Theory and Applications of Categories,
Vol. 31, No. 10, 2016, pp. 257329.
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS:
REPRESENTATIONS AND COMPOSITIONS OF
2-MORPHISMS
MATTEO TOMMASINI
In this paper we investigate the construction of bicategories of fractions
originally described by D. Pronk: given any bicategory C together with a suitable class of
morphisms W, one can construct a bicategory C [W−1 ], where all the morphisms of W
are turned into internal equivalences, and that is universal with respect to this property.
Most of the descriptions leading to this construction were long and heavily based on the
axiom of choice. In this paper we considerably simplify the description of the equivalence
relation on 2-morphisms and the constructions of associators, vertical and horizontal
compositions in C [W−1 ], thus proving that the axiom of choice is not needed under
certain conditions. The simplied description of associators and compositions will also
play a crucial role in two forthcoming papers about pseudofunctors and equivalences
between bicategories of fractions.
Abstract.
1. Introduction
In [P1996] Dorette Pronk introduced the notion of (right ) bicalculus of fractions, generalizing to the framework of bicategories the concept of (right) calculus of fractions,
originally described by Pierre Gabriel and Michel Zisman in [GZ1967] in the framework of
categories. To be more precise, given any bicategory
C
and any class
W
of
1-morphisms
in it, one considers the following set of axioms:
(BF1):
for every object
(BF2): W
A
of
C,
the
1-identity idA
belongs to
W;
is closed under compositions;
I would like to thank Dorette Pronk for several interesting discussions about her work on bicategories
of fractions, and for some useful suggestions about this paper. I am also very thankful to the anonymous
referee for carefully reading the rst version of this paper, and for lots of useful suggestions, in particular
for pointing out that Lemmas 3.1, 3.2 and 3.4 could be used as a foundational part of this work, instead
of being only described as a side result at the end of the paper (compare with the previous version of this
work at http://arxiv.org/abs/1410.3990v2). I am also grateful to Stefano Maggiolo for his program
Commu : it provides a simple interface for shortening the construction of diagrams with the package tikz.
If you are interested, you can download it for free from http://poisson.phc.unipi.it/~maggiolo/
Software/commu/ (only for Linux distributions and Mac). This research was mainly performed at the
Mathematics Research Unit of the University of Luxembourg, thanks to the grant 4773242 by Fonds
National de la Recherche Luxembourg.
Received by the editors 2015-03-02 and, in nal form, 2016-04-14.
Transmitted by Ieke Moerdijk. Published on 2016-04-19.
2010 Mathematics Subject Classication: 18A05, 18A30, 22A22.
Key words and phrases: bicategories of fractions, bicalculus of fractions, pseudofunctors.
c Matteo Tommasini, 2016. Permission to copy for private use granted.
257
MATTEO TOMMASINI
258
(BF3):
w : A → B in W and for every morphism f : C → B , there
0
0
are an object D , a morphism w : D → C in W, a morphism f : D → A and an
0
0
invertible 2-morphism α : f ◦ w ⇒ w ◦f ;
for every morphism
(BF4):
w : B → A in W, any pair of morphisms f 1 , f 2 : C → B
1
2
and any 2-morphism α : w ◦f ⇒ w ◦f , there are an object D , a morphism
v : D → C in W and a 2-morphism β : f 1 ◦ v ⇒ f 2 ◦ v, such that the following
(a) given any morphism
two compositions are equal:
B
f1
v
D
⇓ α
C
f2
(b) if
α
w
v
A,
in (a) is invertible, then so is
f1
⇓ β
D
v
w
B
C
w
B
A;
f2
C
β;
(D0 , v0 : D0 → C, β 0 : f 1 ◦ v0 ⇒ f 2 ◦ v0 ) is another triple with the same
properties as (D, v, β) in (a), then there are an object E , a pair of morphisms
u : E → D and u0 : E → D0 , and an invertible 2-morphism ζ : v ◦ u ⇒ v0 ◦ u0 ,
such that v ◦ u belongs to W and the following two compositions are equal:
(c) if
u
D
⇓ ζ
E
u0
C
v0
v0
(BF5):
if
w : A → B is a morphism
2-morphism v ⇒ w,
invertible
f1
⇓ β0
D0
in
C.
u
B,
D
v
⇓ ζ
E
u0
f2
C
D0
f1
⇓ β
B;
f2
C
v0
W, v : A → B is any morphism
v belongs to W.
and if there is an
then also
For simplicity of exposition, in axioms
of
C
v
v
(BF4a)
and
(BF4c)
we omitted the associators
Also in the rest of this paper we will omit all the associators of
C,
as well as the
right and left unitors (except for the few cases where we cannot ignore them discussing
coherence results); the interested reader can easily complete the proofs with the missing
associators and unitors (by coherence, any two ways of lling a diagram of
2-morphisms
C
with such
will give the same result). In particular, each statement in the rest of this
paper (except Corollaries 4.4 and 8.1) is given without mentioning the associators of
(as if
C
is a
A pair
2-category ),
(C , W)
but holds also when
C
is simply a bicategory.
is said to admit a (right ) bicalculus of fractions if all the axioms
are satised (actually in [P1996, Ÿ 2.1] condition
C,
(BF)
(BF1) is slightly more restrictive than the
version stated above, but it is not necessary for any of the constructions in that paper).
−1
Under these conditions, Pronk proved that there are a bicategory C [W ] (called (right )
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
bicategory of fractions ) and a pseudofunctor
UW : C → C [W−1 ],
259
satisfying a universal
property (see [P1996, Theorem 21]). In order to describe the main results of this paper,
−1
we need to recall briey the construction of C [W ] as described in [P1996, Ÿ 2].
−1
The objects of C [W ] are the same as those of C . A morphism from A to B in
−1
0
C [W ] is any triple (A , w, f ), where A0 is an object of C , w : A0 → A is an element of
W and f : A0 → B is a morphism of C . In order to compose such triples, we need to x
the following set of choices:
C(W):
C
for every set of data in
as follows
f
A0
with
W
v
in
and
W,
f 0,
B
v
B0
(1)
(BF3) we choose an object A00 , a pair
invertible 2-morphism ρ in C , as follows:
using axiom
and an
of morphisms
v0
in
A00
v0
A0
f
ρ
⇒
B
Then, given any pair of morphisms from
A
f0
v
to
B0.
(2)
B
and from
B
to
C
in
C [W−1 ]
as
follows
f := A
w
A
0
f
B
and
g := B
v
B
0
g
C
(3)
(with both
w
and
v
in
W),
in order to get data as in (2); after having done that, one sets
Since axiom
(BF3)
(f, v) in the set C(W),
g ◦ f := (A00 , w ◦ v0 , g ◦ f 0 ).
one has to use the choice for the pair
does not ensure uniqueness in general, the set of choices
C(W)
in
general is not unique; therefore dierent sets of choices give rise to dierent compositions
of morphisms, hence to dierent bicategories of fractions. Such dierent bicategories are
equivalent by [P1996, Theorem 21] (using the axiom of choice).
m
m
m
Given any pair of objects A, B and any pair of morphisms (A , w , f ) : A → B
−1
1
1
1
2
2
2
for m = 1, 2 in C [W ], a 2-morphism from (A , w , f ) to (A , w , f ) is an equivalence
MATTEO TOMMASINI
260
class of data
(A3 , v1 , v2 , α, β)
in
C
as follows
A1
w1
⇓ α
A
f1
v1
⇓ β
A3
v2
w2
f2
A2
such that
w1 ◦ v1
B,
(4)
belongs to W and such that α is invertible in C (in [P1996, Ÿ 2.3] it is
w2 ◦ v2 belongs to W, but this follows from (BF5)). Any other set of
also required that
data
A1
w1
⇓ α0
A
f1
v01
A03
⇓ β0
v02
w2
B
f2
A2
(5)
1
01
0
(such that w ◦ v
belongs to W and α is invertible) represents the same 2-morphism
−1
4
0
1
2
in C [W ] if and only if there is a set of data (A , z, z , σ , σ ) in C as in the following
diagram
A1
v01
A03
v02
z0
σ1
⇒
A4
σ2
⇐
2
A
such that:
• w1 ◦ v1 ◦ z
belongs to
• σ1
are both invertible;
and
σ2
W;
v1
z
A3 ,
v2
(6)
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
•
the compositions of the following two diagrams are equal:
z0
z0
v01
A03
z
A4
•
261
⇓ σ
v1
1
v01
w1
⇓ α
A3
⇓ σ
A1
A,
A4
z0
A1
w1
⇓ α0
A03
A;
2
v2
03
A
v02
A2
w2
v02
A2
w2
(7)
the compositions of the following two diagrams are equal:
z0
z
A4
z0
v01
A03
⇓ σ
v1
1
A03
f1
⇓ β
A3
⇓ σ
A1
v01
B,
A4
z0
A1
f1
⇓ β0
A03
B.
2
v2
v02
A2
f2
v02
A2
f2
(8)
w1 ◦ v01 ◦ z0 belongs to
w1 ◦ σ 1 : w1 ◦ v01 ◦ z0 ⇒
For symmetry reasons, in [P1996, Ÿ 2.3] it is also required that
W, but this follows from (BF5), using the invertible 2-morphism
w1 ◦ v1 ◦ z, so we will always omit this unnecessary technical condition.
The previous
relation is an equivalence relation. We denote by
h
i 3
1
2
1
1
1
2
2
2
A , v , v , α, β : A , w , f =⇒ A , w , f
the class of any data as in (4); these classes are the
2-morphisms
(9)
of the bicategory of
fractions. With an abuse of notation, we will say that diagram (4) is equivalent to (5)
3
1
2
03
01
02
0
0
meaning that the data (A , v , v , α, β) are equivalent to the data (A , v , v , α , β ).
3
1
2
Analogously, we will say that (4) represents (9) meaning that the data (A , v , v , α, β)
−1
represent (9). We denote the morphisms of C [W ] by f , g, · · · and the 2-morphisms
−1
by Γ, ∆, · · · . Even if C is a 2-category, in general C [W ] will only be a bicategory
(with trivial right and left unitors, but non-trivial associators if the set of choices
C(W)
is non-unique). From time to time we will have to take care explicitly of the associators
−1
C ,W
of C [W ]; whenever we will need to write them we will use the notation Θ•
.
Until now we have not described how associators and vertical/horizontal compositions
of
2-morphisms are constructed in C [W−1 ].
For such constructions, in [P1996, Ÿ 2.2, Ÿ 2.3
and Appendix] it is required to x an additional set of choices:
D(W):
w : B → A in W, any pair of morphisms f 1 , f 2 : C → B and
1
2
any 2-morphism α : w ◦f ⇒ w ◦f , using axiom (BF4a) we choose a morphism
v : D → C in W and a 2-morphism β : f 1 ◦ v ⇒ f 2 ◦ v, such that α ◦ v = w ◦β .
for any morphism
MATTEO TOMMASINI
262
Having xed also this additional set of choices, the descriptions of associators and
vertical/horizontal compositions in [P1996] are very long and they do not allow much
freedom on some additional choices that are done at each step of the construction (see
the explicit descriptions in the next pages). Dierent sets of choices
set of choices
C(W))
D(W)
(with a xed
might appear to lead to dierent bicategories of fractions, having
the same objects, morphisms,
2-morphisms
and unitors, but (a priori) dierent associa-
tors and vertical/horizontal compositions. The bicategories obtained with dierent sets
of choices
D(W)
(with a xed set of choices
C(W))
are obviously equivalent (using the
already mentioned [P1996, Theorem 21]), but one cannot get better results using only the
statements of [P1996].
In this paper we will prove that the choices
dierent sets of choices
D(W)
D(W)
are actually not necessary because
(with a xed set of choices
C(W))
give the same bicate-
gory of fractions instead of simply equivalent ones. In the process of proving this fact,
we will also considerably simplify the descriptions of associators and vertical/horizontal
−1
compositions in C [W ], thus providing a useful set of tools for explicit computations in
any bicategory of fractions.
In order to prove these results, we will rst need a simple way of comparing (representatives of ) associators and vertical/horizontal compositions induced by dierent sets
of choices
D(W).
More generally, since the current way of comparing representatives of
2-morphisms (based on the existence of a set of data (A4 , z, z0 , σ 1 , σ 2 )) is too long, we will
prove (and then use) the following comparison result for any pair of
−1
the same source and target in C [W ].
1.1. Proposition.
2-morphisms
with
(comparison of 2-morphisms in C [W−1 ]) Let us x any pair
(C , W) satisfying conditions (BF), any pair of objects A, B , any pair of morphisms f m :=
(Am , wm , f m ) : A → B for m = 1, 2 and any pair of 2-morphisms Γ, Γ0 : f 1 ⇒ f 2 in
C [W−1 ]. Then there are an object A3 , a morphism v1 : A3 → A1 in W, a morphism
v2 : A3 → A2 , an invertible 2-morphism α : w1 ◦ v1 ⇒ w2 ◦ v2 and a pair of 2-morphisms
γ, γ 0 : f 1 ◦ v1 ⇒ f 2 ◦ v2 , such that:
h
i
h
i
0
3
1
2
0
Γ = A3 , v1 , v2 , α, γ
and Γ = A , v , v , α, γ .
(10)
In other words, given any pair of 2-morphisms with the same source and target in
C [W−1 ], they admit representatives which only dier (possibly ) in the last variable.
1
2
0
Moreover, given any pair of 2-morphisms Γ, Γ : f ⇒ f with representatives satisfying (10), the following facts are equivalent:
(i )
Γ = Γ0 ;
(ii ) there are an object
and a morphism
A4 and a
γ ◦ z = γ 0 ◦ z.
(iii ) there are an object
and such that
A4
morphism
z : A4 → A3
in
z : A4 → A3 ,
W,
such that
such that
v1 ◦ z
γ ◦ z = γ 0 ◦ z;
belongs to
W
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
263
UW : C → C [W−1 ] mentioned
only 2-full and 2-faithful modulo
This result implies that the universal pseudofunctor
above in general is not
morphisms of
W
2-full,
neither
2-faithful,
but
(for more details, see Remark 3.5).
Using Proposition 1.1 we will show some simple procedures (still long, but shorter
than the original ones in [P1996]) in order to compute:
(i)
the associators of C [W−1 ]:
in Proposition 4.1 we show how to compute associa-
tors in a way that does not depend on the set of choices
(ii)
vertical composition of 2-morphisms:
in Proposition 5.1 we will prove that this
construction does not depend on the set of choices
D(W).
D(W);
C(W)
nor on the set of choices
In other words, vertical compositions are the same in any bicategory of
fractions for
(C , W)
(a priori we only knew that objects, morphisms,
and unitors are the same in any bicategory of fractions for
(iii)
2-morphisms
(C , W));
horizontal composition of 2-morphisms with 1-morphisms on the left (Proposition 6.1) and on the right (Proposition 7.1): in both cases we will prove that the
composition does not depend on the set of choices
D(W).
We omit here the precise statements of the four propositions mentioned above since
they are rather long and technical. Note that the explicit construction of representatives
for associators and vertical/horizontal compositions in a bicategory of fractions will still
depend:
•
on a choice of representatives for the
2-morphisms
that we want to compose (in
Propositions 5.1, 6.1 and 7.1);
•
on some additional choices of data of
C
(these are the choices called
(F1)
(F10)
in the next sections).
However, any two representatives (for an associator or a vertical/horizontal composition), even if they are constructed in dierent ways and hence appear dierent, are
actually representatives for the same
2-morphism.
If you need to use the constructions
described in Propositions 4.1, 5.1, 6.1 or 7.1 for associators and vertical/horizontal compositions but you don't remember the precise statements, the general philosophy behind
all such results is the following:
(a) rstly, you have to use the set of choices C(W) in order to construct the compositions
−1
on the level of 1-morphisms of C [W ] (see (3));
(b) now you need to construct a representative of a
2-morphism
between the morphisms
constructed in (a). If you need to compute associators, jump to point (c) below; if
you need to compute compositions of the form (ii) and (iii), then the diagram that
you want to construct can be partially lled with
chosen representatives for the
2-morphisms
2-morphisms of C , coming from any
that we are composing;
MATTEO TOMMASINI
264
(c) the missing holes in the diagram that you are constructing can be lled using axioms
(BF3)
and
(BF4)
nitely many times (an example of such an iterative procedure can
be found in the remarks following each proposition mentioned above);
(d) in general, axioms
(BF3)
and
(BF4)
do not ensure uniqueness:
in (c) you might
get dierent llings for the holes, and hence dierent diagrams.
The results in
Propositions 4.1, 5.1, 6.1 and 7.1 ensure that dierent choices in (b) and (c) induce
−1
diagrams representing the same 2-morphism in C [W ]. So you do not have to worry
about the choices made above: you simply take the equivalence class of the diagram
that you constructed, and you are done.
Each horizontal composition in any bicategory can be obtained as a suitable combination of compositions of the form (ii) and (iii). Thus Propositions 5.1, 6.1 and 7.1 prove
−1
immediately that horizontal compositions in C [W ] do not depend on the set of choices
D(W).
This, together with (i) and (ii), implies at once the main result of this paper:
1.2. Theorem.
tions
(BF).
(the structure of C [W−1 ]) Let us x any pair (C , W) satisfying condi-
Then the construction of
i.e., dierent sets of choices
C(W)
C [W−1 ]
depends only on the set of choices
(for any xed set of choices
C(W))
C(W),
give the same
bicategory of fractions, instead of only equivalent ones.
In particular, we get immediately:
1.3. Corollary. Let us suppose that for each pair
a unique choice of
(A00 , v0 , f 0 , ρ)
as in
C(W).
(f, v)
with
v
in
W
Then the construction of
as in (1) there is
C [W−1 ]
does not
depend on the axiom of choice. The same result holds if unique choice above is replaced
by canonical choice (i.e., when there is a canonical choice of pullback diagrams or iso
comma squares for axiom
(BF3)).
In Section 8 we will show some explicit applications of the results mentioned so far.
