weak double categories?

weak double categories?

[John Baez]

Dear Categorists -

If you weaken the notion of 2-category you get the notion of
bicategory.  Has anyone tried to correspondingly weaken the
notion of double category, so that a bicategory is a special
sort of "weak double category" in analogy to the ways in which
a 2-category is a special sort of double category?  Did anyone
succeed?

Best,
jb

Re: weak double categories?

[Tom Leinster]

On Wed, 2005-10-26 at 13:08 -0700, John Baez wrote:

> If you weaken the notion of 2-category you get the notion of
> bicategory.  Has anyone tried to correspondingly weaken the
> notion of double category, so that a bicategory is a special
> sort of "weak double category" in analogy to the ways in which
> a 2-category is a special sort of double category?

At least three parties have done this:

- Bob Pare and collaborators

- at least one Australian of the Sean Carmody/Dominic Verity/Steve Lack
generation (calling them something like "double bicategories")

- me (section 5.2 of book).

There's the question of whether you weaken in just one direction or in
both.  I believe that parties 1 and 3 weaken in just one direction.  But
you write

> in analogy to the ways [*plural!*] in which a 2-category is a special
> sort of double category

so I guess you're after weakening in both directions.

Tom

Re: weak double categories?

[Richard Garner]

--On 26 October 2005 13:08 John Baez wrote:

> If you weaken the notion of 2-category you get the notion of
> bicategory.  Has anyone tried to correspondingly weaken the
> notion of double category, so that a bicategory is a special
> sort of "weak double category" in analogy to the ways in which
> a 2-category is a special sort of double category?  Did anyone
> succeed?

Yes, this has been done; I believe Dom Verity 
is the first person to do this, in his thesis. 
Grandis and Paré are the only people to have 
developed extensively aspects of their theory ([1] 
& [2]). Tom Leinster mentions them in passing (in 
[3] for example) -- they are the `representable' 
fc-multicategories, standing in the same relation 
to them as monoidal categories do to plain 
multicategories.

On my website [4] is my thesis "Polycategories" 
which contains a fair bit more on weak double 
categories, both further aspects of their theory 
and some applications; for those of a terser 
inclination, the edited highlights can be found in 
the two preprints "Double clubs" and 
"Polycategories via pseudo-distributive laws" on 
the same page.

Richard Garner

-----

[1] Marco Grandis & Robert Paré
Limits in double categories
Cah. Topol. Géom. Différ. Catég. 40 (1999), no. 3, 162--220; MR1716779 (2000i:18007)

[2] Marco Grandis & Robert Paré
Adjoints for double categories
Cah. Topol. Géom. Différ. Catég. 45 (2004), no. 3, 193--240.

[3] Tom Leinster
Higher operads, higher categories
http://arxiv.org/abs/math.CT/0305049

[4] Richard Garner
http://www.dpmms.cam.ac.uk/~rhgg2

Re: weak double categories?

[Marco Grandis]

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The definition of weak (or pseudo) double category is what you would =20
expect, once you know the strict case and the definition of =20
bicategory; think of a "pseudocategory object in Cat". It has =20
probably been in the categorical folklore since quite a while.

You can find it (including the machinery of lax double functors, =20
horizontal and vertical transformations, etc.) in two papers in =20
Cahiers, where part of the general theory of weak double categories =20
has been developed

    M. Grandis - R. Pare, Limits in double categories, Cah. Topol. =20
Geom. Diff. Categ. 40 (1999), 162-220,

    - -, Adjoints for double categories,  Cah. Topol. Geom. Diff. =20
Categ. 45 (2004), 193-240.

Other papers on (weak) double categories, many of them by Bob Pare et =20=

al., are referred to in the articles above.
In book form

    Tom Leinster, Higher operads, higher categories, Cambridge Un. =20
Press 2004,

has Section 5.2, devoted to weak double categories.

Best regards

Marco

Marco Grandis
Dipartimento di Matematica
Universit=E0 di Genova
Via Dodecaneso, 35
16146 Genova
Italy

e-mail: grandis@dima.unige.it
tel: +39 010 353 6805
http://www.dima.unige.it/~grandis/

On 26 Oct 2005, at 22:08, John Baez wrote:

> Dear Categorists -
>
> If you weaken the notion of 2-category you get the notion of
> bicategory.  Has anyone tried to correspondingly weaken the
> notion of double category, so that a bicategory is a special
> sort of "weak double category" in analogy to the ways in which
> a 2-category is a special sort of double category?  Did anyone
> succeed?
>
> Best,
> jb
>
>
>
>
>
>


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Re: weak double categories?

[John Baez]

Dear Categorists -

Thanks for all the helpful replies!

Tom Leinster wrote:

> I guess you're after weakening in both directions.

Yes, I need weakening in both directions for my particular
application (to quantum gravity, in fact).  So, I need to
get ahold of Dominic Verity's thesis.

Best,
jb

Re: weak double categories?

[Ross Street]

[Note from moderator: message resent, may have been transmitted
incorrectly.]

Dominic Verity's PhD thesis did that (amongst other things) for some
very good reasons.
---Ross

On 27/10/2005, at 6:08 AM, John Baez wrote:

> If you weaken the notion of 2-category you get the notion of
> bicategory.  Has anyone tried to correspondingly weaken the
> notion of double category, so that a bicategory is a special
> sort of "weak double category" in analogy to the ways in which
> a 2-category is a special sort of double category?  Did anyone
> succeed?