Bicategories of Fractions Revisited Dorette Pronk 1 with Laura Scull - - PowerPoint PPT Presentation
Bicategories of Fractions Revisited Dorette Pronk 1 with Laura Scull 2 1 Dalhousie University, Halifax, NS 2 Fort Lewis College, Durango, CO FMCS 2019 University of Calgary, Kananaskis Field Station May 2019 Outline Localization 1 Weaker
Bicategories of Fractions Revisited
Dorette Pronk1 with Laura Scull2
1Dalhousie University, Halifax, NS 2Fort Lewis College, Durango, CO
FMCS 2019 University of Calgary, Kananaskis Field Station May 2019
Outline
1
Localization
2
Weaker Bicalculus of Fractions Conditions
3
2-Cell Representatives
4
Application to Orbifolds
Localization
Localization of a Bicategory
Let W be a clas of arrows in a bicategory B. A localization of B with respect to W is given by a pseudofunctor JW : B ! B(W1) such that for each w 2 W,
JW(w) is an internal equivalence in B(W1); JW is universal: composition with JW gives an equivalence, PsW(B, C) ' Ps(B(W1), C).
Bicategories of Fractions Revisited May, 2019 3 / 39
Localization
(Bi)Categories of Fractions
For any class W, a representation of this bicategory can be obtained as follows:
1
Objects are those of B;
2
Arrows are given by finite zig-zags
w1
/
w2
wn
/
where all wi are in W;
3
2-Cells are equivalence classes of diagrams that are formal pastings of cells from B such that this would be a valid pasting diagram if all arrows from W were reversed. This bicategory is horribly complicated!
Bicategories of Fractions Revisited May, 2019 4 / 39
Localization
With a Calculus of Fractions
If W satisfies the conditions to admit a bicalculus of fractions, the bicategory B(W1) is given by:
1
Objects are those of B;
2
Arrows are spans
w
/
with w in W;
3
2-Cells are equivalence classes of diagrams
w
{
f
#
α u
O
u0
✏
β w0
c
f 0
;
where wu, w0u0 2 W.
Bicategories of Fractions Revisited May, 2019 5 / 39
Localization
Better, but not good enough
The hom-categories can still have proper classes as objects. The equivalence classes of 2-cell diagrams are a priori a bit mysterious and difficult to work with.
Bicategories of Fractions Revisited May, 2019 6 / 39
Localization
Solutions
To obtain small hom-categories require that W be locally small: for each object A in B, the collection of arrows in W with codomain A is small. In order to make W locally small, it is desirable to work with a weaker set of conditions to obtain a bicalculus of fractions. To obtain a better handle on the 2-cells, we will consider conditions under which there are canonical representatives.
Bicategories of Fractions Revisited May, 2019 7 / 39
wu
Localization
Solutions
To obtain small hom-categories require that W be locally small: for each object A in B, the collection of arrows in W with codomain A is small. In order to make W locally small, it is desirable to work with a weaker set of conditions to obtain a bicalculus of fractions. To obtain a better handle on the 2-cells, we will consider conditions under which there are canonical representatives.
Bicategories of Fractions Revisited May, 2019 7 / 39
we un
Localization
Solutions
To obtain small hom-categories require that W be locally small: for each object A in B, the collection of arrows in W with codomain A is small. In order to make W locally small, it is desirable to work with a weaker set of conditions to obtain a bicalculus of fractions. To obtain a better handle on the 2-cells, we will consider conditions under which there are canonical representatives.
Bicategories of Fractions Revisited May, 2019 7 / 39
uh
we
Weaker Bicalculus of Fractions Conditions
Conditions WBF1-2
The class W ✓ B1 admits a bicalculus of fractions if it satisfies: WBF1=BF1 All identities are in W. BF2 For B
v
/C
w
/D and v, w 2 W, wv 2 W. WBF2 For B
v
/C
w
/D and v, w 2 W, there is A
u
/B in B such that wvu 2 W.
Bicategories of Fractions Revisited May, 2019 8 / 39
Weaker Bicalculus of Fractions Conditions
Conditions WBF1-2
The class W ✓ B1 admits a bicalculus of fractions if it satisfies: WBF1=BF1 All identities are in W. BF2 For B
v
/C
w
/D and v, w 2 W, wv 2 W. WBF2 For B
v
/C
w
/D and v, w 2 W, there is A
u
/B in B such that wvu 2 W.