In particular, we will describe a simple procedure for checking the invertibility of a
2-
morphism in any bicategory of fractions:
1.4. Proposition.
(invertibility of 2-morphisms in C [W−1 ]) Let us x any pair
(C , W) satisfying conditions (BF), any pair of morphisms f m := (Am , wm , f m ) : A → B
1
2
−1
in C [W ] for m = 1, 2 and any 2-morphism Γ : f ⇒ f . Then the following facts are
equivalent:
C [W−1 ];
(i )
Γ
is invertible in
(ii )
Γ
has a representative as in (4), such that
(iii ) given any representative (4) for
3
A
in
W,
such that
β◦u
, such that
v ◦u
belongs to
is invertible in
there are an object
is invertible in
A4
C;
and a morphism
u : A4 →
C;
Γ, there are an object A4 and a morphism u : A4 →
W and such that β ◦ u is invertible in C .
(iv ) given any representative (4) for
3
1
A
Γ,
β
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
Note that in (4)
α
is always invertible by the denition of
265
2-morphism
in
As another application, we obtain a nice description of some associators in
terms of the associators in the original bicategory
C.
C [W−1 ].
C [W−1 ] in
For the precise statement we refer
directly to Corollary 8.1.
We are going to apply all the results mentioned so far in the next two papers [T2014(a)]
and [T2014(b)], where we will investigate the problem of constructing pseudofunctors and
equivalences between bicategories of fractions.
In the Appendix of this paper we will give an alternative description of
2-morphisms
in any bicategory of fraction, inspired by the results of Proposition 1.1. This alternative
description is not further exploited in this paper since it does not interact well with
vertical and horizontal compositions; it does seem interesting for other purposes as it
is considerably simpler than the description given above:
equivalence where each representative is a collection of
5
instead of having classes of
(A3 , v1 , v2 , α, β) as in (4),
data
we have classes of equivalence where each representative is a collection of
3
data of
C.
In
addition, the equivalence relation in this alternative description is much simpler than the
one recalled above.
2. Notations and basic lemmas
We mainly refer to [L] and [PW, Ÿ 1] for a general overview on bicategories, pseudofunctors (i.e., homomorphisms of bicategories), lax natural transformations and modications.
For simplicity of exposition, each composition of
−1
denoted by ◦, both in C and in C [W ].
1-morphisms
and
2-morphisms
In the rest of this paper we will often use the following four easy lemmas.
will be
Even if
each of them is not dicult for experts in this area, they may be harder for inexperienced
readers; so we give a detailed proof for each of them.
(C , W) satisfying conditions (BF). Let us x any morw : B → A in W, any pair of morphisms f 1 , f 2 : C → B and any pair of
2-morphisms γ, γ 0 : f 1 ⇒ f 2 , such that w ◦ γ = w ◦ γ 0 . Then there are an object E and a
0
morphism u : E → C in W, such that γ ◦ u = γ ◦ u.
2.1. Lemma. Let us x any pair
phism
Proof. We set:
α := w ◦ γ : w ◦f 1 =⇒ w ◦f 2 .
Then condition
(BF4a)
is obviously satised by the set of data:
D := C,
v := idC ,
β := γ : f 1 ◦ v =⇒ f 2 ◦ v .
MATTEO TOMMASINI
266
Since we have also
α = w ◦ γ0
D0 := C,
Then by
W)
(BF4c)
by hypothesis,
v0 := idC ,
(BF4a)
β 0 := γ 0 : f 1 ◦ v0 =⇒ f 2 ◦ v0 .
E , a pair of morphisms u, u0 : E → C
2-morphism ζ : u ⇒ u0 , such that:
γ 0 ◦ u0 ◦ f 1 ◦ ζ = f 2 ◦ ζ ◦ γ ◦ u .
there are an object
and an invertible
Using the coherence axioms on the bicategory
0
implies that γ ◦ u = γ ◦ u.
C
The next two lemmas prove that if conditions
(BF4a)
and
(BF4b)
is also satised by the data:
and the fact that
(BF)
ζ
w
morphism z
belongs to
z ◦ w belongs to W for some
(BF3), (BF4a) and (BF4b) when
in
we choose
W
in
hold, then conditions
(BF3),
w.
To be
it is sucient to impose that
(as a special case, one gets back again
as a
1-identity).
(BF). Let us choose any
0
quadruple of objects A, B, B , C and any triple of morphisms w : A → B , z : B → B
and f : C → B , such that both z and z ◦ w belong to W. Then there are an object D , a
0
0
morphism w in W, a morphism f and an invertible 2-morphism α as follows:
2.2. Lemma. Let us x any pair
(C , W)
z
W,
u
is invertible, this
hold under less restrictive conditions on the morphism
more precise, instead of imposing that
(with
satisfying conditions
0
D
⇑
w0
C
z◦w
f0
A
α
f
w
B
z
B0.
W, we can apply axiom (BF3) to the pair of morphisms
(z ◦f, z ◦ w), so we get an object E , a morphism t : E → C in W, a morphism g : E → A
and an invertible 2-morphism γ : z ◦f ◦ t ⇒ z ◦ w ◦ g . Since z belongs to W, we can
apply (BF4a) and (BF4b) to the invertible 2-morphism γ . So there are an object D , a
morphism r : D → E in W and an invertible 2-morphism α : f ◦ t ◦ r ⇒ w ◦ g ◦ r, such
0
0
that γ ◦ r = z ◦ α. Then we set f := g ◦ r : D → A and w := t ◦ r : D → C ; this last
morphism belongs to W by construction and (BF2). This suces to conclude.
Proof. Since
belongs to
(BF). Let us choose any
0
quadruple of objects A, A , B, C and any quadruple of morphisms w : B → A, z : A → A
1
2
and f , f : C → B , such that both z and z ◦ w belong to W. Moreover, let us x any
2-morphism α : w ◦f 1 ⇒ w ◦f 2 . Then there are an object D, a morphism v : D → C
1
2
in W and a 2-morphism β : f ◦ v ⇒ f ◦ v, such that α ◦ v = w ◦β . Moreover, if α is
invertible, so is β .
2.3. Lemma. Let us x any pair
0
(C , W)
satisfying conditions
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
f1
v
D
z◦w
C
v
w
⇓ α
C
f2
Proof. Since
B
z
A
A0 ,
w
f1
⇓ β
D
B
v
B
W,
we can apply
(BF4a)
w
A
z
A0 .
f2
C
belongs to
267
to the
2-morphism
z ◦ α : (z ◦ w) ◦ f 1 =⇒ (z ◦ w) ◦ f 2 .
So there are an object
f 2 ◦ t,
E,
a morphism
t:E→C
in
W
and a
2-morphism γ : f 1 ◦ t ⇒
such that:
z ◦ α ◦ t = z ◦ w ◦ γ.
Since
in
W,
z belongs to W, by Lemma 2.1 there are an object D and a morphism r : D → E
such that:
α ◦ t◦r = w◦γ ◦ r.
(11)
v := t ◦ r : D → C ; this morphism belongs to W by construction
1
2
and (BF2). Moreover, we set β := γ ◦ r : f ◦ v ⇒ f ◦ v. Then from (11) we get that
α ◦ v = w ◦ β . Moreover, if α is invertible, then by (BF4b) so is γ , hence so is β .
Then we dene
(C , W) satisfying conditions (BF), any triple of objects
A, B, C and any pair of morphisms w : C → B and z : B → A, such that both z and z ◦ w
belong to W. Then there are an object D and a morphism v as below, such that w ◦ v
belongs to W:
2.4. Lemma. Let us x any pair
D
v
C
w
z
B
A.
(BF3) on the pair (z, z ◦ w). Since z ◦ w belongs to
object R, a morphism p in W, a morphism q and an
Proof. First of all, we apply axiom
W
by hypothesis, there are an
invertible
2-morphism α
as below:
R
p
B
z
α
⇒
A
q
z◦w
C.
(BF4a) and (BF4b) to the invertible 2-morphism α : z ◦ p ⇒
z ◦(w ◦ q). Since z belongs to W by hypothesis, there are an object D, a morphism
r : D → R in W and an invertible 2-morphism β : p ◦ r ⇒ w ◦ q ◦ r, such that α ◦ r = z ◦β .
Now let us apply axioms
MATTEO TOMMASINI
268
p and r belong to W, hence also p ◦ r belongs to W by axiom (BF2).
Since β is invertible, then by (BF5) we conclude that also w ◦ q ◦ r belongs to W. Then
in order to conclude it suces to dene v := q ◦ r : D → C .
By construction both
3. Comparison of
2-morphisms
in a bicategory of fractions
In this section we are going to prove Proposition 1.1. First of all, we need the following
A = A0 and z = idA , but
lemma. In most of this paper this result will be applied with
we need to state it in this more general form since the presence of a non-trivial
z
will be
crucial in a couple of points in the paper.
(C , W) satisfying conditions (BF), and any set of data
z, z ◦ w1 , z ◦ w2 , z ◦ w1 ◦ p and z ◦ w1 ◦ r all belong to W, and
3.1. Lemma. Let us x any pair
as follows in
such that
ς
C , such that
η are both invertible:
and
A1
w1
A0
z
p
⇓ ς
A
A1
w1
z
A0
E
r
⇓ η F.
A
q
s
w2
w2
A2
A2
Then there are:
(1 ) a pair of objects
G
and
A3 ;
(2 ) a morphism
t1 : G → E ,
(3 ) a morphism
t2 : G → F ;
(4 ) a morphism
t3 : A3 → G
(5 ) a pair of invertible
such that
in
p ◦ t1
belongs to
W;
W;
2-morphisms ε
and
κ
in
C
as follows:
A3
G
t1
E
p
ε
⇒
A1
t2
r
t3
F,
G
s ◦ t2
κ
⇒
A2
t3
q ◦ t1
G,
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
such that
ς ◦ t1 ◦ t3
is equal to the composition of the following diagram:
t1
G
t3
E
⇓ ε
F
t2
⇓ κ
p
r
A1
⇓ η
A2
s
A3
269
t3
G
E
t1
q
w1
w2
A.
(12)
The choice of symbols in Lemma 3.1 may seem curious at rst. It makes sense once
we combine this lemma with Lemma 3.2 below.
0
First of all, let us prove the special case when A = A and z = idA .
1
2
1
1
In this case the hypotheses imply that w , w , w ◦ p and w ◦ r all belong to W. Since
1
1
0
both w and w ◦ p belong to W, by Lemma 2.4 there are an object E and a morphism
v : E 0 → E , such that p ◦ v belongs to W.
Proof.
Step 1.
Since both w
1
phism e
t :G→
1
w1 ◦ r belong to W, by Lemma 2.2 there are an object G, a morE 0 in W, a morphism t2 : G → F and an invertible 2-morphism ε as
and
follows:
G
1
e
t
E0
We set
(BF2)
the
v
E
p
F.
r
1
1
t1 := v ◦ et : G → E . By construction p ◦ v and et both belong to W;
1
1
morphism p ◦ t = p ◦ v ◦ e
t belongs to W, so condition (2) is veried.
By hypothesis and construction, both
belongs to
t2
ε
⇒
A1
W.
So using axiom
(BF5)
w1
and
p ◦ t1
belong to
on the invertible
so by
W, hence w1 ◦ p ◦ t1
also
2-morphism
ς −1 ◦ t1 : w2 ◦ q ◦ t1 =⇒ w1 ◦ p ◦ t1 ,
w2 belongs to W by hypothesis, using
1
Lemma 2.2 we conclude that there are an object H , a morphism z : H → G in W, a
2
morphism z : H → G and an invertible 2-morphism φ as follows:
we get that
w2 ◦ q ◦ t1
also belongs to
W.
Since
H
z1
G
s ◦ t2
φ
⇒
A2
z2
q ◦ t1
G.
MATTEO TOMMASINI
270
Now let us denote by
α : (w2 ◦ q ◦ t1 ) ◦ z2 ⇒ (w2 ◦ q ◦ t1 ) ◦ z1
z2
H
t1
G
⇓ φ−1
G
z1
q
E
A2
⇓ η −1
s
t2
F
⇓ ε−1
E
t1
the following composition:
w2
A1
⇓ ς
r
p
w1
A.
w2
2
A
q
(13)
Since all the 2-morphisms above are invertible, so is α; moreover we already said that
w ◦ q ◦ t1 belongs to W. Therefore using axioms (BF4a) and (BF4b) on α, we get an
3
3
3
2
3
object A , a morphism z : A → H in W and an invertible 2-morphism β : z ◦ z ⇒
z1 ◦ z3 , such that α◦z3 = (w2 ◦ q ◦ t1 )◦ β . The previous identity together with (13) implies
2
that the following composition
z1
A3
z3
E
⇓ ε
F
t2
H
⇓ β
H
z3
t1
G
z2
p
r
⇓ φ
G
z1
A1
⇓ η
s
E
t1
A2
q
w1
w2
A
(14)
is equal to the following one:
A3
Since both
(BF2).
z1
and
z3
z3
H
z1
belong to
G
W
t1
p
A1
⇓ ς
w1
q
2
w2
E
A.
A
t3 := z1 ◦ z3 : A3 → G by
the composition of φ and β
by construction, so does
Then in order to conclude it suces to dene
κ
as
as on the left hand side of (14). So we have proved the lemma in the special case when
A = A0 and z = idA .
Step 2.
Let us prove the general case: for that, we consider the diagrams:
z ◦ w1
A0
idA0
A0
A1
z ◦ w1
p
⇓ z◦ς E
A0
idA0
A0
A2
r
⇓ z◦η F.
q
z ◦ w2
A1
s
z ◦ w2
A2
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
271
Using Step 1, there are:
(1) a pair of objects
G
and
e3 ;
A
(2) a morphism
t1 : G → E ,
(3) a morphism
t2 : G → F ;
(4) a morphism
e3 → G
et 3 : A
such that
in
belongs to
W;
W;
2-morphisms ε
(5) a pair of invertible
p ◦ t1
and
κ
e
in
C
as follows:
G
t1
E
such that
ε
⇒
A1
p
3
z ◦ ς ◦ t1 ◦ et
e3
A
t2
r
F,
G
t1
3
t2
⇓ κ
e
E
⇓ ε
F
Since
So there
e
t
3
e
t
s ◦ t2
q ◦ t1
G,
p
r
s
e3
A
κ
e
⇒
A2
is equal to the composition of the following diagram:
G
e
t
e
t
3
G
3
t1
E
q
A1
⇓ η
A2
w1
w2
z
A
A0 .
z belongs to W by hypothesis, we can apply Lemma 2.1 to the previous identity.
3
3
e3 in W, such that ς ◦ t1 ◦ et 3 ◦ u
are an object A and a morphism u : A → A
coincides with the following composition:
t1
G
e
t
3
t2
⇓ κ
e
A3
u
E
⇓ ε
F
p
r
s
e3
A
3
e
t
Then it suces to set
G
3
t3 := et ◦ u
t1
and
E
κ := κ
e◦u
q
A1
⇓ η
A2
w1
w2
A.
in order to prove the general case.
MATTEO TOMMASINI
272
(C , W) satisfying conditions (BF), any pair of objects
A, B , any pair of morphisms f := (Am , wm , f m ) : A → B for m = 1, 2, and any pair of
2-morphisms Γ, Γ0 : f 1 ⇒ f 2 in C [W−1 ]. Let us x any representative (E, p, q, ς, ψ) for
Γ and (F, r, s, η, µ) for Γ0 as follows:
3.2. Lemma. Let us x any pair
m
A1
w1
p
⇓ ς
A
f1
⇓ ψ
E
A1
w1
B,
⇓ η
A
f1
r
⇓ µ
F
q
B.
s
w2
w2
f2
2
f2
2
A
A
Let us x any set of data (1 ) (5 ) as in Lemma 3.1. Then
(15)
Γ,
respectively
Γ0 ,
has a
representative as follows:
A1
w1
p ◦ t1 ◦ t3
⇓ ς ◦ t1 ◦ t3
A
A1
A3
w1
⇓ ψ ◦ t1 ◦ t3
q ◦ t1 ◦ t3
w2
f1
B,
A
⇓ ς ◦ t1 ◦ t3
⇓ ϕ
A3
q ◦ t1 ◦ t3
w2
f2
f1
p ◦ t1 ◦ t3
A2
B,
f2
A2
(16)
where
ϕ
is the following composition:
t1
G
E
⇓ ε
t2
t3
p
r
F
⇓ κ
s
A
3
Proof. Since
since
Γ
and
0
Γ
G
t3
f1
and
are
f2
E
t1
q
A1
⇓ µ
A2
f1
f2
B.
(17)
C [W−1 ], w1 and w2 belong to W. Moreover,
1
1
bicategory, w ◦ p and w ◦ r also belong to W.
are morphisms in
2-morphisms
in such a
So a set of data (1) (5) as in Lemma 3.1 exists. Now we claim that:
(a) the following two diagrams are equivalent, i.e., they represent the same
−1
of C [W ]:
2-morphism
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
A1
w1
A1
f1
p
⇓ ς
A
273
E ⇓ ψ
w1
B
A
⇓ ς ◦ t1 ◦ t3 A3 ⇓ ψ ◦ t1 ◦ t3
q
w2
f2
f1
p ◦ t1 ◦ t3
q ◦ t1 ◦ t3
w2
A2
B;
f2
A2
(b) the following two diagrams are equivalent:
A1
w1
A1
⇓ η F
A
f1
r
⇓ µ
w1
B
A
⇓ ς ◦ t1 ◦ t3 A3
f2
⇓ ϕ
q ◦ t1 ◦ t3
s
w2
f1
p ◦ t1 ◦ t3
w2
A2
B.
f2
A2
In order to prove such claims, let us consider the following set of data:
A1
p ◦ t1 ◦ t3
A3
idA3
q ◦ t1 ◦ t3
ip ◦ t1 ◦ t3
⇒
A3
iq ◦ t1 ◦ t3
⇐
A1
p ◦ t1 ◦ t3
p
t1 ◦ t3
E,
A3
q
idA3
q ◦ t1 ◦ t3
A2
ε ◦ t3
⇒
A3
κ
⇐
r
t2 ◦ t3
F.
s
A2
(18)
Then claim (a) follows at once using the set of data on the left hand side of (18) and
−1
from the description of 2-cells of C [W ]. Claim (b) follows easily using the set of data
on the right hand side of (18) together with the identities of (12) and of (17).
f m := (Am , wm , f m ) : A → B for
C [W−1 ]. Moreover, let us x any set of
3.3. Corollary. Let us x any pair of morphisms
m = 1, 2 and any 2-morphism Φ : f 1 ⇒ f 2
data in C as follows
in
MATTEO TOMMASINI
274
A1
w1
p
⇓ ς
A
E,
q
w2
A2
w1 ◦ p belongs to W and ς is invertible. Then there are an object A3 , a morphism
t : A → E such that p ◦ t belongs to W, and a 2-morphism ϕ : f 1 ◦ p ◦ t ⇒ f 2 ◦ q ◦ t in
C , such that Φ is represented by the following diagram:
such that
3
A1
w1
⇓ ς ◦ t A3
A
f1
p◦t
⇓ ϕ
B.
q◦t
w2
f2
A2
(19)
Proof. The proof follows the same ideas mentioned in the proof of Lemma 3.2 for
it suces to set
t := t1 ◦ t3 .
