

Ti0q

44

An abstract Segal condition

A general theorem by Mark Weber axiomatizing a theorem by Clemens Berger on higher dimensional categories

45

Motivating example: the free category monad

∆O

∆0 /

/

Cat O

Graph

46

Morphism between categories with arity A morphism between categories with arity (F, `)

(A, i0)

:

−→

(B, i1)

is defined as a pair of functors (F, `) making the diagram ΘO 1

i1

`

/

Θ0

BO F

i0 /

A

commute.

47

Morphism between categories with arity This induces a commutative diagram i∗1

P Θ1 o `∗ 

em

P Θ0 o

PB F∗

id 

i∗0

PA

which is required to be an exact square in the sense of Guitart.

48

Morphism between categories with arity This means that the Beck-Chevalley condition holds, which states that the canonical natural transformation

P Θ1

∀i

1 /

PB F∗

`∗ 

P Θ0



∀i

/

0



PA

is reversible.

49

Proposition A. For every morphism (F, `) between categories with arity (F, `)

:

(A, i0)

−→

(B, i1)

the adjunction i∗1 a ∀i1 induces an adjunction between – –

the full subcategory of presheaves of B whose restriction along F is representable in A, the full subcategory of presheaves of Θ1 whose restriction along ` is representable along i0.

Moreover, this adjunction defines an equivalence when the functor F is essentially surjective.

50

Proposition B. Every monad T with arity i0 induces a commutative diagram ΘO T

iT /

AO T

`

Θ0

F i0 /

A

where the pair (F, `) defines a morphism (F, `)

:

(A, i0)

−→

(AT , iT )

of categories with arity.

51

Algebraic theories with arity

A 2-dimensional approach to Lawvere theories

52

Algebraic theory with arity An algebraic theory L with arity i0 : Θ0 −→ A is an identity-on-object functor :

L

Θ0

−→

ΘL

such that the endofunctor

P Θ0

∃L /

P ΘL

L∗ /

P Θ0

maps a presheaf representable along i0 to a presheaf representable along i0.

53

Algebraic theories with arity

Set

D Z44

44 44 44 44 44 /7 44 44 44 44 44 44 4



























Kan extension

A(i0 −,X)

op

Θ0

/

L

op

ΘL

54

Model of an algebraic theory A model A of a Lawvere theory L is a presheaf A

:

op

ΘL

Set

−→

such that the induced presheaf op

Θ0

i0 /

op

ΘL

A /

Set

is representable along i0.

55

Main theorem the category of algebraic theories with arity (A, i0) is equivalent to the category of monads with arity (A, i0)

56