The tricategory of bicategories

The tricategory of bicategories

[Jamie Vicary]

Dear categorists,

Suppose you have categories A, B, C, and functors S,S': A-->B, T,T':
B-->C, and natural transformations alpha: S==>S', beta: T==>T'.
Suppose we want to see these as part of a 2-category of categories;
then we had better know the horizontal composite of alpha and beta.
There are two possible ways to evaluate this composite: as the natural
transformation having components beta_{S'X}.T(alpha_X), and as the
natural transformation having components T'(alpha_X).beta_{S(X)}. But
these are equal, since beta is a natural transformation. So we have no
difficulty uniquely defining our horizontal composite, and obtaining a
canonical 2-category of categories.

But now suppose that A, B, C are bicategories, S,S',T,T' are
pseudofunctors, and alpha and beta are pseudonatural transformations.
Then the two possible definitions for the horizontal composite of
alpha and beta will not necessarily be equal, although of course they
will be related by an invertible modification. But then we have a
problem forming the tricategory of bicategories, pseudofunctors,
pseudonatural transformations and modifications: there is no longer a
canonical choice available for horizontal composition of pseudonatural
transformations.

Presumably this choice can be made, and a tricategory is the result,
and different choices yield equivalent tricategories. But it bothers
me that there seems to be no canonical tricategory of bicategories.
Should it? Or is my reasoning flawed?

Jamie.


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Re: The tricategory of bicategories

[Steve Lack]

Dear Jamie,

I agree that there is no canonical choice of horizontal composition of
pseudonatural transformations, but that the various possible choices
are, in a suitable sense, equivalent. 

Whether or not you should be bothered by this I can't really say. But 
perhaps it's worth pointing out that there are various different possible
descriptions of the structure of weak 3-category, and not all of them include
a chosen horizontal composition of pseudonatural transformations. Some,
particularly, the simplicial approaches, include *no* choices of compositions.
Others include some choices of composition, but not this particular one. 
For example, the notion of Gray-category does include chosen composition
of 1-cells, and vertical composition of 2-cells, but does not include a chosen
horizontal composition of 2-cells. 

Best wishes,

Steve Lack.

 
On 21/10/2011, at 9:17 PM, Jamie Vicary wrote:

> Dear categorists,
> 
> Suppose you have categories A, B, C, and functors S,S': A-->B, T,T':
> B-->C, and natural transformations alpha: S==>S', beta: T==>T'.
> Suppose we want to see these as part of a 2-category of categories;
> then we had better know the horizontal composite of alpha and beta.
> There are two possible ways to evaluate this composite: as the natural
> transformation having components beta_{S'X}.T(alpha_X), and as the
> natural transformation having components T'(alpha_X).beta_{S(X)}. But
> these are equal, since beta is a natural transformation. So we have no
> difficulty uniquely defining our horizontal composite, and obtaining a
> canonical 2-category of categories.
> 
> But now suppose that A, B, C are bicategories, S,S',T,T' are
> pseudofunctors, and alpha and beta are pseudonatural transformations.
> Then the two possible definitions for the horizontal composite of
> alpha and beta will not necessarily be equal, although of course they
> will be related by an invertible modification. But then we have a
> problem forming the tricategory of bicategories, pseudofunctors,
> pseudonatural transformations and modifications: there is no longer a
> canonical choice available for horizontal composition of pseudonatural
> transformations.
> 
> Presumably this choice can be made, and a tricategory is the result,
> and different choices yield equivalent tricategories. But it bothers
> me that there seems to be no canonical tricategory of bicategories.
> Should it? Or is my reasoning flawed?
> 
> Jamie.
> 



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Re: The tricategory of bicategories

[Richard Garner]

Dear Jamie,

Let me add to Steve's remarks that:

If one wishes to work in a purely algebraic setting, then it is
possible to give a notion of tricategory which is such that the
bicategories together with their homomorphisms, strong natural
transformations and modifications comprise a canonical instance of
this structure. This can be done by using this latter requirement to
inform the very definition of tricategory that one gives.

More specifically, I mean to suggest that one works in the setting of
Michael Batanin's globular operads. Such a globular operad encodes a
notion of higher category by describing the forms of composition that
each instance of this notion must come equipped with. There are a SET
of such composition operations for each reasonable shape of diagram
which one might wish to compose; in particular, for the diagram whose
shape is that of a pair of horizontally composable 2-cells.

We might derive such a set of operations as follows. Consider the
category M whose objects are given by a triple of finitely presentable
bicategories A, B, C, homomorphisms f, g : A -> B and h, k : B -> C,
and a pair of strong natural transformations a : f => g and b : h =>
k; and whose morphisms (A,B,C,f,g,h,k,a,b) ->
(A',B',C',f',g',h',a',b') are given by a triple of STRICT
homomorphisms u:A -> A', v:B -> B', w:C -> C' commuting with all the
other data.

There is a corresponding category N whose objects are given by a
single strong transformation a : f => g : A -> B between homomorphisms
between finitely presentable bicategories, and whose morphisms are
given by compatible strict homomorphisms A->A' and B->B'. We now
consider the set of all functors M -> N which on objects send
(A,B,C,f,g,h,k,a,b) to a strong natural transformation of the form fh
=> gk and which on morphisms send (u,v,w) to (u,w).

This defines a set of composition operations for diagrams whose shape
is that of a pair of horizontally composable 2-cells. We may in a
similar manner define sets of composition operations for other shapes
of diagrams of 2- and 3-cells, in this way obtaining a globular
operad. (At the 0- and 1-cell level, we take it that there is a unique
composition operation of each shape; we do this because we have a
canonical, strictly unital and associative, way of composing
homomorphisms of bicategories).

This construction does indeed yield a 3-dimensional globular operad as
it is really just the construction of the endomorphism operad of a
globular object in a monoidal globular category, which you may find
described in Michael's paper on the subject. (In fact this is not
quite true as we are doing something slightly different at the 0- and
1-cell level but that is easily accounted for).

The point is that this globular operad is, unless something is
terribly wrong with the world, contractible, meaning that it provides
a fully sensible notion of weak 3-category. And almost by definition,
the totality of bicategories and their weak cells comprise a canonical
instance of this structure.

Richard


On 24 October 2011 09:59, Steve Lack <steve.lack@mq.edu.au> wrote:
> Dear Jamie,
>
> I agree that there is no canonical choice of horizontal composition of
> pseudonatural transformations, but that the various possible choices
> are, in a suitable sense, equivalent.
>
> Whether or not you should be bothered by this I can't really say. But
> perhaps it's worth pointing out that there are various different possible
> descriptions of the structure of weak 3-category, and not all of them include
> a chosen horizontal composition of pseudonatural transformations. Some,
> particularly, the simplicial approaches, include *no* choices of compositions.
> Others include some choices of composition, but not this particular one.
> For example, the notion of Gray-category does include chosen composition
> of 1-cells, and vertical composition of 2-cells, but does not include a chosen
> horizontal composition of 2-cells.
>
> Best wishes,
>
> Steve Lack.
>

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