The fact that
p ◦ t belongs to W
follows from
Γ0 := Φ;
(BF2) together
with conditions (2) and (4) in Lemma 3.1.
(C , W) satisfying conditions (BF), any pair of objects
A, B , any pair of morphism f := (Am , wm , f m ) : A → B for m = 1, 2 and any pair of
2-morphisms Γ, Γ0 : f 1 ⇒ f 2 in C [W−1 ]. Let us suppose that there are an object A3 ,
1
2
1
1
a pair of morphisms v and v (such that w ◦ v belongs to W) and 2-morphisms α, γ
0
and γ in C (with α invertible ) as below, such that the following diagrams represent Γ,
0
respectively Γ :
3.4. Lemma. Let us x any pair
m
A1
w1
w2
A3
v2
A2
⇓ γ
A1
w1
v1
⇓ α
A
f1
B,
⇓ α
A
⇓ γ0
A3
v2
w2
f2
f1
v1
A2
B.
f2
Then the following facts are equivalent:
(i )
Γ = Γ0 ;
(ii ) there are an object
and a morphism
A4 and a
γ ◦ z = γ 0 ◦ z.
(iii ) there are an object
and such that
A4
morphism
z : A4 → A3
in
z : A4 → A3 ,
W,
such that
such that
v1 ◦ z
γ ◦ z = γ 0 ◦ z;
belongs to
W
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
275
2-cells
(E , r , r , σ, ρ) in
Proof. Let us suppose that (i) holds and let us prove (iii). Using the description of
of
C
C [W ]
−1
that we recalled in the Introduction, there is a set of data
1
1
2
as in the internal part of the following diagram
A1
v1
r2
A3
v1
σ
⇒
E1
ρ
⇐
v2
r1
A3 ,
v2
A2
such that:
• w 1 ◦ v 1 ◦ r1
• σ
and
• α ◦ r2
ρ
belongs to
W;
are both invertible in
is equal to the following composition:
A3
⇓ σ
r2
r1
E1
r2
• γ 0 ◦ r2
C;
v1
v1
A3
⇓ ρ
A3
A1
w1
⇓ α
A;
v2
A2
v2
w2
(20)
is equal to the following composition:
A3
⇓ σ
r2
r1
E1
r2
v1
v1
A3
⇓ ρ
A3
A1
f1
⇓ γ
B.
v2
v2
A2
f2
(21)
2-morphism in C [W−1 ], w1 and w1 ◦ v1 both belong to W; so
2
3
2
1
Lemma 2.3 applied to σ there are an object E , a morphism r : E → E in W and
invertible 2-morphism σ
e : r2 ◦ r3 ⇒ r1 ◦ r3 , such that v1 ◦ σ
e = σ ◦ r3 .
Since
Γ
is a
w1 ◦ v1 belongs to W, by (BF5) applied α−1 we get that also w2 ◦ v2
W; since also w2 belongs to W, by Lemma 2.3 applied to ρ ◦ r3 there are an
Since
by
an
belongs to
3
object E ,
MATTEO TOMMASINI
276
4
3
2
a morphism r : E → E in W and an invertible 2-morphism ρ
e : r1 ◦ r3 ◦ r4 ⇒ r2 ◦ r3 ◦ r4 ,
2
such that v ◦ ρ
e = ρ ◦ r3 ◦ r4 . Using (20), this implies that α ◦ r2 ◦ r3 ◦ r4 is equal to the
following composition:
r3
E2
r4
E
r3
E1
⇓ σ
e
A1
r1
E1
⇓ ρe
3
E
r4
2
r3
v1
r2
E
⇓ α
A3
r2
w1
A.
w2
v2
1
2
A
1
1
Since α is invertible by hypothesis, the previous identity implies that w ◦ v ◦ σ
e ◦ r4 =
1
−1
1
1
4
w ◦ v ◦ ρe . Since w ◦ v belongs to W, by Lemma 2.1 there are an object E and a
5
4
3
morphism r : E → E in W, such that
1
σ
e ◦ r4 ◦ r5 = ρe−1 ◦ r5 .
From (21) we get that
γ 0 ◦ r2 ◦ r3 ◦ r4 ◦ r5
r3
E2
r4
E
4
r5
E
r3
is equal to the following composition:
E1
⇓ σ
e
E1
⇓ ρe
3
r4
E
2
r3
(22)
A1
r1
E
f1
⇓ γ
A3
r2
1
v1
r2
B.
f2
v2
2
A
(23)
From (22) and (23) we conclude that
γ 0 ◦ r2 ◦ r3 ◦ r4 ◦ r5 = γ ◦ r2 ◦ r3 ◦ r4 ◦ r5 .
(24)
1
1
1
1
1
2
By construction we have that w ◦ v ◦ r belongs to W, so also w ◦ v ◦ r belongs to
W (using (BF5) on w1 ◦ σ ). By construction r3 , r4 and r5 also belong to W. Using (BF2),
1
1
2
3
4
5
we conclude that also the morphism w ◦ (v ◦ r ◦ r ◦ r ◦ r ) belongs to W.
1
4
By hypothesis w also belongs to W. So by Lemma 2.4 there are an object A and
6
4
4
1
2
3
4
5
6
a morphism r : A → E , such that the morphism v ◦ r ◦ r ◦ r ◦ r ◦ r also belongs to
W. We set z := r2 ◦ r3 ◦ r4 ◦ r5 ◦ r6 : A4 → A3 (so that v1 ◦ z belongs to W). Then (24)
0
implies that γ ◦ z = γ ◦ z, so (iii) holds.
1
Now let us assume (iii) and let us prove (ii). By hypothesis v ◦ z belongs to W; more1
1
−1
over w belongs to W because f is a morphism in C [W ]. Then by axiom (BF2) also
w1 ◦ v1 ◦ z belongs to W. Again by hypothesis w1 ◦ v1 belongs to W. So by Lemma 2.4
04
0
04
4
0
there are an object A and a morphism z : A → A , such that z ◦ z belongs to W. By
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
(iii) we have
γ ◦ z = γ 0 ◦ z,
hence
γ ◦ (z ◦ z0 ) = γ 0 ◦ (z ◦ z0 ),
so (ii) holds.
Lastly, if (ii) holds, then (i) is obviously satised using the denition of
−1
in C [W ] and axiom (BF2).
3.5. Remark. Let us x any pair of objects
277
2-morphism
A, B , any pair of morphisms f 1 , f 2 : A → B
2-morphism ρ : f 1 ⇒ f 2 in C . As explained in [P1996, Ÿ 2.4], the universal
−1
m
m
pseudofunctor UW : C → C [W ] is such that UW (f ) is the triple (A, idA , f ) : A → B
for each m = 1, 2, and UW (ρ) is the 2-morphism represented by the following diagram:
and any
A
idA
A
f1
idA
⇓ iidA
⇓ ρ
A
idA
idA
B.
f2
A
(25)
2-morphism Φ : UW (f 1 ) ⇒ UW (f 2 ). Using Corollary 3.3 with
(A , A , E, w , w , p, q, ς) := (A, A, A, idA , idA , idA , idA , iidA ), there are an object A0 , a
0
1
2
morphism t : A → A in W and a 2-morphism ϕ : f ◦ t ⇒ f ◦ t, such that Φ is
Now let us x any
1
2
1
2
represented by the following diagram:
A
idA
⇓ it
A
f1
t
⇓ ϕ
A0
B.
t
idA
f2
A
(26)
Therefore the following facts are equivalent:
(a)
Φ
is in the image of
(b) there is a
Since the
UW ;
2-morphism ρ : f 1 ⇒ f 2 in C , such that (26) is equivalent to (25).
h
i
0
class of (25) is equal to A , t, t, it , ρ ◦ t , by Lemma 3.4 we get that
(b) is
equivalent to:
2-morphism ρ : f 1 ⇒ f 2
W, such that ϕ ◦ z = ρ ◦ t ◦ z.
(c) there are a
in
in
C,
an object
A00
UW
and a morphism
z : A00 → A0
2-full, but only 2-full
modulo morphisms of W. It is 2-full if and only if the pair (C , W) satises the following
condition (in addition to the usual set of axioms (BF1) (BF5)):
Therefore in general the universal pseudofunctor
is not
MATTEO TOMMASINI
278
f 1 , f 2 : A → B , for every morphism t : A0 → A in W
1
2
00
and for every 2-morphism ϕ : f ◦ t ⇒ f ◦ t, there are an object A , a morphism
00
0
1
2
z : A → A in W and a 2-morphism ρ : f ⇒ f , such that ϕ ◦ z = ρ ◦ t ◦ z.
(BF6):
for every pair of morphisms
Now let us x another
2-morphism ρ 0 : f 1 ⇒ f 2 .
Using (25) and Lemma 3.4, the
following facts are equivalent:
• UW (ρ) = UW (ρ 0 );
•
e and
A
e→A
u:A
in
W,
This shows that in general the universal pseudofunctor
UW
there are an object
a morphism
2-faithful modulo morphisms of
such that
is not
ρ ◦ u = ρ 0 ◦ u.
2-faithful,
but only
W.
Now we are able to give the following proof:
Proof of Proposition 1.1. Let us x any representatives (15) for
Γ
and
Γ0
as in
Lemma 3.2, any set of data (1) (5) as in Lemma 3.1. Using Lemma 3.2, there are
0
representatives for Γ and Γ as in the rst part of Proposition 1.1: it suces to set
1
3
1
α := ς ◦ t ◦ t , γ := ψ ◦ t ◦ t3 , γ 0 := ϕ, v1 := p ◦ t1 ◦ t3 and v2 := q ◦ t1 ◦ t3 in diagram
(16).
1
1
Using the data (2) and (4), we get that v belongs to W. Moreover, w belongs to W
1
−1
1
1
because f is a morphism of C [W ]. Hence w ◦ v also belongs to W, so we can apply
Lemma 3.4 in order to get the second part of Proposition 1.1.
3.6. Remark. By induction and using the same ideas mentioned in this section, one
2-morphisms Γ1 , · · · , Γn in C [W−1 ], all dened
3
1
2
1
n
1
between the same pair of morphisms, there are data A , v , v , α, γ , · · · , γ , such that v
m
3
1
2
m
belongs to W, α is invertible and Γ = [A , v , v , α, γ ] for each m = 1, · · · , n. In other
words, given nitely many 2-morphisms with the same source and target, each of them
can also prove that given nitely many
admits a representative that diers from the other ones only in the last variable at most.
3
1
2
However, in general the common data (A , v , v , α) depend on the 2-morphisms chosen:
n+1
if we add another 2-morphism Γ
to the collection above, then the new common data
1
n+1
4
1
2
4
for Γ , · · · , Γ
in general is of the form (A , v ◦ z, v ◦ z, α ◦ z) for some object A and
4
3
some morphism z : A → A in W.
4. The associators of a bicategory of fractions
As we mentioned in the Introduction, in order to prove Theorem 1.2 we need to show
that the constructions of associators and vertical/horizontal compositions do not depend
on the set of xed choices
D(W), but only on the choices C(W) (at most).
the description of associators in any bicategory of fractions:
We start with
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
(associators of C [W−1 ]) Let us x any triple of 1-morphisms in
4.1. Proposition.
C [W−1 ]
279
as follows:
f := A
g := B
h := C
u
A
f
0
v
B0
w
C0
g
h
B
C
D
,
,
.
(27)
Let us suppose that the set of xed choices
C(W)
gives data as in the upper part of
1
2
1
3
the following diagrams (starting from the ones on the left ), with u , u , v and u in W
and
δ, σ, ξ
and
η
invertible:
A1
u1
A0
B0
δ
⇒
B
f
v1
g
f1
B0,
v
B2
ξ
⇒
C
u2
A1
g◦f 1
g1
w
u3
C 0,
A0
f
A2
σ
⇒
C
A3
η
⇒
B
l
C 0,
w
f2
v ◦ v1
B2,
(28)
so that by [P1996, Ÿ 2.2] one has
h◦ g◦f = A
u ◦ u1 ◦ u2
A
2
h◦l
D ,
(29)
u ◦ u3
h◦g ◦f = A
A3
h◦g 1 ◦f 2
D
.
(30)
Then let us x any set of data in
(F1):
(F2):
as follows:
u4 : A4 → A2 such that u ◦ u1 ◦ u2 ◦ u4 belongs to W (for
4
5
4
3
example, using (BF2) this is the case if u belongs to W), a morphism u : A → A
1
2
4
3
5
and an invertible 2-morphism γ : u ◦ u ◦ u ⇒ u ◦ u ;
an object
A4 ,
C
an invertible
a morphism
2-morphism ω : f 1 ◦ u2 ◦ u4 ⇒ v1 ◦f 2 ◦ u5 ,
such that
v◦ω
is equal to
MATTEO TOMMASINI
280
the following composition:
u2
f1
A1
A2
u4
(F3):
an invertible
⇓ η
3
A
f
A0
u3
u5
v
⇓ δ −1
u1
⇓ γ
A4
B0
B
f2
B;
v
B0
2
v1
(31)
2-morphism ρ : l ◦ u4 ⇒ g 1 ◦ f 2 ◦ u5 ,
such that
w◦ρ
is equal to the
following composition:
l
A2
u4
⇓ ω
A4
u5
v1
B2
f2
Then the associator
f1
A3
,W
ΘCh,g,f
w
⇓ σ −1
B0
⇓ ξ
A1
u2
C0
g
C.
w
C0
g1
(32)
from (29) to (30) is represented by the following diagram:
A2
u ◦ u1 ◦ u2
u4
⇓ u◦γ
A
A4
u5
u ◦ u3
A3
(F1)
Given any other data as in
(F3),
h◦l
⇓ h◦ρ
D.
h◦g 1 ◦f 2
(33)
the diagram (33) induced by the new data is
equivalent to (33).
In particular:
•
by denition of composition of
struction of
h ◦ (g ◦ f )
and of
1-morphisms in a bicategory of fractions, the con(h ◦ g) ◦ f depends on the four choices in the set
C(W) giving the four triangles in (28), hence inducing the compositions in (29) and
(30) (dierent sets of choices C(W) give dierent, but equivalent, bicategories of
fractions);
•
apart from the four choices in
h ◦ (g ◦ f )
to
(h ◦ g) ◦ f
C(W)
does not depend on any additional choice
nor on any choice of type
D(W);
C [W−1 ] from
of type C(W),
needed above, the associator in
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
•
281
hence the construction of associators in any bicategory of fractions does not depend
on the set of choices
D(W).
As we will mention below, each set of choices
induces a specic set of data
(F1)
(F3),
D(W) gives a diagram
2-morphism in C [W−1 ].
other set of choices
the same
D(W)
hence a specic diagram (33), but any
that is equivalent to (33), i.e., it induces
4.2. Remark. Before proving Proposition 4.1, we remark that even if such a statement is
long, the construction described there is simpler than the original construction in [P1996].
(F1)
As a consequence of the axioms, a set of data as in
(F3)
is easy to construct, as
shown in Remark 4.3 below.
Proof.
Step 1.
Following [P1996, Appendix A.2], one gets immediately a set of data
(F1) (F3) and inducing the desired associator as in (33). In [P1996]
D(W), so they satisfy also some
4
properties that we don't need to recall here in full details (for example, u
W; using (BF2) this condition is slightly stronger than condition (F1)). The
satisfying conditions
these data are induced by the additional xed choices
additional
belongs to
aim of this proof is to show that all these additional properties are actually not necessary,
because for any data satisfying
(F1)
(F3)
(even if not induced by the choices
C ,W
the induced diagram (33) is a representative for the associator Θh,g,f .
D(W)),
(F1) (F3), in
2 dierent sets of data as in (F1)
Since in [P1996] the associator is induced by a particular set of data
order to prove the claim it is sucient to show that any
(F3)
induce diagrams of the form (33) that are equivalent. So let us x any other set
of data satisfying
(F1)0 :
an object
morphism
(F2)0 :
(F1)
(F3),
as follows:
e4 , a morphism e
e4 → A2 such that u ◦ u1 ◦ u2 ◦ e
A
u4 : A
u4 belongs to W,
e4 → A3 and an invertible 2-morphism γ
e
u5 : A
e : u1 ◦ u2 ◦ e
u4 ⇒ u3 ◦ e
u5 ;
an invertible
2-morphism ω
e : f 1 ◦ u2 ◦ e
u4 ⇒ v1 ◦f 2 ◦ e
u5 ,
such that
the following composition:
u2
2
A
e4
u
e5
u
an invertible
A
3
f2
v
f
A0
u3
⇓ η
B
is equal to
B0
⇓ δ −1
u1
⇓ γ
e
e4
A
(F3)0 :
A
f1
1
v◦ω
e
a
2
v1
2-morphism ρe : l ◦ e
u4 ⇒ g 1 ◦ f 2 ◦ e
u5 ,
B;
v
B
0
(34)
such that
w ◦ ρe is
equal to the
MATTEO TOMMASINI
282
following composition:
u
e
l
A2
4
⇓ ω
e
u
e
A
f2
B
w
g
B0
f1
v1
3
5
⇓ σ −1
A1
u2
e4
A
C0
C.
⇓ ξ
w
C0
2
g1
(35)
Then proving the claim is equivalent to proving that (33) is equivalent to:
u ◦ u1
A2
◦ u2
⇓ u◦γ
e
A
⇓ h ◦ ρe
e4
A
u
e
u ◦ u3
A
Step 2.
h◦l
u4
e
D.