Bicategories of Fractions Revisited May, 2019 8 / 39
Weaker Bicalculus of Fractions Conditions
Condition WBF3
WBF3=BF3 For each A
w
✏
C
f
/ B
with w 2 W, there is an invertible 2-cell, D
h
/
v
✏
α
=)
A
w
✏
C
f
/ B
with v 2 W.
Bicategories of Fractions Revisited May, 2019 9 / 39
Weaker Bicalculus of Fractions Conditions
Condition WBF4
WBF4=BF4 For each B
w (
A
α+ f
6
g
(
B0 with w 2 W B
w
6
there exists a lifting A
f
(
A0
˜ α+ ˜ w
6
˜ w
(
B with ˜ w 2 W A
g
6
such that α ˜ v = w ˜ α. The collection of pairs ( ˜ w, ˜ α) needs to be suitably compatible.
Unit BF4 as BF4, but require ˜ w = id. Unit Co-BF4 If α: fu ) gu and u 2 W, then there is a 2-cell ˜ α: f ) g such that ˜ αu = α.
Bicategories of Fractions Revisited May, 2019 10 / 39
Weaker Bicalculus of Fractions Conditions
Condition WBF4
WBF4=BF4 For each B
w (
A
α+ f
6
g
(
B0 with w 2 W B
w
6
there exists a lifting A
f
(
A0
˜ α+ ˜ w
6
˜ w
(
B with ˜ w 2 W A
g
6
such that α ˜ v = w ˜ α. The collection of pairs ( ˜ w, ˜ α) needs to be suitably compatible.
Unit BF4 as BF4, but require ˜ w = id. Unit Co-BF4 If α: fu ) gu and u 2 W, then there is a 2-cell ˜ α: f ) g such that ˜ αu = α.
Bicategories of Fractions Revisited May, 2019 10 / 39
Weaker Bicalculus of Fractions Conditions
Condition WBF4
WBF4=BF4 For each B
w (
A
α+ f
6
g
(
B0 with w 2 W B
w
6
there exists a lifting A
f
(
A0
˜ α+ ˜ w
6
˜ w
(
B with ˜ w 2 W A
g
6
such that α ˜ v = w ˜ α. The collection of pairs ( ˜ w, ˜ α) needs to be suitably compatible.
Unit BF4 as BF4, but require ˜ w = id. Unit Co-BF4 If α: fu ) gu and u 2 W, then there is a 2-cell ˜ α: f ) g such that ˜ αu = α.
Bicategories of Fractions Revisited May, 2019 10 / 39
Weaker Bicalculus of Fractions Conditions
Condition WBF5
WBF5=BF5 When w 2 W and there is an invertible 2-cell α: v ) w, then v 2 W.
Bicategories of Fractions Revisited May, 2019 11 / 39
Weaker Bicalculus of Fractions Conditions
Adjusted Horizontal Composition
Instead of composing by
¯ w2
¯ f1
w1
f1
f2
¯ w2u
¯ f1u
w1
f1
f2
w2u 2 W (with u a chosen arrow as in BF2).
Bicategories of Fractions Revisited May, 2019 12 / 39
Weaker Bicalculus of Fractions Conditions
Key Technical Lemma
Given arrows
w2
/
w1
¯ w1
✏
¯ f
/
α w1
✏
˜ w1
✏
˜ f
/
β w1
✏
w2
/
w2
/
with w2 ¯ w1, w2 ˜ w1 2 W, there is a diagram,
¯ w1
w
¯ f
'
δ s
O
t
✏
ε ˜ w1
g
˜ f
7
with w2 ¯ w1s, w2 ˜ w1t 2 W, δ, ε invertible 2-cells and
¯ f
$
¯ w1
.
s
i
t
(
ε δ s
(
˜ f
/
˜ w1 ✏ β w2
✏
⌘
¯ f
/
¯ w1 ✏ α w1
✏
f
/
f
/
Bicategories of Fractions Revisited May, 2019 13 / 39
Weaker Bicalculus of Fractions Conditions
Remarks
This result enables us to define the associativity isomorphisms. It also enables us to use similar techniques to the one given for horizontal composition of arrows to define vertical composition of 2-cell diagrams and whiskering of 2-cells with arrows.
