[PPT] - A Double Approach to Variation and Enrichment for Bicategories PowerPoint Presentation

SLIDE 1

A Double Approach to Variation and Enrichment for Bicategories

Susan Niefield (with J.R.B. Cockett and R.J. Wood) June 2012

SLIDE 2

CT95 Moncat/V

ev

mod Mon V

Kelly: works for bicategories

(N)

1997 Generalize to bicategories?

(NW)

Relate to Cat/B ≃ Fun(B, Span) ≃ FunN(B, Prof)? 2005 (1) Fun(B, S) ≃ FunN(B, Mod S)

(CNW)

(2) LDF/B ≃ Fun(Bco, Span) ≃ FunN(Bco, Prof)

(1)

2011 ▲ax(❇, ❙) double category for nice ❇ and ❙

(Par´ e)

▲ax((❱B)op, ❙pan) ≃ ❈at/ /B Note: Fun(Bco, Span) ≃ H▲ax((❱B)op, ❙pan) Idea: (1) for double categories and vertical structure for (2)

SLIDE 3

Double Categories

Weak category objects ❉1 ×❉0 ❉1 ❉1

π2

❉1 ×❉0 ❉1

❉1

µ

❉1 ×❉0 ❉1

❉1

π1

❉1

❉0

d0 ❉0

❉1

∆

❉1

❉0

d1

in CAT

Objects: objects of ❉0 Horizontal morphisms: morphisms of ❉0, D

D′

Vertical morphism: objects of ❉1, D

¯

D Cells: morphisms of ❉1,

¯

D ¯ D′

D

¯ D

D

D′

¯ D′

Note: V❉ is a bicategory and H❉ is a 2-category

SLIDE 4

Examples

❙pan: sets, functions, spans, . . . ❈at: categories, functors, profunctors, . . . ❱B: vertically B, a bicategory (horizontally discrete) ▲ax(❇, ❙): lax functors, transformations, modules, modulations

(horizontal) (CKSW)

▼od ❉: monads in V❉, homomorphisms, modules, . . . ❉/ /B: (❉/ /B)0 = ❉0/B, (❉/ /B)1 = ❉1/id•

B

❙pan\ \1 = ❙pan∗, pointed sets

SLIDE 5

The Double Category ▲ax(❇, ❙) (Par´ e)

F : ❇

❙

β

¯ B ¯ B′

¯ f

B

¯ B

b

B B′

f

B′

¯ B′

b′

→
Fβ

F ¯ B F ¯ B′

F¯ f

FB

F ¯ B

Fb

FB FB′

Ff FB′

F ¯ B′

Fb′

lax functor

horiz functorial, vert lax, . . .

F ◦

B : id• FB

Fid•

B,

Fb,¯

b : F ¯

b · Fb

F(¯

b · b) t : F

F ′

B

¯ B

b

→

tb

F ¯ B F ′ ¯ B

t¯

B

FB

F ¯ B

Fb

FB F ′B

tB F ′B

F ′ ¯ B

F ′b

transformation

horiz natural, vert functorial, . . .

SLIDE 6

The Double Category ▲ax(❇, ❙), cont.

F

G

m

β

¯ B ¯ B′

¯ f

B

¯ B

b

B B′

f

B′

¯ B′

b′

→
mβ

G ¯ B G ¯ B′

G¯ f

FB

G ¯ B

mb

FB FB′

Ff FB′

G ¯ B′

mb′

FB

G ˜ B

G ¯

B G ˜ B

G¯ b

FB

G ¯ B

mb

FB

F ¯ B

Fb F ¯

B G ˜ B

m¯ b

module
µ

G G ′

u

F G

m

F F ′

t F ′

G ′

m′

B

¯ B

b

→

µb

G ¯ B G ′ ¯ B

u¯

B

FB

G ¯ B

mb

FB F ′B

tB F ′B

G ′ ¯ B

m′b

modulation

Define: ❋un(Bco, Span) = ▲ax((❱B)op, ❙pan)

SLIDE 7

The Equivalence ▲ax(❇, ❙) Mon ▲axN(❇, ▼od ❙)