5
h◦g 1 ◦f 2
3
(36)
We want to construct a pair of diagrams that are equivalent to (33), respec-
tively to (36), and that share a common left hand side. For this, we follow the procedure
1
explained in Lemmas 3.1 and 3.2. Using Lemma 3.1, there are a pair of objects F and
2
1
2
3
4
1
3
F , a triple of morphisms t , t and t , such that both u ◦ t and t belong to W, and a
pair of invertible
2-morphisms ε
and
κ
in
C
as follows
F1
t1
A4
such that
ε
⇒
A2
u4
u ◦ γ ◦ t1 ◦ t3
F2
t2
u4
e
t3
e4 ,
A
F1
F1
t2
t3
A4
⇓ ε
e4
A
u4
u2
t3
F
1
e4
u
u
e
4
t1
A
A1
A2
⇓ κ
F
t3
u5 ◦ t1
F 1,
is equal to the following composition:
t1
2
u5 ◦ t2
e
κ
⇒
A3
5
A0
⇓ γ
e
3
u5
u1
u
A.
u3
A
(37)
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
283
Using Lemma 3.2 and the previous identity, diagram (33) is equivalent to
A2
u ◦ u1 ◦ u2
h◦l
u4 ◦ t1 ◦ t3
⇓ u ◦ γ ◦ t1 ◦ t3 F 2 ⇓ h ◦ ρ ◦ t1 ◦ t3
A
u5
u ◦ u3
◦ t1
◦ t3
B
h◦g 1 ◦f 2
A3
(38)
and diagram (36) is equivalent to
A2
u ◦ u1 ◦ u2
⇓ u ◦ γ ◦ t1 ◦ t3 F 2
A
u5
u ◦ u3
A
where
ζ
h◦l
u4 ◦ t1 ◦ t3
⇓ h◦ζ
◦ t1
◦ t3
B,
h◦g 1 ◦f 2
3
(39)
is the following composition:
t1
F1
t3
F2
t2
⇓ κ
F1
t3
A4
⇓ ε
e4
A
A4
t1
u4
A2
e4
u
l
⇓ ρe
5
u
e
u5
A3
B2
f2
g1
C 0.
(40)
Step 3.
Now the claim is equivalent to proving that (38) and (39) are equivalent.
1
3
Since they have already a common left hand side, it suces to prove that h ◦ ρ ◦ t ◦ t
and
h◦ζ
are equal if pre-composed with a suitable morphism of
W; then we will conclude
using Lemma 3.4(ii).
Step 4.
u ◦ γ ◦ t1 ◦ t3 coincides with (37), by Lemma 2.1 there are
t : F 3 → F 2 in W, such that γ ◦ t1 ◦ t3 ◦ t4 is equal to
Since
and a morphism
t1
F1
t2
t3
A4
⇓ ε
e4
A
u4
u2
F
t4
F
2
t3
F
1
e4
u
u5
e
4
t1
A
A1
A2
⇓ κ
3
an object
F3
4
A0 .
⇓ γ
e
3
u5
u1
u3
A
(41)
MATTEO TOMMASINI
284
Let us denote by
φ
the composition below:
t1
F
t4
3
F
F
t3
v◦φ
4
F
4
A
⇓ ε
e4
A
t2
F
t3
1
u4
u2
F3
F2
t3
F1
t1
f2
B
v1
2
e4
u
e5
u
v◦φ
u5
⇓ γ
e
A
v
⇓ δ −1
A0
⇓ η
u1
u3
B
f2
f
B.
v
B0
2
v1
coincides also with the following composition:
A
u5
B0
⇓ δ
u1
⇓ γ
A4
f1
A1
u2
2
u4
B0
A
3
A
t1
f1
1
A2
4
Then using (41), we get that
t4
B0.
⇓ ω
e
(42)
⇓ κ
F
f1
is equal to the following composition:
t3
2
u2
A1
e4
u
u5
A
t1
F1
t4
A2
A3
1
t1
3
e5
u
⇓ κ
2
Using (34), we get that
t2
F1
t3
u4
A4
⇓ ε
e4
A
f2
f
A0
u3
A3
v
−1
⇓ η
B2
v
B0
v1
B.
v ◦ φ coincides also with v ◦ ω ◦ t1 ◦ t3 ◦ t4 . Since g =
(B , v, g) is a morphism in C [W ], v belongs to W. So by Lemma 2.1 there are an
4
5
4
3
5
1
3
4
5
object F and a morphism t : F → F in W, such that φ ◦ t = ω ◦ t ◦ t ◦ t ◦ t . Using
1
3
4
5
(42), this implies that ω ◦ t ◦ t ◦ t ◦ t is equal to the following composition:
Using (31), we conclude that
0
−1
t1
F1
t3
F4
t5
Step 5.
F3
t4
F2
t2
⇓ κ
t3
Using (40) and (35),
F1
t1
w ◦ ζ ◦ t4 ◦ t5
A4
⇓ ε
e4
A
u4
u5
e
A2
A4
u5
A1
u4
e
A3
u2
f1
B0.
⇓ ω
e
f2
B2
v1
(43)
is equal to the following composition:
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
t1
t3
F
4
t5
F
3
t4
F
t2
F1
u4
A4
⇓ ε
e4
A
u5
e
⇓ κ
2
F
t3
4
w ◦ ζ ◦ t4 ◦ t5
F4
F3
t4
F2
t3
F1
t1
⇓ ω
e
⇓ σ −1
B0
⇓ ξ
v1
f2
B
2
C
g1
l
w
g
C.
w
0
⇓ ω
3
f2
C0
⇓ σ −1
B0
⇓ ξ
C0
A1
u2
A
A1
f1
A2
A4
u5
C0
coincides also with the composition
u4
t5
u
e
u5
A
t1
Using (43) we conclude that
u2
A3
1
l
A2
4
285
v1
B2
f1
g1
w
g
C.
w
1
3
4
5
Using the previous identity and (32), we conclude that w ◦ ρ ◦ t ◦ t ◦ t ◦ t = w ◦ ζ ◦
5
5
t ◦ t . Using Lemma 2.1, this implies that there are an object F and a morphism
t6 : F 5 → F 4 in W, such that ρ ◦ t1 ◦ t3 ◦ t4 ◦ t5 ◦ t6 = ζ ◦ t4 ◦ t5 ◦ t6 . This implies that:
4
(h ◦ ρ ◦ t1 ◦ t3 ) ◦ (t4 ◦ t5 ◦ t6 ) = (h ◦ ζ) ◦ (t4 ◦ t5 ◦ t6 ).
Step 6.
By construction the morphisms
composition also belongs to
W
by
(BF2).
t4 , t5
and
t6
(44)
all belong to
W,
so their
So (44) and Lemma 3.4(ii) imply that (38) and
(39) are equivalent, so we conclude by Step 3.
4.3. Remark. In order to nd a set of data
(A4 , u4 , u5 , γ, ω, ρ)
as in
(F1)
(F3)
you can
follow the construction in [P1996, Appendix A.2]. However, such a procedure is long: a
shorter construction is the following one. First of all, we use axiom (BF3) in order to get
u4 in W and γ invertible:
data as in the upper part of the following diagram, with
E
u4
A2
Then we use
in
W
(BF4a)
and an invertible
and
u1 ◦ u2
(BF4b)
2-morphism
γ
⇒
A0
u5
u3
A3 .
F , a morphism z : F → E
ω : f ◦ u ◦ u ◦ z ⇒ v ◦f 2 ◦ u5 ◦ z, such that v ◦ ω
in order to get an object
4
1
2
1
is equal to the following composition:
MATTEO TOMMASINI
286
A2
u4
z
F
B0
⇓ δ −1
A0
⇓ η
u1
⇓ γ
E
u3
u5
r:
f1
A1
u2
A3
B2
f2
v
f
B.
v
B0
v1
4
Then we use again (BF4a) and (BF4b) in order to get an object A , a morphism
A4 → F in W and an invertible 2-morphism ρ : l ◦ u4 ◦ z ◦ r ⇒ g 1 ◦ f 2 ◦ u5 ◦ z ◦ r, such
that
w◦ρ
is equal to the following composition:
u4 ◦ z
A4
r
l
A2
A1
u2
⇓ ω
F
u5 ◦ z
f1
v1
A3
B2
f2
C0
⇓ σ −1
B0
⇓ ξ
w
g
C.
w
C0
g1
4
4
5
Then it suces to dene u := u ◦ z ◦ r, u :=
4
Note that u belongs to W by construction and
u5 ◦ z ◦ r, γ := γ ◦ z ◦ r and ω := ω ◦ r.
(BF2). Since u, u1 and u2 belong to
W by hypothesis or construction, (F1) holds by (BF2). Also (F2) and (F3) are easily
veried. Above we used axioms (BF3) and (BF4), hence the data that we found are in
general non-unique.
In the following Corollary, dierently from the previous statements, we need to explicitly use the associators and the right and left unitors for
C.
We denote these
2-morphisms
θa,b,c : a◦(b◦c) ⇒ (a◦b)◦c (for any triple of composable morphisms a, b, c), respectively
by πa : a ◦ idA ⇒ a, respectively by υa : idB ◦ a ⇒ a (for any morphism a : A → B ).
by
satisfying conditions (BF), any triple of
0
0
as in (27) and let us suppose that B = B , C = C , v = idB and
4.4. Corollary. Let us x any pair
morphisms
f , g, h
(C , W)
w = idC .
Moreover, let us suppose that the set of xed choices C(W) gives data as in the
3
upper part of the following diagram, with u in W and η invertible:
A3
u3
A0
Then
f
η
⇒
B
f2
idB ◦ idB
B.
(45)
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
(u ◦ idA0 )◦idA0
h◦ g◦f = A
h◦(g◦f )
A0
D
287
,
(46)
u ◦ u3
h◦g ◦f = A
(h◦g)◦f 2
A3
D
(47)
,W
ΘCh,g,f
and the associator
from (46) to (47) is represented by the following diagram:
A0
(u ◦ idA0 )◦idA0
⇓ α
A
α
⇓ β
A3
idA3
u ◦ u3
where
h◦(g◦f )
u3
(48)
is the following composition
u ◦ idA0
A0
⇓ πu
⇓ πu
A0
u3
⇓
πu−1◦ u3
3
u
A
u
u ◦ u3
3
A
β
(h◦g)◦f 2
A3
idA0
and
D,
A
idA3
is the following one:
u3
g◦f
A0
⇓ θh,g,f
f ◦u3
A
3
⇓ η
idB ◦ idB
B
⇓ υf 2
⇓ υidB
f2
⇓
B
idB
f
−1
θh◦g,f,u
3
B
C
h
h◦g
h◦g
D.
−1
⇓ π(h◦g)◦f
2
f2
idA3
C
A3
(h◦g)◦f 2
2-category, then the associators θ• and the unitors π• and υ• are
all trivial; moreover, idB ◦ idB = idB . We recall that the xed choices C(W) imposed by
Pronk on any pair (f, v) assume a very simple form in the case when either f or v are
3
3
2
an identity (see [P1996, p. 256]), so the quadruple (A , u , f , η) of (45) coincides with
0
(A , idA0 , f, if ). So if C is a 2-category, then the morphisms (46) and (47) coincide, and
the associator (48) is the 2-identity of this morphism.
4.5. Remark. If
is a
MATTEO TOMMASINI
288
Proof of Corollary 4.4. This is the rst proof where we explicitly need to use asso-
ciators and unitors of
of a
2-category
C
because we cannot prove the statement only in the special case
and then appeal to coherence results for the general case. First, anyway,
we prove this special case.
By the already mentioned [P1996, p. 256], since both
that the
4
v
and
w
are identities, one gets
diagrams of (28) (chosen from left to right) assume this simple form:
A1 := A0
u1 :=idA0
A0
A2 := A0
f
u2 :=idA0
f 1 :=f
δ := if
⇒
B v=id
B0 = B,
B
A1 = A0
σ := ig◦f
⇒
C w=id
g◦f 1 =g◦f
B0 = B
g
C0 = C,
C
A3 := A0
B 2 := B
v1 :=idB
l:=g◦f
ξ := ig
⇒
C w=id
g 1 :=g
u3 :=idA0
C0 = C,
A0
η := if
⇒
B
v ◦ v1 =idB
f
C
f 2 :=f
B2 = B.
(49)
Then identities (46) and (47) follow at once from (29) and (30). In order to compute
the associator, according to Proposition 4.1 we have to x any set of data as in
(F3).
(F1)
For that, we choose:
• A4 := A0 , u4 := idA0 , u5 := idA0
and
γ := iidA0 ;
• ω := if ;
• ρ := ig◦f .
Then the claim follows from Proposition 4.1. In the general case when
C
is a bicate-
gory, by [P1996, p. 256] the rst diagram in (49) is given by
A1 := A0
u1 :=idA0
A0
f
δ := υf−1 ◦ πf
⇒
B v=id
f 1 :=f
B0 = B,
B
and analogously for the second diagram and for the third one in (49); the fourth diagram
must be replaced by (45). Indeed, if C is simply a bicategory, in general we cannot write
v ◦ v1 = idB ◦ idB in a simpler form, hence we cannot say anything more precise about the
data in diagram (45) (it is completely determined as one of the xed choices in the set
C(W),
so we have no control over it). The data of
(F1)
(F3)
above have to be changed
according to these choices; then the statement follows again from Proposition 4.1.
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
289
5. Vertical composition
Following the plan explained in the Introduction, in this section we will prove that the
−1
vertical composition in C [W ] does not depend on the set of choices D(W) (actually,
it does not depend on
C(W)
either). In doing that, we will also provide a simple way of
1
2
computing a representative for the vertical composition of any pair of 2-morphisms Γ , Γ
1
2
in any bicategory of fractions, having xed representatives for Γ and Γ .
5.1. Proposition.
(vertical composition) Let us x any pair of objects A, B , any
m
triple of morphisms f
:= (Am , wm , f m ) : A
1
2
2
3
1
2
morphisms Γ : f ⇒ f and Γ : f ⇒ f in
Γ1 , respectively for Γ2 , as follows:
A1
w1
f1
A4
⇓ β1
u2
w2
B,
w3
in
C
C,
⇓ β2
B.
f3
3
A
Then let us x any set of data in
an object
A5
u4
A
(F4):
f2
u3
⇓ α2
A
f2
2
A2
w2
u1
⇓ α1
A
→ B for m = 1, 2, 3 and any pair of 2C [W−1 ]; let us x any representative for
a morphism
t1
in
C
(50)
as follows:
W,
a morphism
t2
and an invertible
2-morphism ρ
as below:
C
ρ
⇒
A2
t1
A4
Then the vertical composition
Γ2 ◦ Γ1
u2
t2
u3
A5 .
is represented by the following diagram
A1
f1
w1
u1 ◦ t1
⇓ ξ
A
⇓ γ
C
u4 ◦ t2
w3
3
A
where
ξ
and
γ
are the following compositions:
B,
f3
(51)
MATTEO TOMMASINI
290
t1
ξ:
t2
u3
A5
u4
t2
u3
A5
u4
Given any other pair of representatives (50) for
as in
(F4),
A2
⇓ α2
A3
w2
A,
w3
(52)
A1
⇓ β
u2
⇓ ρ
C
w1
1
u1
A4
t1
A1
⇓ α
u2
⇓ ρ
C
γ:
u1
A4
f1
1
A2
⇓ β2
A3
f2
B.
f3
(53)
Γ1
Γ2 ,
and
and any other set of data
the diagram (51) induced by the new data represents the same
2-morphism
as
(51).
In particular, the vertical composition in
C(W) nor
C [W−1 ] does not depend on the set of choices
D(W), hence it is the same in any bicategory of fractions
(C , W).
on the set of choices
constructed from the pair
5.2. Remark. Since
Γ2
is a
2-morphism
in
Therefore by Lemma 2.2 a set of choices as
C [W−1 ], both w2 and w2 ◦ u3 belong to W.
in (F4) always exists, but in general it is not
unique.
The proof of Proposition 5.1 mostly relies on the following:
5.3. Lemma. Let us assume the same notations as Proposition 5.1.
Moreover, let us
choose:
(F4)0 :
an object
µ
D,
a morphism
p1
in
W,
a morphism
p2
and an invertible
as follows:
D
p1
A4
Let
ϕ
and
δ
u2
µ
⇒
A2
be the following compositions:
p2
u3
A5 .
2-morphism
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
ϕ:
p2
u3
A5
u4
p2
u3
A5
u4
w2
A,
w3
(54)
A1
⇓ β
u2
⇓ µ
D
w1
1
A2
⇓ α2
A3
u1
A4
p1
A1
⇓ α
u2
⇓ µ
D
δ:
u1
A4
p1
291
f1
1
A2
⇓ β2
A3
f2
B.
f3
(55)
Then the following diagram
A1
f1
w1
u1 ◦ p1
⇓ ϕ
A
⇓ δ
D
B,
u4 ◦ p2
w3
f3
3
A
represents the same
Proof.
Step 1.
2-cell
of
C [W−1 ]
(56)
as diagram (51).
Let us apply Lemma 3.1 on the following pair of diagrams:
A4
u2
t1
⇓ ρ
A2
p1
⇓ µ D.
A2
C
p2
t2
u3
A4
u2
u3
A5
A5
1
2
1
2
3
Then there are a pair of objects E and E , a triple of morphisms q , q and q , such
1
1
3
that both t ◦ q and q belong to W, and a pair of invertible 2-morphisms ε and κ in C
as follows:
E1
q1
C
t1
ε
⇒
A4
E2
q2
p1
q3
D,
E1
p2 ◦ q2
κ
⇒
A5
q3
t2 ◦ q1
E 1,
MATTEO TOMMASINI
292
such that
ρ ◦ q1 ◦ q3
is equal to the composition of the following diagram:
q1
E1
q3
C
⇓ ε
D
q2
⇓ κ
E2
Step 2.
E1
q3
ξ
p1
A4
⇓ µ
A5
p2
C
q1
Using the denition of
t1
t2
u2
A2 .
u3
(57)
in (52) together with (57), we get that
ξ ◦ q1 ◦ q3
is
equal to the following composition:
t1
C
q1
p1
⇓ ε
E
q3
1
q2
E
E
q3
1
q1
u2
p2
5
C
ϕ
w2
A.
w3
3
A
t2
w1
A2
⇓ α2
u3
Using (58) and the denition of
A1
⇓ α1
⇓ µ
D
⇓ κ
2
u1
A4
A
u4
(58)
in (54), we conclude that
ξ ◦ q1 ◦ q3
is also equal to
the following composition:
E2
Step 3.