Bicategories of Fractions Revisited May, 2019 14 / 39
Weaker Bicalculus of Fractions Conditions
Associativity Isomorphism
Bicategories of Fractions Revisited May, 2019 15 / 39
a
02
associativityisomorphism
Weaker Bicalculus of Fractions Conditions
Vertical Composition
Vertical compositon of 2-cell diagrams
u1
w
f1
'
u2
w
f2
'
+α1 v1
O
v2
✏
+β1
and
+α2 v3
O
v4
✏
+β2 u2
g
f2
7
u3
g
f3
7
. is given by
u1
v
f1
(
v1
O
α1 β1 v2
v
v2
(
u2
˜ δ v0
3¯
˜ u2
O
¯ ˜ δ v0
2¯
˜ u2
✏
f2
/
v3
h
α2 v4
✏
β2 v3
6
u3
h
f3
6
Bicategories of Fractions Revisited May, 2019 16 / 39
Weaker Bicalculus of Fractions Conditions
Left Whiskering
To calculate
u1
u
f1
)
α s1
O
s2
✏
β v
/
u2
i
f2
5
we construct
¯ f1
%
¯ v1
)
s0
1˜
u1
O
δ1 ˜ v1
)
f1
)
γ1w1 t1
O
t2 ✏ δ3 s1
O
s2
✏
β v
/
˜ v2
5
δ2 s0
2˜
u2 ✏ f2
5
γ2w2 ¯ v2
5
¯ f2w2
:
and lift with respect to v.
Bicategories of Fractions Revisited May, 2019 17 / 39
Weaker Bicalculus of Fractions Conditions
Left Whiskering
This gives
¯ v1
t
¯ f1w1
*
u1
t
δ1
x
s0
1¯
˜ u1
O
α s1
f
s2
x
δ3˜ v t1˜ v
O
t2˜ v
✏
˜ β g
/
u2
j
δ2
f
s0
2¯
˜ u2
✏
¯ v2
j
¯ f2w2
4
Bicategories of Fractions Revisited May, 2019 18 / 39
Weaker Bicalculus of Fractions Conditions
Right Whiskering
v1
u
g1
)
u
/
α s1
O
s2
✏
β v2
i
g2
5
is constructed as
¯ v1
⌅
¯ f1
$
t1
O
r1
u
r1
)
s0
1
u
ε1 ¯ p O ρ1 ρ1 ¯ r1
)
¯ r1
u
f 0
1
)
ϕ1 g1
)
u
α p
O
q
✏
ρ3x ρ3x ˜ r2x
O
˜ r1x
✏
p
O
q
✏
τ s1
O
s2
✏
β s0
2
i
ε2 ¯ q
✏
ρ2 ρ2 ¯ r2
5
¯ r2
i
f 0
2
5
ϕ2 g2
5
r2
i
t2
✏
r2
5
¯ v2
[
¯ f2
:
Bicategories of Fractions Revisited May, 2019 19 / 39
Weaker Bicalculus of Fractions Conditions
Bicategories of Fractions
Write hWi for the class of arrows generated by W under composition. If W satisfies the conditions WBF1-5, then hWi satisfies BF1-BF5. The universal properties give us an equivalence of bicategories, B(W1) ' B(hWi1).
Bicategories of Fractions Revisited May, 2019 20 / 39
2-Cell Representatives
2-Cells in the Bicategory of Fractions
For a bicategory B with a class of arrows W satisfying conditions WBF1-WBF5, the bicategory of fractions B(W1) has as 2-cells equivalence classes of diagrams C1
u1
α+o
D
v1
O
v2
✏
β+
B C2
u2
^
f2
@
Bicategories of Fractions Revisited May, 2019 21 / 39
2-Cell Representatives
The Equivalence Relation
C1
u1
f1
C1
u1
f1
A
α+o
D
v1
O
v2 ✏ β+
B ⇠ A
α0+o
D0
v1
O
v2 ✏ β0+
B C2
u2
_
f2
?