Given F : ❇

❙, define Mon F : ❇ ▼od ❙ by

B → (FB

Fid•

B

FB,

Fid•

B,id• B, F ◦

B)

Ff homomorphism, Fb module, Fβ equivariant (since F is lax) Mon: transformations, modules, modulations → same (Mon)−1 is composition with U : ▼od ❙

❙

▲ax(❇, ❙pan) ≃ ▲axN(❇, ❈at), ❋un(B, S) ≃ ❋unN(B, Mod S) Note: loosely related to ev ⊣ mod

W

V

✶W

Mod V

in Moncat normal in Bicat

SLIDE 8

Variation for Bicategories

Given (❱B)op

F ❙pan, consider the projection BF P B, where

bjects of BF: (B, x ∈ FB)

morphisms of BF: (B, x)

(b,s)

(¯

B, ¯ x), with FB F ¯ B Fb FB

Fb

F ¯ B

1

FB

x

1

F ¯ B

¯ x

1

Fb

s

cells of BF: (B, x) (¯ B, ¯ x)

(b,s)

(B, x) (¯ B, ¯ x)

(b′,s′)

β
, with

1 Fb′

s′

1

Fb

s

Fb′

Fb

Fβ

Note: ❱BF is Par´

e’s “elements of F” ❊l F, for F : (❱B)op

❙pan

SLIDE 9

Local Discrete Fibrations

A lax functor P : X

B is a local discrete fibration (LDF) if

X(X, ¯ X)

B(PX, P ¯

X) is a discrete fibration, for all X, ¯ X Proposition: BF

P B is an LDF strict functor with small fibers

Proof: BF B

P

(B, x)

(¯ B, ¯ x)

(b,Fβs′)

(B, x)

(¯ B, ¯ x)

(b′,s′)

β
B

¯ B

b

B

¯ B

b′

β
Remark:

Bco Span

Bco

F

Bco

Bco

F

Span∗

Span

pb

in the category of bicats and lax functors

SLIDE 10

Local Discrete Fibrations, cont.

A transformation t : F

F ′ : (❱B)op ❙pan

B

¯ B

b

→

tb

F ¯ B F ′ ¯ B

t¯

B

FB

F ¯ B

Fb

FB

F ′B

tB F ′B

F ′ ¯ B

F ′b

induces an LDF functor Bt : BF

BF ′ over B defined by

(B, x) (¯ B, ¯ x)

(b,s)

(B, x) (¯ B, ¯ x)

(b′,s′)

β
→

(B, tBx) (¯ B, t¯

B¯

x)

(b,tbs)

(B, tBx) (¯ B, t¯

B¯

x)

(b′,tb′s′)

β
since the following diagram commutes when the triangle does

by horiz naturality of t

1 Fb′

s′

1

Fb

s

Fb′

Fb

Fβ

Fb

F ′b

tb

Fb′

Fb

Fb′

F ′b′

tb′ F ′b′

F ′b

F ′

β

SLIDE 11

Local Discrete Fibrations, cont.

A module m: F

G : (❱B)op

❙pan is given by a lax functor

M : (❱(B × ✷))op

❙pan s.t. M(−, 0) = F and M(−, 1) = G

Thus, m induces an LDF functor (B × ✷)M

B × ✷, together

with a diagram (B × ✷)M B × ✷

BF

(B × ✷)M

LDF

BF B

PF

B

B × ✷

(−,0)

BG

B

PG

(B × ✷)M

BG

LDopF

(B × ✷)M B × ✷

B × ✷

B

(−,1)

pb pb

SLIDE 12

Local Discrete Fibrations, cont.

