* A double bicategory of cobordisms with boundary
@ 2006-12-18 8:23 John Baez
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From: John Baez @ 2006-12-18 8:23 UTC (permalink / raw)
To: categories
Dear Categorists -
Here's an excerpt from "week242".
.....................................................................
Also available as http://math.ucr.edu/home/baez/week242.html
December 2, 2006
This Week's Finds in Mathematical Physics (Week 242)
John Baez
This week I'd like to talk about a paper by Jeffrey Morton. Jeff
is a grad student now working with me on topological quantum field
theory and higher categories. I've already mentioned his work on
categorified algebra and quantum mechanics in "week236". He'll be
be finishing his Ph.D. thesis in the spring of 2007 - and as usual,
that means he's already busy applying for jobs.
[stuff deleted]
10) Jeffrey Morton, A double bicategory of cobordisms with corners,
available as math.CT/0611930.
People have been talking a long time about topological quantum field
theory and higher categories. The idea is that categories, 2-categories,
3-categories and the like can describe how manifolds can be chopped into
little pieces - or more precisely, how these little pieces can be glued
together to form manifolds. Then the problem of doing quantum field
theory on some manifold can be reduced to the problem of doing it on
these pieces and gluing the results together. This works easiest if
the theory is "topological", not requiring a background metric.
There's a lot of evidence that this is a good idea, but getting the details
straight has proved tough, even at the 2-category level. This is what
Morton does, in a rather clever way. Very roughly, his idea is to use
something I'll call a "weak double category", and prove that these:
(n-2)-dimensional manifolds
(n-1)-dimensional manifolds with boundary
n-dimensional manifolds with corners
give a weak double category called nCob_2. The proof is a cool mix of
topology and higher category theory. He then shows that this particular
weak double category can be reinterpreted as something a bit more
familiar - a "weak 2-category".
In the rest of his thesis, Jeff will use this formalism to construct
some examples "extended TQFTs", which are roughly maps of weak 2-categories
Z: nCob_2 -> 2Vect
where 2Vect is the weak double category of "2-vector spaces". He's
focusing on some extended TQFTs called the Dijkgraaf-Witten models,
coming from finite groups.
But, he's also thought about the case where the finite group is
replaced by the Lie group SU(2). In this case we get something a
lot like an extended TQFT, but not quite, called the Ponzano-Regge
model. In this case of 3d spacetime, this is a nice theory of
quantum gravity. And, as I hinted back in "week232", we can let 2d
space in this model be a manifold with *boundary* by poking little
holes in space - and these holes wind up acting like particles!
So, we get a relation like this:
(n-2)-dimensional manifolds MATTER
(n-1)-dimensional manifolds with boundary SPACE
n-dimensional manifolds with corners SPACETIME
which is really quite cool.
It would be fun to talk about this. However, to understand Morton's
work more deeply, you need to understand a bit about "weak double
categories". He explains them quite nicely, but I think I'll spend
the rest of this Week's Finds giving a less detailed introduction,
just to get you warmed up.
This chart should help:
BIGONS SQUARES
LAWS HOLDING strict strict
AS EQUATIONS 2-categories double categories
LAWS HOLDING weak weak
UP TO ISOMORPHISM 2-categories double categories
2-categories are good for describing how to glue together 2-dimensional
things that, at least in some abstract sense, are shaped like *bigons*.
A "bigon" is a disc with its boundary divided into two halves. Here's
my feeble ASCCI rendition of a bigon:
f
--->---
/ \
/ || \
X o ||B o Y
\ \/ /
\ /
--->---
g
The big arrow indicates that we think of the bigon B as "going from"
the top semicircle, f, to the bottom semicircle, g. Similarly, we
think of the arcs f and g as going from the point X to the point Y.
Similarly, double categories are good for describing how to glue together
2-dimensional gadgets that are shaped like *squares*:
f
X o---->----o X'
| |
g v S v g'
| |
Y o---->----o Y'
f'
Both 2-categories and double categories come in "strict" and "weak"
versions. The strict versions have operations satisfying a bunch of
laws "on the nose", as equations. In the weak versions, these laws
hold up to isomorphism whenever possible.