C2
u2
_
f2
?
if there is a diagram with invertible 2-cells, C1
σ1
)
D
v1
7
v2
'
E
s
/
σ2
)
D0
v0
1
g
v0
2
w
C2 such that u1v1s 2 W and
Bicategories of Fractions Revisited May, 2019 22 / 39
d
E D
J
A
D
i
B
2-Cell Representatives
The Equivalence Relation
E
s
/
t ✏
σ2
(
D
v2 ✏ v1
/
α
(
C1
u1
✏
=
E
s
/
t ✏
σ2
(
D
v1
✏
D0
v0
2
/ C2
u2
/ A
D0
v0
1
/
v0
2 ✏ α0
(
C1
u1
✏
C2
u2
/ A
and E
s
/
t ✏
σ2
(
D
v2 ✏ v1
/
β
(
C1
f1
✏
=
E
s
/
t ✏
σ2
(
D
v1
✏
D0
v0
2
/ C2
f2
/ B
D0
v0
1
/
v0
2 ✏ β0
(
C1
f1
✏
C2
f2
/ B
Bicategories of Fractions Revisited May, 2019 23 / 39
2-Cell Representatives
Representatives
Given a 2-cell C1
u1
{
f1
"
A
α+o
D
v1
O
v2 ✏ β+
B C2
u2
c
f2
;
we may think of α as providing the frame for β to be defined. So if there is another 2-cell C1
u1
{
A
α0+o
D
v0
1
O
v0
2 ✏
C2
u2
c
with u1v0
1 2 W, we may ask whether there is an equivalent diagram with
left hand 2-cell α0.
Bicategories of Fractions Revisited May, 2019 24 / 39
2-Cell Representatives
Representatives
Using our technical lemma we find that There is an equivalent 2-cell with α0 as left hand side if W satisfies the unit co-BF4 condition. If the liftings for this condition are unique and W satisfies 3-for-2, the representative is unique.
Bicategories of Fractions Revisited May, 2019 25 / 39
2-Cell Representatives
Pseudo Pullback Representatives
If W is closed under pseudo pullbacks, representatives of the form C1
u1
f1
A
ρu1,u2+o Pu1,u2 p1
O
p2
✏
β+
B C2
u2
_
f2
_
are canonical and behave well with respect to whiskering (as we will see).
Bicategories of Fractions Revisited May, 2019 26 / 39
2-Cell Representatives
Whiskering with an Arrow on the Left
The whiskering B1
v1
~
g1
A X
u
/ B
ρv1,v2+o Pv1,v2 s1
O
s2
✏
β+
C B2
v2
_
g2
>
is constructed as follows
Bicategories of Fractions Revisited May, 2019 27 / 39
2-Cell Representatives
Whiskering Composite
The composition 2-cell diagram becomes
uv1w1
w
f 1w1
/
g1
%
ρuv1,uv2
Puv1w1,uv2w2
h
/
s0
1
O
s0
2
✏
Pv1,v2
s1
O
s2
✏
β uv2w2
g
f 2w2
/
g2
9
where h is the unique arrow induced by the diagram
v1w1
)
f 1w1
/
γv1,f w1 v1
)
Puv1,uv2
s0
1
4
s0
2
*
˜ ρuv1,uv2 f
/ B
v2w2
5
f 2w2
/
γv2,f w2 v2
5
Bicategories of Fractions Revisited May, 2019 28 / 39
2-Cell Representatives
Whiskering with an Arrow on the Right
The whiskering A1
u1
f1
A
ρu1,u2+o Pu1,u2 p1
O
p2
✏
β+
B Y
w
/ C
A2
u2
_
f2
_
is constructed as follows
Bicategories of Fractions Revisited May, 2019 29 / 39
2-Cell Representatives
The Construction of the 2-Cells
Let h: Pu1 ¯
w,u2 ¯ w ! Pu1,u2 be the unique arrow induced by the two
pseudo pullback squares. Let ˜ β be the lifting with respect to w of the following
w¯ f1
w
'
γw,f1 f1
#
Pu1 ¯
w,u2 ¯ w s1
O
s2
✏
h
/
= =
Pu1,u2
p1
O
p2
✏
β
B
f2
;
¯ w
7
γw,f2 w¯ f2
I
Bicategories of Fractions Revisited May, 2019 30 / 39
2-Cell Representatives
Whiskering Composite
The composition 2-cell diagram becomes
u1 ¯ w
x
g¯ f1
&
A
ρu1 ¯
w,u2 ¯ w+o Pu1 ¯
w,u2 ¯ w s1
O
s2
✏
g˜ β+
C
u2 ¯ w
f
g¯ f2
8
Bicategories of Fractions Revisited May, 2019 31 / 39
Application to Orbifolds
Orbifolds as Groupoids
Orbigroupoids are proper étale groupoids internal to the Spaces
To obtain the category structure, we take OrbiGpds = ProperEtaleGpds(E1) with respect to the class E of essential equivalences.