1
q2
E1
q3
u1 ◦ t1
C
q1
⇓ u ◦ε
4
u4 ◦ p2
E1
u4
γ
Using the denition of
◦ t2
w1
◦ p1
⇓ ϕ
D
⇓ u ◦κ
q3
A1
u1
A.
w3
A3
◦ q1
in (53) together with (57), we get that
γ ◦ q1 ◦ q3
is equal to the following composition:
C
q1
t1
p1
⇓ ε
E
q3
1
q2
E
q3
E
1
q1
p2
C
δ
5
t2
A
f1
f2
A2
⇓ β2
u3
Using (59) and the denition of
the following composition:
⇓ µ
D
A1
⇓ β1
u2
⇓ κ
2
u1
A4
B.
f3
3
u4
in (55), we conclude that
A
(59)
γ ◦ q1 ◦ q3
is also equal to
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
E1
q3
E2
u1 ◦ t1
C
q1
1
⇓ u ◦ε
q2
E1
f1
◦ p1
⇓ δ
D
⇓ u4 ◦ κ
q3
A1
u1
u4 ◦ p2
B.
f3
A3
u4 ◦ t2 ◦ q1
293
Step 4.
Using Steps 2 and 3 together with Lemma 3.2 on the (classes of the) diagrams
1
2 1 2 3
1
4
(51) and (56), with the data (1) (5) given by (E , E , q , q , q , u ◦ ε, u ◦ κ), we conclude
that the classes of the diagrams (51) and (56) have representatives that are equal. In other
−1
terms, (51) and (56) represent the same 2-cell of C [W ], as we wanted to prove.
Proof of Proposition 5.1. Following [P1996, p. 258], the vertical composition
can be computed by choosing any pair of representatives for
Γ1
and
Γ2
Γ2 ◦ Γ1
(as we did in (50)).
Having xed such representatives, the construction of the vertical composition in [P1996]
is a special case of the construction of Proposition 5.1.
In particular, the construction
(F4), induced by the xed
C(W) and D(W). Lemma 5.3 proves that any other set of data as in (F4)
−1
induces the same 2-cell of C [W ]. Since the construction in [P1996], does not depend
1
2
on the representatives for Γ and Γ , the construction in Proposition 5.1 also does not
1
2
depend on the chosen representatives for Γ and Γ , so we conclude.
in [P1996] a priori holds only for a specic set of data as in
set of choices
6. Horizontal composition with
1-morphisms
on the left
(horizontal composition with 1-morphisms on the left) Let us
f := (A0 , w, f ) : A → B , any pair of morphisms g m := (B m , vm , g m ) :
B → C for m = 1, 2, and any 2-morphism ∆ : g 1 ⇒ g 2 in C [W−1 ]. Let us x also any
representative for ∆ as below:
6.1. Proposition.
x any morphism
B1
g1
v1
u1
⇓ α
B
B3
u2
v2
B
⇓ β
C.
g2
2
Let us suppose that the xed set of choices
(60)
C(W)
gives data as in the upper part of
MATTEO TOMMASINI
294
the following diagrams, with
w1
w2
and
in
W,
and
ρ1
and
D1
w1
A0
f1
(F5):
C
w2
B1,
v1
(so that by [P1996, Ÿ 2.2] we have
let us x any set of data in
A0
f
B2
v2
(61)
g m ◦ f = (Dm , w ◦ wm , g m ◦ f m )
for
m = 1, 2).
D1
f1
Then
as follows:
D3
σ1
⇒
B1
t1
and
t2
in
and
ϕ
W,
D4
h1
t2
B3,
u1
D2
σ2
⇒
B2
f2
h2
B3;
u2
(62)
t3
in
v 1 ◦ u1 ◦ δ
is
we choose any set of data as in the upper part of the following diagram, with
W
invertible:
t3
D3
(F7):
f2
ρ2
⇒
B
we choose data as in the upper part of the following diagrams, with
1
2
and σ and σ invertible:
t1
(F6):
invertible:
D2
ρ1
⇒
B
f
ρ2
we choose any invertible
w1 ◦ t1
D5
ϕ
⇒
A0
t4
w 2 ◦ t2
D4 ;
2-morphism δ : h1 ◦ t3 ⇒ h2 ◦ t4 ,
such that
equal to the following composition:
B3
h1
t3
D
t1
D3
⇓ (σ 1 )−1
D1
f1
w1
⇓ ϕ
5
u1
B1
⇓ (ρ1 )−1
A0
⇓ ρ2
w2
t4
t2
D4
D2
⇓ σ2
h2
B
3
f2
B2
⇓ α−1
u2
u1
v1
f
B.
v2
v1
B1
(63)
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
Then
∆◦f
is represented by the following diagram
D1
w ◦ w1
C,
g 2 ◦f 2
D2
(64)
is the following composition:
t3
D
t1
3
h1
⇓ δ
D5
D
f1
1
4
t2
B1
D
⇓ σ1
u1
B3
h2
t4
⇓ (σ 2 )−1
D2
Given any other representative (60) for
(F7),
⇓ ξ
D5
t2 ◦ t4
w ◦ w2
ξ
g 1 ◦f 1
t1 ◦ t3
⇓ w◦ϕ
A
where
295
∆,
g1
⇓ β
C.
u2
B
f2
2
g2
(65)
and any other set of data as in
the diagram (64) induced by the new data represents the same
Therefore, each composition of the form
∆◦f
2-morphism
(F5)
as (64).
does not depend on the set of choices
D(W).
6.2. Remark. Since (60) represents a
2-morphism
in a bicategory of fractions, both
v1
v1 ◦ u1 belong to W, so by Lemma 2.2 there are data as on the left hand side of (62).
2
2
2
Since α is an invertible 2-morphism of C , by (BF5) also v ◦ u belongs to W; since g is
−1
2
a morphism in C [W ], v belongs to W. So using again Lemma 2.2 there are data as
on the right hand side of (62), hence we can always nd data satisfying condition (F5).
2
2
By axiom (BF2) the composition w ◦ t belongs to W, hence a set of data as in (F6)
1
1
exists by (BF3). Since v ◦ u belongs to W, a set of data as in (F7) exists by (BF4), up
to replacing ϕ in (F6) with the composition of ϕ with a suitable morphism in W with
5
target in D (this is analogous to the procedure explicitly described in Remark 4.3). So
a set of data as in (F5) (F7) exists, but in general it is not unique.
and
The proof of Proposition 6.1 relies on the following:
6.3. Lemma. Let us assume the same notations as Proposition 6.1. Moreover, let us x
any set of data in
(F5)0 :
C
as follows:
we choose data as in the upper part of the following diagrams, with
z1
and
z2
in
MATTEO TOMMASINI
296
W,
and
γ1
and
γ2
invertible:
E3
z1
D1
(F6)0 :
f1
γ1
⇒
B1
E4
l1
z2
B3,
u1
D2
l2
γ2
⇒
B2
f2
B3;
u2
(66)
z3
in
v1 ◦ u1 ◦ η
is
we choose any set of data as in the upper part of the following diagram, with
W
and
ν
invertible:
E5
z3
E3
(F7)0 :
we choose any invertible
ν
⇒
A0
w 1 ◦ z1
z4
w2 ◦ z2
E 4;
2-morphism η : l1 ◦ z3 ⇒ l2 ◦ z4 ,
such that
equal to the following composition:
B3
l1
z3
z1
E3
z2
E4
l2
ω
f1
w1
B1
⇓ (ρ1 )−1
f2
D2
⇓ γ2
B2
⇓ α−1
u2
B3
v1
f
A0
⇓ ρ2
w2
z4
Let
⇓ (γ )
D1
⇓ ν
E5
u1
1 −1
B.
v2
v1
B1
u1
(67)
be the following composition:
z3
l1
⇓ η
E5
z1
E3
B3
l2
z4
E
D1
⇓ γ1
4
z2
⇓ (γ 2 )−1
D2
f1
u1
B1
g1
⇓ β
C.
u2
f2
B2
g2
(68)
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
297
Then the following diagram
D1
w ◦ w1
⇓ w◦ν
A
Proof.
Step 1.
2-cell
of
⇓ ω
E5
z2 ◦ z4
w ◦ w2
represents the same
g 1 ◦f 1
z1 ◦ z3
g 2 ◦f 2
D2
C [W−1 ]
C
(69)
as (64).
Let us apply Lemma 3.1 on the following pair of diagrams:
D1
w1
t1 ◦ t3
⇓ ϕ D5
A0
z1 ◦ z3
⇓ ν E 5.
A0
t2 ◦ t4
w2
D1
w1
z2 ◦ z4
w2
D2
D2
1
2
1
2
Then there are a pair of objects F and F , a triple of morphisms p , p and
1
3
1
3
that both t ◦ t ◦ p and p belong to W, and a pair of invertible 2-morphisms
in
C
p3 , such
ε and κ
as follows:
F1
p1
D5
such that
F2
p2
ε
⇒
D1
t1 ◦ t3
ϕ ◦ p1 ◦ p3
p3
E 5,
z1 ◦ z3
F1
z2 ◦ z4 ◦ p2
p3
t2 ◦ t4 ◦ p1
F 1,
is equal to the following composition:
p1
F1
p2
p3
D5
⇓ ε
E5
⇓ κ
t1 ◦ t3
z1 ◦ z3
z2 ◦ z4
F2
κ
⇒
D2
p3
F1
p1
D5
t2 ◦ t4
D1
⇓ ν
D2
w1
w2
A0 .
(70)
MATTEO TOMMASINI
298
Step 2.
Using (70) we get that
(w ◦ ϕ) ◦ p1 ◦ p3
p1
F1
p3
is equal to the following composition:
D5
⇓ ε
p2
z1 ◦ z3
E5
⇓ κ
t1 ◦ t3
z2 ◦ z4
F2
F1
p3
D5
p1
t2 ◦ t4
D1
⇓ w◦ν
D2
w ◦ w1
A.
w ◦ w2
(71)
Therefore, using Lemma 3.2 and (71), diagram (64) is equivalent to
D1
w ◦ w1
⇓ w ◦ ϕ ◦ p1 ◦ p3
A
g 1 ◦f 1
t1 ◦ t3 ◦ p1 ◦ p3
⇓ ξ ◦ p1 ◦ p3
F2
t2 ◦ t4 ◦ p1 ◦ p3
w ◦ w2
C
g 2 ◦f 2
D2
(72)
and diagram (69) is equivalent to
D1
w ◦ w1
⇓ w ◦ ϕ ◦ p1 ◦ p3
A
g 1 ◦f 1
t1 ◦ t3 ◦ p1 ◦ p3
⇓ ζ
F2
t2 ◦ t4 ◦ p1 ◦ p3
w ◦ w2
C,
g 2 ◦f 2
D2
where
ζ
(73)
is the following composition:
p1
p3
F1
t3
D5
⇓ ε
p2
F2
⇓ η
⇓ κ
F
z4
1
p1
D5
l1
t4
z2
t2
f1
⇓ γ1
B3
l2
E4
D4
D1
z1
E3
z3
E5
p3
t1
D3
B1
u1
u2
⇓ (γ 2 )−1
D2
g1
⇓ β
B2
C
g2
f2
(74)
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
(i.e., the vertical composition of the
2-morphism ω ,
299
dened in (68), with
ε
and
κ).
Step 3.
The claim is now equivalent to proving that (72) and (73) represent the
−1
1
3
same 2-cell of C [W ]. For this, it will be sucient to prove that ξ ◦ p ◦ p and ζ are
equal if pre-composed with a suitable morphism of
W;
then we will conclude by applying
Lemma 3.4(ii).
Step 4.
ϕ ◦ p1 ◦ p3
By construction
F2
p3
p1
F1
D5
⇓ ε
E5
p2
coincides with (70). Therefore the composition
t3
D3
E3
z3
D1
w1
z1
A0
⇓ ν
z4
E4
t1
w2
D2
z2
(75)
is equal to the following one:
F1
p3
p3
F1
p2
p3
p1
D3
t4
E5
Now let us denote by
F1
t3
D5
⇓ κ−1
F2
Step 5.
p1
E4
D5
F1
p2
E5
A0 .
t2
z2
D2
w2
(76)
t4
h2
D4
z4
E4
B3
⇓ (σ 2 )−1
t2
D2
⇓ γ2
z2
p3
w1
the following composition:
⇓ κ−1
F2
D1
⇓ ϕ
D4
z4
ς
t1
l2
f2
B3
u2
B2.
u2
(77)
∆ is a 2-morphism in C [W−1 ] with representative (60), v1 ◦ u1 belongs to W.
2
2
2
Using (BF5) on α, we get that v ◦ u also belongs to W. Moreover, v also belongs
2
2
2 2
−1
to W because the triple g = (B , v , g ) is a morphism in C [W ]. Therefore we can
3
4
3
2
apply Lemma 2.3 to ς , so there are an object F , a morphism p : F → F in W and a
2-morphism
Since
ςe : h2 ◦ t4 ◦ p1 ◦ p3 ◦ p4 =⇒ l2 ◦ z4 ◦ p2 ◦ p3 ◦ p4
(78)
MATTEO TOMMASINI
300
C , such that ς ◦ p4 = u2 ◦ ςe. ς is invertible since each 2-morphism in (77) is
in C ; then using again Lemma 2.3, we get that also ς
e is invertible. Using the
4
2
of ς in (77) and the fact that ς ◦ p = u ◦ ς
e, we get that the composition
in
F
p3
p4
F3
p1
1
D5
p3
F1
p2
denition
t4
D4
⇓ κ−1
F2
invertible
E5
t2
E4
z4
f2
D2
⇓ γ2
z2
l2
B
B2
u2
3
(79)
is equal to the following one:
D5
p1
F1
p3 ◦ p4
F3
Step 6.
Let
µ
p2
E5
p4
F2
z4
h2
E4
l2
f2
D2
⇓ σ2
B2.
u2
B3
(80)
be the following composition:
p1
F3
t2
D4
⇓ ςe
F1
p3 ◦ p4
t4
p3
t3
D5
⇓ ε
F1
z3
p2
E5
E3
⇓ η
E4
t1
D1
z1
l1
l2
f1
⇓ γ1
B3
B1.
u1
(81)
1
we use (67) and we simplify γ (coming from
1
(81)) with its inverse (coming from (67)), so we get that v ◦ µ is equal to the following
Let us consider the
composition:
2-morphism v1 ◦ µ:
z4
D3
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
p4
F3
p3
F2
p1
F1
D5
t3
D3
⇓ ε
p2
E
5
t1
w1
w2
z4
E
4
D
⇓ γ
l2
v1
f
A0
⇓ ν
z2
B1
⇓ (ρ1 )−1
z1
E3
z3
f1
D1
301
⇓ ρ2
f2
2
B
2
B3
v2
2
v1
⇓ α
u2
B.
−1
B1
u1
In the previous diagram we can replace (75) with (76), so we get that
v1 ◦ µ
coincides
also with the following composition:
p1
F1
t3
D5
t1
D3
D1
f1
B1
p3
F
3
t4
p4
F
−1
⇓ κ
2
D
w1
⇓ ϕ
4
w2
p3
F
1
p2
E
5
z4
E
4
2
⇓ ρ
f2
D
⇓ γ2
z2
f
A0
t2
B
l2
B3
B.
2
v2
2
v1
⇓ α
u2
In the previous diagram we can replace (79) with (80).
v1
⇓ (ρ1 )−1
−1
B1
u1
So we conclude that
v1 ◦ µ
coincides also with the following composition:
D3
t1
D1
f1
t3
D
p1
p3 ◦ p4
F
F
1
A0
⇓ ρ2
w2
t4
1
w1
⇓ ϕ
5
B1
⇓ (ρ1 )−1
D
⇓ ςe
4
t2
2
h2
f2
F
p3 ◦ p4
p2
E
5
z4
E
4
l2
D
⇓ σ2
B
f
B.
v2
2
B
⇓ α−1
u2
3
v1
3
u1
v1
B1
MATTEO TOMMASINI
302
v1 ◦ µ
If we compare this diagram with (63), we conclude that
is also equal to the
following composition:
p3
p1
F1
◦ p4
p3 ◦ p4
F1
Since the triple
D4
E5
p2
t1
D3
g 1 = (B 1 , v1 , g 1 )
⇓ σ
F4
p5
◦ p4
p1
F1
p3 ◦ p4
F1
(82)
C [W−1 ], v1 belongs to W. Since
4
5
an object F and a morphism p :
is a morphism in
Using the denition of
µ
t3
t4
D4
h1
D1
⇓ σ
B3
h2
1
u1
f1
B1.
l2
E4
z4
composition:
t1
D3
⇓ δ
E5
p2
B.
v1
l2
E4
D5
⇓ ςe
F3
B1
u1
v1 ◦ µ coincides with (82), by Lemma 2.1 there are
F 4 → F 3 in W, such that µ ◦ p5 is equal to the following
p3
f1
1
B3
h2
z4
D1
h1
⇓ δ
t4
⇓ ςe
F3
t3
D5
(83)
in (81), we conclude that (83) is equal to the following
composition:
p1
F4
p5
F3
p4
F2
p3
t3
D5
⇓ ε
F1
z3
p2
z4
Step 7.
Let us consider the
E4
2-morphism ζ ◦ p4 ◦ p5 ,
D1
z1
E3
⇓ η
E5
t1
D3
f1
⇓ γ1
l1
B3
B1.
u1
l2
(84)
where
ζ
is dened in (74). If we
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
replace (84) with (83), we conclude that
ζ ◦ p4 ◦ p5
is equal to the following composition:
D3
t3
F2
p4
F4
p5
p3
p1
F1
t4
⇓ ςe
F3
p4
p3
F2
p2
F1
p3
E5
h1
h2
l2
⇓ κ
B1
u1
u2
⇓ (γ 2 )−1
z2
t4
f1
B3
E4
D5
D1
⇓ σ1
D4
z4
F1
p1
t1
⇓ δ
D5
303
g1
⇓ β
C.
g2
B2
f2
D2
t2
D4
(85)
4
u ◦ ςe with ς ◦ p (see (78)), then we replace ς with
γ 2 and κ−1 (coming from (77)) with their inverses
4
5
conclude that ζ ◦ p ◦ p coincides also with the following
In the diagram above we replace
diagram (77).
2
Lastly, we simplify
(coming from (85)).
