Van Kampen diagrams are bicolimits Pawel Sobocinski, University of - - PowerPoint PPT Presentation

SLIDE 1

Van Kampen diagrams are bicolimits

Pawel Sobocinski, University of Southampton (joint work with Tobias Heindel) CT ’08, Calais, 23/06/08

SLIDE 2

• Internal cat in = monad in
• takes internal functors to monad

morphisms

and

Γ : C → Span(C)

C

f

− → D − → C ← C

f

− → D

Γ

C Span(C)

(Benabou)

in this talk category = category with pullbacks

C Span(C)

SLIDE 3

Plan of talk

• Extensive cats, adhesive cats, VK diags
• Bicolimits
• The theorem & rough proof
• Example and future work

SLIDE 4

Extensive Cats

• A category is extensive when it has

coproducts and, for any coproduct diag and comm diag below TFAE:

(Lawvere; Schanuel; Carboni, Lack & Walters)

i: top row is a coproduct diagram ii: the two squares are pullbacks “coproducts exist and are well-behaved”

X

• Z
• Y
• A

i1

A + B B

i2

SLIDE 5

Adhesive Cats

• A category is adhesive when, pushouts along

monos exist, and for any pushout along mono, and comm cube with rear faces pullbacks, TFAE:

(Lack & Sobocinski)

i: top row is a pushout diagram ii: the two squares are pullbacks

C′

m′

• f ′
• c
• A′

a

• g′
• B′

b

• n′
• D′

d

• C

m

• f
• A

g

• B

n

• D

“pushouts along monos exist and are well-behaved”

SLIDE 6

Van Kampen diagrams

a functor and cartesian nat. trans. γ :

→

F′

F

is a VK diagram when a cocone κ : F → C for all functors F′ : J → C, cocones κ′ : F′ → C′ st

F′

i κ′

i

• F′

u F′

j κ′

j

• C′

Fi

κi

• Fu

Fj

κj

• C

γi

• γj
• c
• TFAE
• i. is a colimit diagram
• ii. are all

pullback diagrams κ′

F′

iC′CFi

(Lack & Sobocinski, Cockett & Guo)

F : J → C

SLIDE 7

VK diagrams

ii ⇒ i: stability i ⇒ ii: “co-stability”

F′

i κ′

i

• F′

u F′

j κ′

j

• C′

Fi

κi

• Fu

Fj

κj

• C

γi

• γj
• c
• tfae
• i. is a colimit diagram
• ii. are all

pullback diagrams κ′

F′

iC′CFi

SLIDE 8

Simple observations

• VK diagrams are colimit diagrams
• sums in extensive cats are VK cospans
• pushouts along monos in adhesive cats are

VK squares

• strict initial objects are VK empty diagrams

SLIDE 9

Bicolimits

M: J → B Mi κu

• κi

Mu Mj κj

• bic M

κidi = 1κi κv◦u = (κv ◦ Mu) • κu (Kelly & Street) bic M ∈ B & pseudo-cocone κ: M → bic M

(pseudo) functor

SLIDE 10

Universal property

for all pseudo cocones

h: bic M → X

exists pseudo mediating morphism: & isos

∃ !

Mi

κu

• κi
• λi
• Mu Mj

κj

• bic M

h

• X

ϕi

Mi

λu

• λi
• Mu Mj

κj

• λj
• bic M

h

• X

ϕj

= ψ : ∆h ◦ κ → ∆h′ ◦ κ (∆ξ) ◦ κ

is for unique st

∃ !

& , any modification

h, h′ : bic M → X

given

λ: M → X ϕ : λi ⇒ (∆h) ◦ κ

ξ : h ⇒ h′

SLIDE 11

Univ prop restated

λ : M → D

for all pseudo cocones

{h : C → D, ϕ}

exists pseudo mediating morphism where is universal for

h κ

is a bicolimit diagram iff:

κ

all arrows h : C → D are universal for κ, D is universal for

h : C → D

when

κ, D h′ : C → D

for all exists ! ξ : h ⇒ h′ st ψ = (∆ξ) ◦ κ a pseudo cone

κ : M → C

i. ii.

, ψ : ∆h ◦ κ ⇒ ∆h′ ◦ κ

SLIDE 12

The theorem

is a bicolimit diagram in

Γκ Span(C)

is Van Kampen in C

κ : F → C

Suppose that has -colimits.

C J

Γ : C → Span(C)

SLIDE 13

Vague idea

• There is a 1-1 correspondence between:
• 2-dimensional “pseudo” diagrams in
• 3-dimensional diagrams (with some

pullbacks) in

Span(C) C

SLIDE 14

Rough proof

stability co-stability

λ : M → D

for all pseudo cocones

{h : C → D, ϕ}

exists pseudo mediating morphism where is universal for

h κ

all arrows h : C → D are universal for κ, D

⇔ ⇔

In the presence of -colimits:

J

SLIDE 15

VK diags are bicolimits

• Being a VK diagram in is a universal

property, in ! (a bigger universe)

• Corollaries:
• init. object in is strict iff preserves it
• is extensive iff it has coproducts and

preserves them

• ...

C Span(C) C Γ C Γ

SLIDE 16

Little example - symmetries

• Consider
• objects are natural

numbers

• arrows are

matrices

Mat(Ordf) k → l k × l

2

!

• !
• 1
• 1
• 1

2 ( 1

1)

• ( 1

1)

• 1

“ id id ”

• 1
• 1

1

• 1

is not VK in Ordf

2 ( 1

1)

• ( 1

1)

• 1

“ id id ”

• 2
• 1

2

• 1

2 ( 1

1)

• ( 1

1)

• 1

( tw

id ) 2

• 1

2

• 1

2 ( 1

1)

• ( 1

1)

• 1

( id

tw ) 2

• 1

2

• 1

2 ( 1

1)

• ( 1

1)

• 1

( tw

tw ) 2

• 1

2

• 1

But there are only 2 mediating morphisms!

Γ

− →

SLIDE 17

Future work

• Quasiadhesive categories don’t seem to be

common in the wild (Johnstone, Lack & S)

• It seems that categories in certain pushouts

are preserved by are more common ( is a smaller universe than )

Γ′ : C → Par(C) Γ′ Span(C) Par(C)