A modulation

µ

G G ′

u

F G

m

F F ′

t F ′

G ′

m′

induces a lax functor

(B × ✷)M

(B × ✷)M′ over B × ✷, and a diagram

(B × ✷)M (B × ✷)M′

BF

(B × ✷)M

BF

BF ′

Bt

BF ′

(B × ✷)M′

BG

BG ′

Bu

(B × ✷)M

BG

(B × ✷)M

(B × ✷)M′

BG ′

pb

pb

SLIDE 13

The Double Category ▲DF/ /B

bjects: X

P B LDF functors with small fibers

morphisms: X B

P

X

X ′

H

X ′

B

P′

M

B × ✷

X

M

LDF

X B

P

B

B × ✷

(−,0)

Y

B

Q

M

Y

LDopF

M B × ✷

B × ✷

B

(−,1)

pb pb horizontal vertical cells: M M′

X

M

X

X ′

M′

Y

Y′

M

Y

M

M′

Y′

pb

pb

ver

B B × ✷

(−,0)

B × ✷

B

(−,1)

SLIDE 14

The Equivalence

Theorem: B− : ▲ax((❱B)op, ❙pan)

▲DF/

/B is an equivalence Proof (sketch): Given P : X

B, define F : (❱B)op ❙pan by

FB = {X | PX = B} and F(B

b ¯

B) = {X

x ¯

X | Px = b} with projections FB d0 Fb

d1 F ¯

B, and constraints FB F ◦ Fid•

B given

by X → id•

X, and F ¯

b ×F ¯

B Fb

F F(¯

bb) by X B

P

X

¯ X

x

X

˜ X

F(x,¯

x)

˜

X ¯ X

¯

x

B

¯ B

b

B

˜ B

¯ bb

˜

B ¯ B

¯

b

id
Horizontal and vertical morphisms of ▲DF/

/B give rise to transformations and modules, and cells induce modulations.

SLIDE 15

A Double Approach to Enrichment for Bicategories

2005 Showed LDF/B ≃ ˆ B-Cat, where ˆ B is the bicategory with

(CNW)

| ˆ B| = |B| and ˆ B(B, ¯ B) = SetsB(B,¯

B)op

For F : B(B, ¯ B)op

Sets and ¯

F : B(¯ B, ˜ B)op

Sets,

¯ F · F : B(B, ˜ B)op

Sets is given for c : B ˜

B by (¯ F · F)(c) = b ¯

b

Fb × ¯ F ¯ b × B(c, ¯ bb) and the identity on B is (−, idB): B(B, B)op

Sets

SLIDE 16

The Double Category ˆ B-❈at: Objects

ˆ B-categories X, i.e., a set |X| together with a function P : |X|

|B|, ˆ

B-morphisms X[X, ¯ X]: PX

P ¯

X, and cells X[ ¯ X, ˜ X] · X[X, ¯ X]

X[X, ˜

X], and idPX

X[X, X] s.t. . . .

Example: For P : X

B an LDF, define

X[X, ¯ X]: B(PX, P ¯ X)op

Sets

b → X[X, ¯ X]b = {X

x ¯

X | Px = b} β∗ : X[X, ¯ X]b′

X[X, ¯

X]b X

X, ¯

X

B
PX, P ¯

X

X

¯ X

β∗x′

X

¯ X

x′

PX

P ¯ X

b

PX

P ¯ X

b′

β

SLIDE 17

The Double Category ˆ B-❈at: Horizontal Morphisms

ˆ B-functors H : X

X ′

|X| |B|

P

|X|

|X ′|

H |X ′|

|B|

P′

PX

P ¯ X

X[X, ¯ X]

PX

P ¯ X

X ′[HX,H ¯ X]

s.t. . . .

Example: For X B

P

X

X ′

H

X ′

B

P′

in ▲DF/

/B, define Hb : X[X, ¯ X]b

X ′[HX, H ¯

X]b by X

x ¯

X → HX

Hx H ¯

X

SLIDE 18

The Double Category ˆ B-❈at: Vertical Morphisms

ˆ B-modules M : X

Y
PX

Q ¯ Y

M[X, ¯ Y]

QY

Q ¯ Y

Y[Y , ¯ Y]

PX

QY

M[X,Y ]

PX

P ¯ X

X[X, ¯ X] P ¯

X Q ¯ Y

M[ ¯ X, ¯ Y]

s.t. . . .

Example: M B × ✷

X

M

i

X B

P

B

B × ✷

Y

B

Q

M

Y

j

M B × ✷

R B × ✷

B

, define M[X, Y ]b = {iX

m jY | Rm = b}