So we
composition:
t3
F4
p4 ◦ p5
F2
p1 ◦ p3
D
t1
3
h1
⇓ δ
D5
D
If we compare this with the denition of
4
ξ
t2
B1
D
⇓ σ1
u1
B3
h2
t4
f1
1
⇓ (σ 2 )−1
D2
g1
⇓ β
C.
u2
f2
B
2
g2
in (65), we conclude that:
ζ ◦ (p4 ◦ p5 ) = (ξ ◦ p1 ◦ p3 ) ◦ (p4 ◦ p5 ).
Step 8.
Since both
p4
and
p5
belong to
W,
so does their composition by
(BF2).
So
the previous identity together with Lemma 3.4(ii) proves that (72) and (73) are equivalent
diagrams, so we have proved the claim (see Step 3).
Proof of Proposition 6.1. Following [P1996, pp. 259261], in order to compute
one can choose freely any representative for
on this choice).
conditions
(F5)
∆
as in (60) (and the result does not depend
Then after a very long construction, one gets a set of data satisfying
(F7)
and some additional properties, that we don't need to recall here.
Such data are uniquely determined by the choices of type
construction of
∆◦f
∆◦f
C(W)
and
D(W).
Then the
in [P1996] is simply a special case of the construction described in
MATTEO TOMMASINI
304
(F5) (F7), Using Lemma
C [W ], so we conclude.
Proposition 6.1, with a special set of data
(F5)
(F7)
2-cell
gives the same
of
7. Horizontal composition with
7.1. Proposition.
6.3, any set of data
−1
1-morphisms
on the right
(horizontal composition with 1-morphisms on the right) Let
g = (B 0 , u, g) : B → C , any pair of morphisms f m := (Am , wm , f m ) :
A → B for m = 1, 2 and any 2-morphism Γ : f 1 ⇒ f 2 in C [W−1 ]. Let us x any
representative for Γ as in (4) and let us suppose that the set of choices C(W) gives data
1
2
1
2
as in the upper part of the following diagrams, with u and u in W, and ρ and ρ
us x any morphism
invertible
D1
u1
A1
D2
h1
ρ1
⇒
B
f1
let us x any set of data in
(F8):
C
B0,
u
(so that by [P1996, Ÿ 2.2] we have
u2
A2
f2
B0
u
(86)
g ◦ f m = (Dm , wm ◦ um , g ◦ hm )
for
m = 1, 2).
we choose data as in the upper part of the following diagrams, with
W, and η 1 and η 2 invertible:
u3
A3
v1
Then
as follows:
D3
(F9):
h2
ρ2
⇒
B
η1
⇒
A1
u3
and
u5
in
in
W
and
η3
D4
u4
u1
u5
D1 ,
A3
v2
η2
⇒
A2
u6
u2
D2 ;
we choose data as in the upper part of the following diagram, with
invertible:
D5
u7
D3
u3
η3
⇒
A3
u8
u5
D4 ;
u7
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
(F10):
we choose any
2-morphism λ : h1 ◦ u4 ◦ u7 ⇒ h2 ◦ u6 ◦ u8 ,
305
such that
u◦λ
is equal
to the following composition:
D
u7
u8
⇓ (η )
u3
g◦Γ
⇓ (ρ )
A1
u1
1 −1
v1
u5
v2
⇓ η2
D4
u
f1
f2
⇓ β
A3
u6
Then
B0
1 −1
3
⇓ η3
D5
h1
D1
u4
B.
A2
⇓ ρ2
u2
D2
u
B0
h2
(87)
is represented by the following diagram:
D1
g◦h1
w1 ◦ u1
u4 ◦ u7
⇓ µ
A
u6 ◦ u8
w2 ◦ u2
D
where
µ
⇓ g◦λ
D5
C,
g◦h2
2
(88)
is the following composition:
u7
D3
⇓ η3
D5
u8
D
u4
D1
⇓ (η 1 )−1
u3
u1
v1
A3
u5
⇓ η2
D2
4
u6
Given any other representative (4) for
Γ,
A1
w1
⇓ α
A.
v2
u2
A2
w2
and any other set of data as in
(89)
(F8) (F10),
the diagram (88) induced by the new data is equivalent to diagram (88).
So the horizontal composition of the form
g◦Γ
does not depend on the set of choices
D(W).
u1
u2
belong to W, hence a set of data as in (F8)
(BF3); analogously also a set of data as in (F9) exists. Since g = (B 0 , u, g) is a
−1
morphism in C [W ], u belongs to W; hence a set of data as in (F10) exists by (BF4),
7.2. Remark. By hypothesis
exists by
and
MATTEO TOMMASINI
306
3
3
up to replacing η with the composition of η with a suitable morphism in
5
in D (analogously to the construction explained in Remark 4.3).
W
with target
The proof of Proposition 7.1 relies on the following:
7.3. Lemma. Let us assume the same notations as Proposition 7.1. Moreover, let us x
any set of data in
(F8)0 :
C
as follows:
we choose data as in the upper part of the following diagrams, with
W, and ξ 1 and ξ 2 invertible:
E3
t3
A3
(F9)0 :
v1
ξ1
⇒
A1
t3
t5
in
and
ξ3
and
E4
t4
t5
D1 ,
u1
A3
ξ2
⇒
A2
v2
t6
u2
D2 ;
we choose data as in the upper part of the following diagram, with
t7
in
W
invertible:
E5
t7
E3
(F10)0 :
we choose any
t3
ξ3
⇒
A3
t8
t5
E 4;
2-morphism ϕ : h1 ◦ t4 ◦ t7 ⇒ h2 ◦ t6 ◦ t8 ,
such that
u◦ϕ
to the following composition:
t4
t7
⇓ ξ3
E5
t8
t3
t5
t6
D2
⇓ (ρ )
A1
u1
v1
⇓ β
A3
⇓ ξ2
E4
B0
1 −1
⇓ (ξ 1 )−1
E3
h1
D1
v2
u
f1
f2
A2
⇓ ρ2
u2
h2
B.
u
B0
is equal
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
Let
δ
307
be the following composition:
t4
E3
t7
⇓ ξ3
E5
t8
E
u1
D1
⇓ (ξ 1 )−1
t3
v1
A3
t5
t6
w1
⇓ α
A.
v2
⇓ ξ2
D2
4
A1
w2
2
u2
A
(90)
Then the following diagram:
D1
w 1 ◦ u1
⇓ δ
A
g◦h1
t4 ◦ t7
⇓ g◦ϕ
E5
t6 ◦ t8
w 2 ◦ u2
C
g◦h2
D2
represents the same
Proof.
Step 1.
2-cell
of
C [W ]
−1
(91)
as (88).
We want to apply Lemma 3.1 for the following pair of diagrams:
D1
u1
A
w1
A1
D1
u1
u4 ◦ u7
⇓ (v1 ◦ η 3 )◦ 5
D
◦((η 1 )−1 ◦ u7 )
A
w1
⇓ (v1 ◦ ξ 3 )◦ 5
E .
◦((ξ 1 )−1 ◦ t7 )
A1
u5 ◦ u8
v1
t4 ◦ t7
v1
A3
t5 ◦ t8
A3
In order to do this, let us prove that all the relevant morphisms belong to
• w1
belongs to
W
since (4) represents the
• u1
belongs to
W
by hypothesis, so by
•
2-cell Γ
of
(BF2) w1 ◦ u1
(92)
W:
C [W−1 ];
also belongs to
W;
2-cell of C [W−1 ], also w1 ◦ v1 belongs to W; moreover u3
1
1
3
belongs to W by (F8); by (BF2) this implies that w ◦ v ◦ u belongs to W. Us1
1
1
1
4
ing (BF5) on the invertible 2-morphism w ◦ η , we conclude that also w ◦ u ◦ u
7
1
1
4
7
belongs to W. By (F9) u belongs to W, so by (BF2) w ◦ u ◦ u ◦ u belongs to
W;
since (4) represents a
MATTEO TOMMASINI
308
•
(F8)0
analogously, using
and
(F9)0
we get that
w1 ◦ u1 ◦ t4 ◦ t7
also belongs to
W.
1
2
So we can apply Lemma 3.1 on (92). Then there are a pair of objects F , F , a triple
1
2
3
4
7
1
3
of morphisms r , r and r , such that both u ◦ u ◦ r and r belong to W, and a pair of
invertible
2-morphisms ω
and
τ
in
C
as follows:
F1
r1
D5
u4 ◦ u7
F2
r2
ω
⇒
D1
t4 ◦ t7
r3
E 5,
F1
t5 ◦ t8 ◦ r2
τ
⇒
A3
r3
u5 ◦ u8 ◦ r1
F 1,
such that the composition
F2
r3
r1
F1
u7
D5
u4
D3
u3
⇓ η3
u8
D1
D4
u1
⇓ (η 1 )−1
A3
u5
A1
v1
(93)
coincides with the following one:
D3
u7
r1
F1
r3
F
2
r3
Step 2.
F
⇓ω
D5
r2
1
r1
t7
E5
⇓ τ
D
E3
⇓ ξ3
E4
t8
5
u8
D
u4
t4
t3
D1
⇓ (ξ 1 )−1
A3
t5
v1
(94)
Let us apply Lemma 3.1 for the following pair of diagrams:
v2 ◦ u5 ◦ u8 ◦ r1 ◦ r3
A
A1 .
u5
4
F2
w2
u1
2
A
v2 ◦ u5 ◦ u8 ◦ r1 ◦ r3
id
⇓ η 2 ◦ u8 ◦ r1 ◦ r3
u2
F2
F
2
u6 ◦ u8 ◦ r1 ◦ r3
D2
2
A
w2
2
A
id
⇓ (ξ 2 ◦ t8 ◦ r2 ◦ r3 )◦ 2
F .
◦(v2 ◦τ −1 )
u2
t6 ◦ t8 ◦ r2 ◦ r3
D2
(95)
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
W by hypothesis, hence also w2 ◦ u2 belongs to W. So in or2
2
5
8
1
3
der to apply the lemma, we need only to verify that w ◦ v ◦ u ◦ u ◦ r ◦ r belongs to W.
Both
w2
and
u2
309
belong to
1
1
4
7
1
The various hypotheses and the construction of Step 1 imply that w , u , u ◦ u ◦ r
3
and r all belong to W; hence also their composition belongs to W by (BF2). Now we
consider the invertible
F2
r3
r1
F1
2-morphism
u8
D5
dened as the following composition:
u5
D4
3 −1
⇓ (η )
u3
The target of this
2-morphism
D1
u4
W,
2-morphisms χ
R1
χ
⇒
F2
q1
F2
and
id
ζ
so by
w2
⇓ α−1
A1
u1
w1
A.
(BF5)
also the source belongs to
W. So we can apply Lemma 3.1 on (95). Therefore there are a pair of objects R1 , R2 , a
1
2
3
1
3
triple of morphisms q , q and q , such that both q and q belong to W, and a pair of
invertible
belongs to
A2
v1
⇓ η1
u7
D3
v2
A3
as follows
R2
q2
q3
F 2,
id
R1
q3
ζ
⇒
D2
2
t6 ◦ t8 ◦ r2 ◦ r3 ◦ q
R1 ,
u6 ◦ u8 ◦ r1 ◦ r3 ◦ q1
such that the composition
R2
q3
R1
q1
A3
u5
r3
F2
F1
r1
D5
u8
v2
⇓ η2
D4
u6
A2
u2
D2
(96)
coincides with the following one:
r1
r3
q1
R1
q3
⇓ χ
F1
D5
⇓ τ −1
F2
r3
q2
R
2
Step 3.
q3
R
1
q1
Let us consider the
F
F1
⇓ ζ
2
2-cell
F
r3
of
r2
E5
u8
u5
D4
A3
t5
t8
⇓ ξ2
E4
t6
1
r1
D
5
u8
D
v2
4
u6
D
2
A2 .
u2
(97)
C
u ◦ λ ◦ r1 ◦ r3 ◦ q1 ◦ q3 .
(98)
MATTEO TOMMASINI
310
u ◦ λ with (87), then we replace (93)
τ with τ −1 . So we get that (98) is equal
We replace
we simplify
q1
R
⇓χ
1
q3
F
2
r3
q2
F
u7
D5
r1
h1
D1
E
t7
⇓ (ρ )
A1
t8
u1
⇓ (ξ 1 )−1
3
t3
⇓ ξ3
E5
v1
v2
⇓ ξ2
E4
F2
q1
F1
r3
D5
r1
(F10)0 ,
Comparing with
q3
⇓χ
R1
F2
r2
⇓ ζ
R1
q1
F2
⇓ ω
t7
r1
D3
u4
B0
h2
D1
h1
t4
E3
⇓ ϕ
E5
t8
F1
r3
u7
F1
q2
R2
q3
r3
D2
u6
u
we get that (98) is also equal to the following composition:
D5
r1
q1
D4
u8
B.
A2
⇓ ρ2
u2
t6
R1
u
f1
f2
⇓ β
A3
t5
q3
B0
1 −1
t4
⇓ ω
⇓ ζ
R2
to the composition
u4
D3
1
r2
with (94) and (96) with (97); lastly
D5
E4
u8
D4
u
B0
t6
B.
h2
D2
u6
(99)
Since the triple
0
g = (B , u, g)
is a morphism in
C [W ], u
−1
belongs to
W. So the
R3 and a
equality of (98) with (99) and Lemma 2.1 imply that there are an object
4
3
2
morphism q : R → R in W, such that
g ◦ λ ◦ r1 ◦ r3 ◦ q1 ◦ q3 ◦ q4
(100)
is equal to the following composition:
r1
q1
q3
R
3
q4
R
⇓χ
R1
q2
R1
q1
F2
r2
r3
F1
⇓ ω
F1
⇓ ζ
2
q3
F2
r3
u7
D5
t7
u4
D5
D1
g◦h1
t4
E3
⇓ g◦ϕ
E5
t8
r1
D3
E4
u8
D4
t6
u6
C.
g◦h2
D2
(101)
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
Step 4.
Now let us consider the
2-morphism
of
311
C
µ ◦ r1 ◦ r3 ◦ q1 ◦ q3 ◦ q4 ,
(102)
where µ dened in (89). Replacing (93) with (94), (96) with (97), and simplifying
τ −1 , we get that (102) is equal to the following composition:
r1
q1
⇓χ
R1
q3
R
3
q4
R
F2
r3
u7
D5
r2
A1
t3
⇓ ξ3
E5
t8
w1
⇓ (ξ 1 )−1
E3
t7
⇓ ζ
2
u1
D1
with
t4
⇓ ω
F1
q2
u4
D3
τ
v1
⇓ α
A3
t5
v2
⇓ ξ2
E4
q3
A.
w2
t6
R1
q1
F2
r3
F1
r1
D5
Comparing with the denition of
D4
u8
δ
D2
u6
A2
u2
in (90), we conclude that (102) is also equal to the
following composition:
r1
q1
q3
R3
q4
⇓χ
R1
F2
r3
r2
⇓ ζ
R2
q3
R
1
q1
F
2
r3
F
t7
E5
r1
D
u1
D1
A1
w1
t4
E3
⇓ δ
t8
1
u4
D3
⇓ ω
F1
q2
u7
D5
E4
5
D
u8
A.
t6
w2
4
D
u6
2
2
A
u2
(103)
Step 5.
Now let
ε
be the composition
r1
q1
q3
R
and let
κ
3
R1
F2
r3
R
2
be the composition
u7
D3
⇓ ω
F1
r2
q4
D5
E5
E3
t7
u4
D1
t4
MATTEO TOMMASINI
312
q1
q3
R
3
q4
R
⇓χ
R1
r3
F1
q2
r2
E5
⇓ ζ
2
q3
F2
t8
E4
R1
q1
F2
r3
F1
r1
D5
Then let us apply Lemma 3.2 on the pair of
of ) (88) and (91), with the choice of
ε
and
κ
D4
u8
2-cells
of
C [W−1 ]
t6
u6
D2 .
given by (the classes
as above. Then the equalities (100)=(101)
and (102)=(103) prove that the classes of (88) and (91) admit representatives that are
equal, hence (88) and (91) are equivalent, as we wanted to prove.
g ◦ Γ in [P1996, p. 259] begins by
Γ; then it is simply a particular case of the construction
Proof of Proposition 7.1. The construction of
choosing any representative (4) for
described in the statement of Proposition 7.1.
To be more precise, the construction
in [P1996] is done by selecting a specic set of data
(F8)
(F10),
uniquely determined
C(W) and D(W). Using Lemma 7.3, any set of data (F8) (F10)
−1
gives the same 2-cell in C [W ], so we conclude. Note that the construction described
in Proposition 7.1 does not depend on the representative (4) chosen for Γ because the
by the choices of type
construction of [P1996] does not depend on this choice.
8. Some useful applications
As we stated in the Introduction, Theorem 1.2 and Corollary 1.3 follow immediately
from Propositions 4.1, 5.1, 6.1 and 7.1. As we remarked in the Introduction, these four
propositions are also interesting in their own, since they allow us to easily calculate various
compositions of
2-cells
in a bicategory of fractions, only knowing the set of xed choices
C(W) (that completely determines the bicategory because of Theorem 1.2), and choosing
freely any representatives of the 2-morphisms that we are composing (plus the additional
data as in (F1) (F10)). For example, having shown a simple procedure for computing
vertical compositions (i.e., Proposition 5.1), we can prove the already mentioned Propo−1
sition 1.4 about invertibility of 2-morphisms in C [W ]:
C [W ],
−1
1
1
f1
is a morphism of
1
1
belongs to W, so does w ◦ v ◦ u. Since
Proof of Proposition 1.4. Let us assume (iv); by hypothesis
belongs to W. Since v ◦ u
1
1
(4) represents Γ, then w ◦ v also belongs to W. So by Lemma 2.4 there are an object
04
0
04
A and a morphism u : A → A4 such that u ◦ u0 belongs to W. Since β ◦ u is invertible,
0
so is β ◦ (u ◦ u ), hence (iii) holds.
hence
w
Let us assume (iii), so let us choose any representative (4) for Γ and let us assume that
4
4
3
there are an object A and a morphism u : A → A in W, such that β ◦ u is invertible
−1
in C . By the description of 2-morphisms in C [W ], we have that Γ is also represented
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
(A4 , v1 ◦ u, v2 ◦ u, α ◦ u, β ◦ u).
representative for Γ.
by the quintuple
with this new
Since
β◦u
is invertible in
313
C,
(ii) holds
Now let us assume (ii), so let us assume that the data in (4) are such that
vertible in
C.