Bicategories of Fractions Revisited May, 2019 32 / 39
BE
Application to Orbifolds
Essential Equivalences
A morphism ϕ: G ! H is an essential equivalence when it is essentially surjective and fully faithful. It is essentially surjective when G0 ⇥H0 H1 ! H0 in G0 ⇥H0 H1
✏ / H1
s
✏
t
/ H0
G0
ϕ0
/ H0
is an open surjection.
H Gobj
Bicategories of Fractions Revisited May, 2019 33 / 39
Application to Orbifolds
Properties of Essential Equivalences
Essential equivalences satisfy conditions BF1-5. Essential equivalences satisfy the unit BF4 condition. Essential equivalences also satisfy the unit co-BF4 condition. Essential equivalences are closed under pseudo pullback. Essential equivalences are not locally small.
Bicategories of Fractions Revisited May, 2019 34 / 39
Application to Orbifolds
Canonical representatives for orbifold 2-cells
Each 2-cell (w, f) ) (w0, f 0) in the bicategory of orbigroupoids has a unique representative of the form
w
|
f
"
πw,w0Pw,w0 p
O
q
✏
β w0
b
f 0
<
where πw,w0 is the pseudo pullback square. Furthermore, horizontal whiskering can be calculated as described before.
Bicategories of Fractions Revisited May, 2019 35 / 39
Application to Orbifolds
Essential Covering Maps
To obtain a subclass of arrows that is locally small we introduce the following. Definition An essential covering map is a groupoid homomorphism of the form ϕU : G⇤(U) ! G, where
1
G⇤(U)0 = `
U2U U, where U is a non-repeating collection of open
subsets of G0 that meets every orbit of G;
2
(ϕU)0 is the inclusion embedding on each component U;
3
G⇤(U)1 is the pullback G⇤(U)1
(ϕU)1
/
(s,t) ✏
G1
(s,t)
✏
G⇤(U)0 ⇥ G⇤(U)0
(ϕU)0
/ G0 ⇥ G0
Bicategories of Fractions Revisited May, 2019 36 / 39
Application to Orbifolds
Properties of Essential Covering Maps
The class C of essential covering maps is locally small. For each essential equivalence ε: G0 ! G there is an arrow ψ: G⇤(U) ! G0 such that G⇤(U)
ψ
/
ϕU
"
G0
ε
G. (We say that the class C covers the class E.) We obtain equivalent bicategories of fractions: ProperEtaleGpds(C1) ' ProperEtaleGpds(hCi1) ' ProperEtaleGpds(E1).
Bicategories of Fractions Revisited May, 2019 37 / 39
Application to Orbifolds
Pseudo Pullback Representatives?
The bad news:
In G⇤(U1)
ϕU1
~
ψ1
! G
ρU1,U2+o PU1,U2 π1
O
π2
✏
β+
H G⇤(U2)
ϕU2
`
ψ2
> the composites ϕUiπi won’t necessarily be essential covering maps.
The good news:
Since essential equivalences satisfy 3-for-2, we may take the pseudo functor ProperEtaleGpds(C1) ! ProperEtaleGpds(E1) to preserve horizontal composition on the nose. So we can use the pseudo pullback representatives in ProperEtaleGpds(C1).
Bicategories of Fractions Revisited May, 2019 38 / 39
Application to Orbifolds
Pseudo Pullback Representatives?
The bad news:
In G⇤(U1)
ϕU1
~
ψ1
! G
ρU1,U2+o PU1,U2 π1
O
π2
✏
β+
H G⇤(U2)
ϕU2
`
ψ2
> the composites ϕUiπi won’t necessarily be essential covering maps.
The good news:
Since essential equivalences satisfy 3-for-2, we may take the pseudo functor ProperEtaleGpds(C1) ! ProperEtaleGpds(E1) to preserve horizontal composition on the nose. So we can use the pseudo pullback representatives in ProperEtaleGpds(C1).
Bicategories of Fractions Revisited May, 2019 38 / 39
Application to Orbifolds
Conclusion and Future Work
ProperEtaleGpds(C1) is a bicategory of orbifolds with small hom-categories and canonical representatives for its 2-cells with simplified whiskering. This has set us up for the construction of mapping orbifolds. To learn about the interaction of continuity with these constructions, attend the CMS-meeting in Regina.
Bicategories of Fractions Revisited May, 2019 39 / 39