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From: Tom Leinster <tl@maths.gla.ac.uk>
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Subject: Re: weak double categories?
Date: Thu, 27 Oct 2005 22:55:51 +0100
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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