Since
α
is invertible in
C
β
is in-
2-morphism in a bicategory
2-morphism [A3 , v2 , v1 , α−1 , β −1 ] : f 2 ⇒ f 1 .
this is a (vertical) inverse for Γ, so (i) holds.
by denition of
of fractions, it makes sense to consider the
Using Proposition 5.1, it is easy to see that
Γ is invertible and let us
−1
x any representative (4) for it; by denition of 2-morphism in C [W ] we have that α is
1
2
invertible, so we can apply Corollary 3.3 with f interchanged with f and (E, p, q, ς, Φ)
3
2
1
−1
−1
3
replaced by (A , v , v , α , Γ ). So there are an object E , a morphism r : E → A such
2
2
2
1
1
−1
that v ◦ r belongs to W, and a 2-morphism ϕ : f ◦ v ◦ r ⇒ f ◦ v ◦ r, such that Γ
is
Now let us prove that (i) implies (iv), so let us assume that
represented by the following diagram:
A2
f2
w2
A
⇓ α−1 ◦ r
v2 ◦ r
⇓ ϕ
E
v1 ◦ r
w1
A
B.
f1
1
Then we have
h
i
E, v2 ◦ r, v2 ◦ r, iw2 ◦ v2 ◦ r , if 2 ◦v2 ◦ r = i(A2 ,w2 ,f 2 ) =
h
i
(∗)
= Γ ◦ Γ−1 = E, v2 ◦ r, v2 ◦ r, iw2 ◦ v2 ◦ r , β ◦ r ◦ ϕ ,
where
object
(∗) is obtained by applying Proposition 5.1. So by Lemma
F and a morphism s : F → E in W, such that:
if 2 ◦v2 ◦ r ◦ s = β ◦ r ◦ s ◦ ϕ ◦ s .
3.4(ii) there are an
(104)
Analogously, we have:
h
i
1
1
F, v ◦ r ◦ s, v ◦ r ◦ s, iw1 ◦ v1 ◦ r ◦ s , if 1 ◦v1 ◦ r ◦ s = i(A1 ,w1 ,f 1 ) =
h
i
(∗)
= Γ−1 ◦ Γ = F, v1 ◦ r ◦ s, v1 ◦ r ◦ s, iw1 ◦ v1 ◦ r ◦ s , ϕ ◦ s ◦ β ◦ r ◦ s ,
where
object
(∗) is obtained using again Proposition 5.1. So by Lemma 3.4(iii) there are an
A4 and a morphism t : A4 → F , such that v1 ◦ r ◦ s ◦ t belongs to W and such that
if 1 ◦v1 ◦ r ◦ s ◦ t = ϕ ◦ s ◦ t ◦ β ◦ r ◦ s ◦ t .
(105)
MATTEO TOMMASINI
314
Then (104) and (105) prove that
β ◦r◦s◦t
C , given by ϕ ◦ s ◦ t.
u := r ◦ s ◦ t : A4 → A3 .
has an inverse in
order to conclude that (iv) holds, it suces to dene
In
A second application of the results of the previous sections is the following corollary,
that will be useful in the next paper [T2014(a)]. In that paper, we will have to compare
the compositions of
3
morphisms of the form (27) and the compositions of
3
morphisms
of the following form
f 0 := A
g 0 := B
0
h := C
u ◦ idA0
A
v ◦ idB 0
f ◦idA0
0
g◦idB 0
B0
w ◦ idC 0
h◦idC 0
C0
B
C
D
,
,
.
(106)
0
In the special case when C is a 2-category we have that f = f and so on, hence
0
0
0
0
0
0
h ◦ (g ◦ f ) = h ◦ (g ◦ f ) and (h ◦ g) ◦ f = (h ◦ g ) ◦ f . However, when C is simply a
1
1
1
bicategory, in general the xed choice (A , u , f , δ) in the set C(W) for the pair (f, v) (see
the rst diagram in (28)) is dierent from the xed choice for the pair
(f ◦ idA0 , v ◦ idB 0 ),
and analogously for all the remaining choices needed to compose the morphisms in (106).
0
0
0
Therefore, we need a tool for comparing h ◦ (g ◦ f ) with h ◦ (g ◦ f ) and analogously for
0
the other pair of compositions. In order to do that, rst of all we compare separately f
with
f.
πa : a ◦ idA ⇒ a the right unitor of C
0
invertible 2-morphism χ(f ) : f ⇒ f as
For that, we recall that we are denoting by
for any morphism
a : A → B.
Then we dene an
the class of the following diagram
A0
u ◦ idA0
idA0
⇓ πu ◦ idA0
A
A0
idA0
u
f ◦idA0
⇓ πf ◦idA0
B,
f
0
A
and analogously for
χ(g)
and
χ(h).
Then we have the following result:
8.1. Corollary. Let us x any pair
6
(C , W)
satisfying conditions
C ,W
morphisms as in (27) and (106). Then the associator Θh,g,f : h ◦
(BF) and any set of
(g ◦ f ) ⇒ (h ◦ g) ◦ f
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
315
is equal to the following composition:
g◦f
−1
⇓ χ(g)
◦ χ(f )
g 0 ◦f 0
A
h
C
−1
⇓ χ(h)−1
h0
⇓ ΘCh0,W
,g 0 ,f 0
f0
⇓ χ(f )
⇓ χ(h) ◦ χ(g)
B
f
D.
h0 ◦g 0
h◦g
(107)
8.2. Remark. As we mentioned above, in the case when
χ(f ), χ(g)
to prove (since
and
χ(h)
C
is a
2-category there is nothing
are all trivial). In the case of a bicategory, things are
a bit more complicated. One could be tempted to do the following:
(a) prove that
gously for
χ(f ) is a right
χ(g) and χ(h);
or left unitor for
f
in the bicategory
C [W−1 ],
and analo-
(b) appeal to coherence results in order to conclude that necessarily the composition in
C ,W
(107) coincides with the associator Θh,g,f .
However, since the identities for
(B, idB , idB )
A
and
B
in
C [W−1 ]
are the triples
(A, idA , idA )
and
respectively, we have:
f ◦ idA = A
idA ◦ u
0
A
f ◦idA0
B
,
idB ◦f = A
u ◦ idA0
A
0
idB ◦f
B
.
Since we are working in a bicategory, in general idA ◦ u 6= u ◦ idA0 and idB ◦f 6= f ◦idA0 ,
0
therefore in general f is dierent from f ◦ idA and from idB ◦f . Therefore in general
χ(f ) : f 0 ⇒ f cannot be a left or right unitor for f in C [W−1 ]. So in general we cannot
use (a) (then (b)) in order to conclude. Instead, we need to rely on Propositions 4.1, 5.1
and 7.1, as we will show below.
Proof of Corollary 8.1. The main problem in the proof comes from the presence of
unitors for
of
C
C
(with trivial unitors, the statement is trivial). The presence of associators
complicates only the exposition of the proof, but it does not introduce additional
problems. So for simplicity of exposition we suppose that
Step 1.
f, g
and
h,
suppose that the set of xed choices
invertible:
C(W),
so that we have identities (29) and (30).
needed for
Moreover, let us
C(W) gives data as in the upper part of the following
u1 , u2 , v1 and u3 in W, and δ , σ , ξ
diagrams (starting from the ones on the left), with
η
has trivial associators.
Let us denote as in (28) the xed choices in the set
composition of
and
C
MATTEO TOMMASINI
316
1
u1
A0
f ◦idA0
A
δ
⇒
B
2
f
1
v ◦ idB 0
u2
B0,
A
1
g◦idB 0 ◦f
2
v1
B0
g◦idB 0
B
ξ
⇒
C
A
σ
⇒
C
1
l
w ◦ idC 0
C 0,
3
g1
w ◦ idC 0
A
η
⇒
B
u3
C 0,
A0
f ◦idA0
f
2
v ◦ idB 0 ◦v1
B
2
,
(108)
so that by [P1996, Ÿ 2.2] one has
u ◦ idA0 ◦u1 ◦u2
h0 ◦ g 0 ◦ f 0 = A
0
0
0
D ,
A
u ◦ idA0 ◦u3
h ◦g ◦f = A
Step 2.
h◦idC 0 ◦l
2
3
h◦idC 0 ◦g 1 ◦f
A
2
D
In this long step we are going to construct a set of data in
be used in order to provide representatives for the various
.
C;
2-morphisms
such data will
appearing in the
claim of Corollary 8.1, in such a way that it will be simple to compute their composition
C ,W
in (107), and to prove that it coincides with Θh,g,f .
Step 2a.
Using
(BF2)
we get that
u1 ◦ u2
W, so by (BF3)
1
1
with r in W and ζ
belongs to
of data as in the upper part of the following diagram,
there is a set
invertible:
E1
r1
A2
Using
(BF4a)
and an invertible
u1 ◦ u2
ζ1
⇒
A0
r2
2
u1 ◦u2
A
.
(109)
(BF4b), there are an object E 2 , a morphism r3 : E 2 → E 1 in W
1
2-morphism ε1 : f 1 ◦ u2 ◦ r1 ◦ r3 ⇒ f ◦ u2 ◦ r2 ◦ r3 in C , such that v ◦ ε1
and
is equal to the following composition:
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
u2
A2
f1
A1
r1
E
E
⇓ δ −1
⇓ ζ
1
f
1
⇓ πf−1
0
A
u1
f ◦idA0
A
r3
E2
r3
E
u2
A2
⇓ ε1
r2
1
1
u2
A
δ −1 ◦ u2 ◦ r1 ◦ r3
r1
E1
2
A
f1
f
1
A
A
u2
1
⇓ δ
u1
⇓ (ζ )
2
v
A1
v
B0
1 −1
r1
⇓ πv
B0
1
is equal to the following composition:
A1
u2
2
f
B.
v ◦ idB 0
⇓ δ
r2
This implies that
v
B0
u1
r3
2
317
−1
⇓ πv−1
v ◦ idB 0
⇓ πf
A0
u1
B.
f ◦idA0
f
(110)
Step 2b.
Now we use again (BF4a) and (BF4b) in order to get an object
3
2
2
1
3
4
morphism r : E → E in W and an invertible 2-morphism ε : l ◦ r ◦ r ◦ r
r2 ◦ r3 ◦ r4 , such that w ◦ ε2 is equal to the following composition:
4
E
3
r4
E
u2
E1
r3
⇓ ε
2
r3
E3
E2
A
r3
⇓ σ −1
g
f
⇓ πg−1
0
r3
E1
r3
r2
C.
g◦idB 0
1
⇓ σ
w ◦ idC 0
2
⇓ πw
C0
l
w
is equal to the following composition:
r1
E1
l
A2
l
2
A
⇓ (ε1 )−1
u2
E1
A2
r1
A
⇓ σ −1
1
u2
f
A1
w
C0
⇓ ε2
r4
E2
1
A
w
C0
f1
B
u2
σ −1 ◦ r1 ◦ r3 ◦ r4
r4
A1
1
E1
r2
Therefore,
l
A2
r1
E 3, a
⇒ l◦
w ◦ idC 0
C.
g◦idB 0
1
f1
⇓ πw−1
B0
⇓ πg
g
(111)
MATTEO TOMMASINI
318
Step 2c.
We use (BF3) in order to get data as in the upper part of the following
5
2
diagram, with r in W and ζ invertible:
E4
r5
A3
Using
(BF4a)
and an invertible
r6
ζ2
⇒
A0
u3
3
u3
A
.
(BF4b), there are an object E 5 , a morphism r7 : E 5 → E 4 in W
2
2-morphism ε3 : v1 ◦f 2 ◦ r5 ◦ r7 ⇒ v1 ◦ f ◦ r6 ◦ r7 in C , such that v ◦ ε3
and
is equal to the following composition:
f2
3
v1
B2
A
⇓ η −1
r5
E
r7
5
E
⇓ ζ
4
u3
2
A
η ◦ r5 ◦ r7
r7
E5
f
B0
v1
⇓ πv
v
is equal to the following composition:
r5
E4
r7
v ◦ idB 0
2
B
2
B.
f ◦idA0
⇓ η
3
A
Therefore,
f
⇓ πf−1
0
u3
r6
v
B0
r6
⇓ (ε3 )−1
E4
u3
A3
⇓ ζ2
A
u3
3
f
r5
A3
⇓ η
2
2
f2
f
A0
B
B2
v1
B0
v1
⇓ πf−1
f ◦idA0
B.
v ◦ idB 0
⇓ πv
v
(112)
Step 2d.
Using
(BF2)
we get that
3
5
7
u ◦r ◦r
belongs to
r8 in
data as in the upper part of the following diagram, with
W; so by (BF3) there
W and ζ 3 invertible:
are
E6
r8
E3
Then we use
E7 → E6
in
W
r9
u3 ◦ r5 ◦ r7
E 5.
(BF4b) in order to get an object E 7 , a morphism r10 :
1
2
2
4
2
3
4
8
10
invertible 2-morphism ε : f ◦ u ◦ r ◦ r ◦ r ◦ r ◦ r
⇒ v1 ◦ f ◦
(BF4a)
and an
u1 ◦ u2 ◦ r1 ◦ r3 ◦ r4
ζ3
⇒
A0
and
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
r6 ◦ r7 ◦ r9 ◦ r10
in
r4
E
r8
E
r10
7
E
C,
r3
E2
v ◦ ε4
such that
E1
is equal to the following composition:
r2
2
6
⇓ ζ
2
u
A
1
1 −1 A
⇓
(ζ
)
A2
u2
A1
r1
3
1
v
B0
⇓ δ
⇓ πv−1
−1
v ◦ idB 0
f ◦idA0
0
A
B.
v ◦ idB 0
⇓ η
u3
⇓ ζ2
r5
r7
u1
u3
A3
E5
f
u1
3
r9
319
A
3
f
B
2
2
⇓ πv
B0
v1
v
r6
E4
(113)
8
Now we use (BF4a) and (BF4b) in order to get an object E , a morphism
r11 : E 8 → E 7 in W and an invertible 2-morphism ε5 : g 1 ◦ f 2 ◦ r5 ◦ r7 ◦ r9 ◦ r10 ◦ r11 ⇒
2
g 1 ◦ f ◦ r6 ◦ r7 ◦ r9 ◦ r10 ◦ r11 , such that w ◦ ε5 is equal to the following composition:
Step 2e.
r5
E4
f2
A3
g1
B2
r7
E
8
r11
E
7
r10
E
r9
6
⇓ ξ −1
v1
E
⇓ ε
5
3
B
E4
r9 ◦ r10 ◦ r11
r9 ◦ r10 ◦ r11
Step 2f.
r12 : A4 →
r7
E4
7
r5
A3
r
⇓ (ε5 )−1
E8
r6
ξ ◦ f 2 ◦ r5 ◦ r7 ◦ r9 ◦ r10 ◦ r11
E5
E5
r7
E4
E4
A
f
2
B
⇓ ξ
2
B2
v1
⇓ ε3
r6
r5
A
A3
f
2
f2
We use again (BF4a) and
E 8 in W and an invertible
g1
w ◦ idC 0
C0
C.
⇓ πw
w
is equal to the following composition:
f2
3
g
g◦idB 0
r7
3
w
⇓ πg−1
0
v1
Therefore
C0
B
v1
2
B2
g1
g1
g
B0
⇓ ξ
C0
⇓ πg−1
g◦idB 0
C.
w ◦ idC 0
⇓ πw
w
(114)
(BF4b) in order to get an object A4 , a morphism
2-morphism ρ : l ◦ r1 ◦ r3 ◦ r4 ◦ r8 ◦ r10 ◦ r11 ◦ r12 ⇒
320
MATTEO TOMMASINI
g 1 ◦ f 2 ◦ r5 ◦ r7 ◦ r9 ◦ r10 ◦ r11 ◦ r12 ,
such that
r1 ◦ r3 ◦ r4
E3
r8
◦ r10
r2
⇓ ε
E7
f
2
◦ r6
◦ r7
◦ r9
w◦ρ
◦ r3
is equal to the following composition:
l
A2
⇓ ε2
2
A
◦ r4
4
B
◦ r10
2
B
⇓ σ −1
f ◦u2
0
⇓ ξ
5 −1
⇓ (ε )
4
A
E
8
E
r9 ◦ r10 ◦ r11
w ◦ idC 0
g◦idB 0
r11
r12
⇓ πw−1
l
1
v1
w
C0
C.
w ◦ idC 0
g1
5
r5 ◦ r7
A
3
B
f2
⇓ πw
C0
2
g1
w
(115)
Step 3.
Now we dene the following set of morphisms and
2-morphisms:
u4 := r1 ◦ r3 ◦ r4 ◦ r8 ◦ r10 ◦ r11 ◦ r12 : A4 −→ A2 ,
u5 := r5 ◦ r7 ◦ r9 ◦ r10 ◦ r11 ◦ r12 : A4 −→ A3 ,
γ := ζ 3 ◦ r10 ◦ r11 ◦ r12 : u1 ◦ u2 ◦ u4 =⇒ u3 ◦ u5 .
Moreover, we dene
ω : f 1 ◦ u2 ◦ u4 ⇒ v1 ◦f 2 ◦ u5
as the following composition:
r1 ◦ r3
E2
r4
A4
◦ r8
r11 ◦ r12
A2
u2
⇓ ε
◦ r10
f
(116)
1
◦u2 ◦r2
A1
1
f1
◦ r3
⇓ ε4
E7
B0.
2
v1 ◦f ◦r6 ◦ r7
v1
r9 ◦ r10
E5
By construction of
u4
and
u5 ,
we have that
ρ
r5 ◦ r7
A3
is dened from
⇓ (ε3 )−1
B2
2
f
l ◦ u4
to
(117)
g 1 ◦ f 2 ◦ u5 .
We
claim that the set of data
A4 , u4 , u5 , γ, ω, ρ
,W
4
(F3) for the computation of the associator ΘCh,g,f
. u belongs
3
to W by construction and (BF2), and γ is invertible because ζ is so, hence condition
(F1) holds. In order to prove (F2), we replace in (31) the denition of u4 , u5 and γ (see
−1
(116)), the expression for δ
◦ u2 ◦ r1 ◦ r3 (obtained in diagram (110)) and the expression
5
7
for η ◦ r ◦ r (obtained in diagram (112)). After simplifying the term πf (coming from
satises conditions
(F1)
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
321
(110)) with its inverse (coming from (112)), we get that the composition in (31) is equal
to the following one:
E1
r3
r
r8
A
4
r10 ◦ r11 ◦ r12
E2
4
E
E
r1
E1
r3
r1
3
6
⇓ ζ
A2
r2
2
u2
⇓ ε1
A
1
1 −1 A
⇓
(ζ
)
A2
u2
A1
r6
E4
r7
E4
f
1
⇓ δ
A
u3
A
3 −1
⇓ (ε )
A3
f
B
2
2
B2
⇓ πv−1
−1
v ◦ idB 0
f ◦idA0
0
3
v
B0
B.
v ◦ idB 0
⇓ η
⇓ ζ2
r5
f1
u1
u3
r9
E5
A1
u1
3
A3
r7
u2
v1
⇓ πv
B0
v
v1
f2
r5
(118)
v ◦ ω . This
(F2) holds. In order to prove (F3), in (32) we replace ω with its deni4
5
−1
tion in (117), u and u with their denition in (116), σ
◦ r1 ◦ r3 ◦ r4 with (111), and
ξ ◦ f 2 ◦ r5 ◦ r7 ◦ r9 ◦ r10 ◦ r11 with (114). After simplifying the terms πg , ε1 and ε3 with
their inverses, we get that the composition in (32) coincides with (115), hence with w ◦ ρ.
Therefore also condition (F3) holds. So using Proposition 4.1 we conclude that the assoC ,W
ciator Θh,g,f is represented by the following diagram:
Using (113) and (117), we get that (118) (hence (31)) coincides with
implies that
A2
u ◦ u1 ◦ u2
r1 ◦ r3 ◦ r4 ◦ r8 ◦
◦ r10 ◦ r11 ◦ r12
⇓ u◦γ
A
A4
r5 ◦ r7 ◦ r9 ◦
◦ r10 ◦ r11 ◦ r12
u ◦ u3
h◦l
⇓ h◦ρ
D.
h◦g 1 ◦f 2
A3
Step 4.
Now we want to compute also the associator
(119)
ΘCh0,W
.
,g 0 ,f 0
For that, we dene the
following set of data:
2
u4 := r2 ◦ r3 ◦ r4 ◦ r8 ◦ r10 ◦ r11 ◦ r12 : A4 −→ A ,
3
u5 := r6 ◦ r7 ◦ r9 ◦ r10 ◦ r11 ◦ r12 : A4 −→ A ,
1
2
ω := ε4 ◦ r11 ◦ r12 : f ◦ u2 ◦ u4 =⇒ v1 ◦ f ◦ u5 .
(120)
MATTEO TOMMASINI
322
Moreover, we dene
γ : u1 ◦ u2 ◦ u4 ⇒ u3 ◦ u5
r3
E2
as the following composition
r2
E1
r4
A2
r1
u2
2
A
⇓ (ζ )
u1
A1
u2
⇓ ζ3
E3
u3
r8
A3
r5
r10
A4
and
◦ r11
◦ r12
E6
2
ρ : l ◦ u4 ⇒ g 1 ◦ f ◦ u5
E5
r9
E8
r9
E6
r4
E5
u3
3
A
(121)
r11
E
r7
r2
2
E1
A
⇓ (ε2 )−1
2
E
r3
E4
r5
1
A3
l
2
r1
f2
A
l
E7
r10
E6
r9
E5
C 0.
g1
B2
B
⇓ ε5
E7
r11
r3
E3
⇓ ρ
r10
r12
E8
A0
⇓ ζ2
r6
E2
r4
r8
E6
E7
r11
r12
u1
as the following composition:
r10
A4
E4
r7
1
1 −1 A
E4
r7
f
3
A
r6
g1
2
2
(122)
We claim that the set of data
A4 , u4 , u5 , γ, ω, ρ
satises conditions
(F1)
(F3)
(123)
for the computation of the associator
ΘCh0,W
,
,g 0 ,f 0
i.e., the
same conditions stated in Proposition 4.1, but with (27) replaced by (106), and with the
four diagrams appearing in (28) replaced by the four diagrams of (108). By construction
u1 , u2 and r1 belong to W. So using (109) together with (BF2) and (BF5), we get that
u1 ◦ u2 ◦ r2 also belongs to W. Moreover, by construction r3 , r4 , r8 , r10 , r11 and r12 belong
1
2
4
1
2
2
3
4
8
10
11
12
to W. So by (BF2) also the morphism u ◦ u ◦ u = u ◦ u ◦ r ◦ r ◦ r ◦ r ◦ r ◦ r ◦ r
W; moreover γ is invertible because all the 2-morphisms in (121) are invertible
construction, so (F1) holds. Using the denition of ω in (120), we get that v ◦ idB 0 ◦ ω
belongs to
by
is equal to the following composition:
r10
A4
r11
◦ r12
E6
r8
E3
r4
E2
r3
E1
r2
2
2 u
A
A
1
f
1
⇓ ε4
E7
r10
E6
r9
E5
r7
E4
B0
3
r6
A
f
2
B
2
v1
B0
idB 0
idB 0
v
v
⇓ πv
⇓ πv−1
B0
B.
v
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
323
4
In the diagram above we replace v ◦ ε with diagram (113), then we simplify all the
−1
terms of the form πv and πv . Using (121), we get that v ◦ idB 0 ◦ ω is equal to the following
composition:
A
u4
A
f
1
u2
1
B0
A
2
⇓ δ
u1
⇓ γ
4
⇓ η
u5
3
A
This proves that condition
(F3).
(F2)
f ◦idA0
A0
u3
prove also
v ◦ idB 0
−1
f
2
B
2
v1
B
B.
v ◦ idB 0
0
holds for the set of data in (123). Now we want to
ρ
If we use the denition of
in (122), then
w ◦ idC 0 ◦ ρ
is equal to the
following composition:
E
r11
E
8
E
r11
A4
ε
2
r12
E8
r11
E3
E2
r4
⇓ ρ
7
r3
r2
2
E1
A
⇓ (ε2 )−1
E1
r3
r1
l
A2
E
r9
6
E
r7
5
E
⇓ ε
7
E7
r10
E6
r5
A
3
B
B
r7
E4
A
r6
3
w
f
w
C0
2
5
E5
r9
4
f2
C0
idC 0
l
g1
r10
r12
r8
E6
r10
E2
r4
2
g1
idC 0
⇓ πw
⇓ πw−1
C
C.
w
0
2
In the diagram above we replace the term w ◦ ρ with diagram (115), and we simplify
5
, ε and all the terms of the form πw with their inverses. Then we get that w ◦ idC 0 ◦ ρ
coincides also with the following composition
l
2
C0
A
u4
u2
⇓ ω = ε4 ◦ r11 ◦ r12
A4
u5
A
f
so
(F3)
f
v1
3
2
B
⇓ σ −1
1
A
1
B0
⇓ ξ
2
g1
w ◦ idC 0
g◦idB 0
C,
w ◦ idC 0
C0
holds. Therefore, using Proposition 4.1 with the data of (106), (108) and (123),
MATTEO TOMMASINI
324
we get that the associator
ΘCh0,W
,g 0 ,f 0
is represented by the following diagram:
A
u ◦ idA0 ◦u1 ◦u2
h◦idC 0 ◦ l
r2 ◦ r3 ◦ r4 ◦ r8
◦ r10 ◦ r11 ◦ r12
⇓ u ◦ idA0 ◦ γ
A
2
⇓ h ◦ idC 0 ◦ ρ
A4
◦ r7 ◦ r9 ◦
◦ r10 ◦ r11 ◦ r12
D.
r6
u ◦ idA0 ◦u3
A
Step 5.
h◦idC 0 ◦g 1 ◦f
2
3
(124)
Until now we have computed the two associators appearing in the claim
of Corollary 8.1.
Using Propositions 5.1 and 7.1 it is not dicult to prove that the
2-morphism
−1
χ(h)
−1
◦ χ(g)
−1
◦ χ(f )
0
0
: h ◦ g ◦ f =⇒ h ◦ g ◦ f
0
appearing in the upper part of (107) is represented by the diagram
A2
u ◦ u1 ◦ u2
⇓ α1
A
A
h◦l
r1 ◦ r3 ◦ r4 ◦ r8 ◦
◦ r10 ◦ r11 ◦ r12
4
⇓ β1
◦ r3 ◦ r4 ◦ r8
◦ r10 ◦ r11 ◦ r12
D,
r2
u ◦ idA0 ◦u1 ◦u2
A
where
α1
and
4
A
β1
A
r1
r3 ◦ r4 ◦ r8 ◦ r10 ◦ r11 ◦ r12
r8 ◦ r10 ◦ r11 ◦ r12
r4
E
E
E2
3
r4
Step 6.
(125)
are the following compositions:
1
r2
4
h◦idC 0 ◦l
2
E
2
r3
u2
A2
A1
⇓ ζ1
A
E1
⇓ ε2
E1
3
r
2
r1
A
u2
A2
0
A
1
u1
r2
A
l
⇓ πu−1
u ◦ idA0
0
⇓ πh−1
D.
h◦idC 0
Using again Propositions 5.1 and 7.1, the composition
A,
h
l
C
2
u
u1
χ(h) ◦ χ(g) ◦ χ(f ) : h0 ◦ g 0 ◦ f 0 =⇒ h ◦ g ◦ f
(126)
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
325
appearing in the lower part of (107) is represented by the diagram
3
u ◦ idA0
A
◦u3
r6 ◦ r7 ◦ r9 ◦
◦ r10 ◦ r11 ◦ r12
4
⇓ α2
A
A
h◦idC 0 ◦g 1 ◦f
⇓ β2
r5 ◦ r7 ◦ r9 ◦
◦ r10 ◦ r11 ◦ r12
u ◦ u3
2
D,
h◦g 1 ◦f 2
3
A
where
α2
and
β2
are the following compositions:
A
3
r6
r7 ◦ r9 ◦ r10 ◦ r11 ◦ r12
4
(127)
E
A
⇓ (ζ 2 )−1
A3
4
r5
A4
r12
r11
E
7
r10
E
6
r9
E8
r11
Step 7.
E
7
r10
E
6
r9
5
r7
4
E
E
5 −1
⇓ (ε )
E5 7 E4
r
r6
3
f
u ◦ idA0
u3
A
B
2
h◦idC 0
g1
⇓ πh
C0
3
A
f2
B
2
g1
D.
h
(128)
Now we have to compute (107), i.e., the vertical composition of:
(a)
χ(h)−1 ◦ (χ(g)−1 ◦ χ(f )−1 ),
(b)
ΘCh0,W
,
,g 0 ,f 0
(c)
(χ(h) ◦ χ(g)) ◦ χ(f ),
represented by (125);
represented by (124);
represented by (127).
We compose (a) and (b) using Proposition 5.1: since the map
2
4
coincides with the map A → A in (124), we can choose
A4 , idA4 , idA4 , ir2 ◦ r3 ◦ r4 ◦ r8 ◦ r10 ◦ r11 ◦ r12
as the data
A,
u
u3
2
A
r5
⇓ πu
0
(C, t1 , t2 , ρ)
needed in
(F4).
2
A4 → A
in (125)
So the composition of (a) and (b) is represented
MATTEO TOMMASINI
326
by the following diagram:
A2
u ◦ u1 ◦ u2
r1 ◦ r3 ◦ r4 ◦ r8 ◦
◦ r10 ◦ r11 ◦ r12
⇓ (u ◦ idA0 ◦ γ) ◦ α1 A4
A
h◦l
⇓ (h ◦ idC 0 ◦ ρ) ◦ β 1
◦ r7 ◦ r9 ◦
◦ r10 ◦ r11 ◦ r12
D.
r6
u ◦ idA0 ◦u3
A
h◦idC 0 ◦g 1 ◦f
2
3
(129)
Using the same ideas on the pair of diagrams (129) and (127), the composition of (a),
(b) and (c) is represented by the following diagram:
A2
u ◦ u1 ◦ u2
A
r1 ◦ r3 ◦ r4 ◦ r8 ◦
◦ r10 ◦ r11 ◦ r12
h◦l
⇓ α2 ◦ (u ◦ idA0 ◦ γ) ◦ α1 A4 ⇓ β 2 ◦ (h ◦ idC 0 ◦ ρ) ◦ β 1
u ◦ u3
◦ r7 ◦ r9 ◦
◦ r10 ◦ r11 ◦ r12
D.
r5
A
h◦g 1 ◦f 2
3
(130)
1
1
Above we replace γ with (121) and ρ with (122). In addition, we replace α , β ,
α2 and β 2 with their denitions in (126) and (128). Then we simplify the terms of the
1 2 2
5
form πu , πh , ζ , ζ , ε and ε with their inverses. So on the left hand side of (130) we get
3
10
11
12
u ◦ ζ ◦ r ◦ r ◦ r = u ◦ γ , and on the right hand side we get h ◦ ρ. In other terms,
(130) coincides with (119). By construction (130) represents the composition of (107),
C ,W
and (119) represents Θh,g,f , so we conclude.
9. Appendix - An alternative description of
2-morphisms
in a bicategory of
fractions
Proposition 1.1 suggests an alternative construction of
2-morphisms in C [W−1 ].
This
new construction is equivalent to the original one of [P1996], but much simpler. Let us
1
1
1
1
suppose that we want to dene a 2-morphism from the morphism f := (A , w , f ) to
2
the morphism f
:= (A2 , w2 , f 2 ) (both dened from A to B ). Since the composition
−1
of 1-morphisms in C [W ] depends on the set of choices C(W), such choices must be
xed before constructing the bicategory of fractions (and such a bicategory depends on
these choices). Since they are xed, there is no harm in using them in order to get data
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
(E, p, q, ς)
C
in
as below, with
p
in
W
and
ς
327
invertible:
E
p
A1
ς
⇒
A
w1
q
A2 .
w2
(131)
Then we give the following denition:
9.1. Definition. Let us x any pair
choices
C(W);
C [W ]
−1
let
(C , W)
satisfying conditions
(BF),
and any set of
be the bicategory of fractions induced by such choices (see
−1
Theorem 1.2). Then an almost canonical representative of a 2-morphism in C [W ]
f 1 to f 2 is any triple (A3 , t, ϕ) where t : A3 → E is a morphism in W, and ϕ is a
2-morphism in C from f 1 ◦ p ◦ t to f 2 ◦ q ◦ t. Given another triple (A03 , t0 : A03 → E, ϕ0 :
f 1 ◦ p ◦ t0 ⇒ f 2 ◦ q ◦ t0 ), we say that it is equivalent to the previous one, and we write
(A3 , t, ϕ) ∼ (A03 , t0 , ϕ0 ) if and only if there are data (A4 , u, u0 , σ) in C as follows
from
E
t0
A03
u0
σ
⇒
A4
t
u
A3 ,
such that:
• u
belongs to
• σ
is invertible;
•
W;
the compositions of the following two diagrams are equal:
u0
u0
t
⇓ σ
u
A4
t0
A03
E
A1
f1
⇓ ϕ
A3
⇓ σ
p
t0
B,
A4
u0
E
p
A1
f1
⇓ ϕ0
A03
B.
−1
t
03
A
t0
E
q
A2
f2
t0
E
q
A2
f2
(132)
A
2-morphism
in
C [W−1 ] is any class of equivalence of data as
h
i A3 , t, ϕ : A1 , w1 , f 1 =⇒ A2 , w2 , f 2 .
above, denoted by
As you can see, this denition is much shorter than the original one (that we recalled
in the Introduction). It is not too hard to prove that:
MATTEO TOMMASINI
328
(a)
∼
is a actually an equivalence relation (so Denition 9.1 is well-posed);
(b) there is a natural bijection between
2-morphisms
according to Denition 9.1 and
2-
morphisms according to the original denition of [P1996], induced by associating to
3
any triple (A , t, ϕ) the equivalence class of the following diagram:
w1
⇓ ς ◦t
A
A1
f1
p◦t
⇓ ϕ
A3
B.
q◦t
w2
A
f2
2
(133)
The result above can be rephrased by saying that each
2-morphism
in
C [W−1 ]
(de-
ned according to [P1996]) has a representative given by a diagram of type (133) for a
3
triple (A , t, ϕ) that is almost canonical (having xed the set of choices C(W) that com−1
pletely determine the structure of C [W ], see Theorem 1.2). Almost canonical here
3
refers to the fact that the triple (A , t, ϕ) inducing a given 2-morphism is almost unique
and varies in a much smaller set if compared to the set of diagrams as (4). We cannot
say canonical since sometimes there is more than one such triple inducing the same
2-morphism
(this is why we need to use the equivalence relation given in Denition 9.1).
The description of
2-cells
of
C [W−1 ]
in terms of almost canonical triples is simpler
than the original one in [P1996], but does not interact well with vertical and horizontal
composition. Therefore we are just mentioning it here in the Appendix, and we leave the
proof of (a) and (b) as an useful exercise for the interested reader (hint: use the axioms
(BF) together with the lemmas stated in Ÿ 2
C [W−1 ] that we recalled in the Introduction).
and Ÿ 3, and the denition of
2-cells
of
References
[L]
Tom Leinster, Higher operads, higher categories. Vol. 298, Cambridge University
Press, 2004, arXiv: math.CT 0305049v1;
[GZ1967] Pierre Gabriel, Michel Zisman, Calculus of Fractions and Homotopy Theory.
Springer-Verlag, New York, 1967;
[P1996] Dorette A. Pronk, Etendues and stacks as bicategories of fractions. Compositio Mathematica 102, 243303, available at
1996__102_3_243_0,
http://www.numdam.org/item?id=CM_
1996;
[PW] Dorette A. Pronk, Michael A. Warren, Bicategorical bration structures and stacks.
Theory and Applications of Categories 29.29 (2014), pp. 836873, arXiv: math.CT
1303.0340v1;
SOME INSIGHTS ON BICATEGORIES OF FRACTIONS
329
[T2014(a)] Matteo Tommasini, Right saturations and induced pseudofunctors between
bicategories of fractions. arXiv: math.CT 1410.5075v2, 2014;
[T2014(b)] Matteo Tommasini, Equivalences of bicategories of fractions. arXiv: math.CT
1410.6395v2, 2014.
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