A few more details might help....
A 2-category has a set of objects, a set of morphisms f: X -> Y going
from any object X to to any object Y, and a set of 2-morphisms T: f => g
going from any morphism f: X -> Y to any morphism g: X -> Y. We can
visualize the objects as dots:
o
X
the morphisms as arrows:
f
X o---->----o Y
and the 2-morphisms as bigons:
f
--->---
/ \
/ || \
X o ||B o Y
\ \/ /
\ /
--->---
g
We can compose morphisms like this:
f g fg
o---->----o---->----o gives o--->---o
X Y Z X Y
We can also compose 2-morphisms vertically:
f f
---->---- --->---
/ S \ / \
/ g \ / \
X o ----->----- o Y gives X o ST o Y
\ T / \ /
\ / \ /
---->---- --->---
h h
and horizontally:
f f' ff'
--->--- --->--- --->---
/ \ / \ / \
/ \ / \ / \
X o S o T o Z gives X o S.T o Y
\ / \ / \ /
\ / \ / \ /
--->--- --->--- --->---
g g' gg'
There are also a bunch of laws that need to hold. I don't want to
list them; you can find them in Jeff's paper (also see "week80").
I just want to emphasize how a strict 2-category is different from
a weak one.
In a strict 2-category, the composition of morphisms is associative
on the nose:
(fg)h = f(gh)
and there are identity morphisms that satisfy these laws on the nose:
1f = f = f1
In a weak 2-category, these equations are replaced by 2-isomorphisms - that
is, invertible 2-morphisms. And, these 2-isomorphisms need to satisfy new
equations of their own!
What about double categories?
Double categories are like 2-categories, but instead of bigons, we have
squares.
More precisely, a double category has a set of objects:
o
X
a set of horizontal arrows:
f
X o---->----o X'
a set of vertical arrows:
X o
|
g v
|
Y o
and a set of squares:
f
X o---->----o X'
| |
g v S v g'
| |
Y o---->----o Y'
f'
We can compose the horizontal arrows like this:
f f' f.f'
o---->----o---->----o gives o--->---o
X Y Z X Y
We can compose the vertical arrows like this:
X o
|
g v o
| |
Y o gives gg' v
| |
g' v o
|
Y o
And, we can compose the squares both vertically:
f
X o---->----o X'
| | f
g v S v g' X o---->----o X'
| | | |
Y o---->----o Y' gives gh v SS' v g'h'
| | | |
h v S' v h' Z o---->----o Z'
| | f'
Z o---->----o Z'
f'
and horizontally:
f Y g f.g
X o---->----o---->----o Z X o---->----o Z
| | | | |
h v S v S' v h' h v S.S' v h'
| | | | |
X' o---->----o---->----o Z' X' o---->----o Z'
f' Y' g' f'.g'
In a strict double category, both vertical and horizontal composition
of morphisms is associative on the nose:
(fg)h = f(gh) (f.g).h = f.(g.h)
and there are identity morphisms for both vertical and horizontal
composition, which satisfy the usual identity laws on the nose.
In a weak double category, we want these laws to hold only up to
isomorphism. But, it turns out that this requires us to introduce
bigons as well! The reason is fascinating but too subtle to explain
here. I didn't understand it until Jeff pointed it out. But, it
turns out that Dominic Verity had already introduced the right concept
of weak double category - a gadget with both squares and bigons - in
*his* Ph.D. thesis a while back:
11) Dominic Verity, Enriched categories, internal categories, and
change of base, Ph.D. dissertation, University of Cambridge, 1992.
Interestingly, if you weaken *only* the laws for vertical composition,
you don't need to introduce bigons. The resulting concept of
"horizontally weak double category" has been studied by Grandis and Pare:
12) Marco Grandis and Bob Pare, Limits in double categories, Cah.
Top. Geom. Diff. Cat. 40 (1999), 162-220.
Marco Grandis and Bob Pare, Adjoints for double categories, Cah.
Top. Geom. Diff. Cat. 45 (2004), 193-240. Also available at
http://www.dima.unige.it/~grandis/rec.public_grandis.html
and more recently by Martin Hyland's student Richard Garner:
13) Richard Garner, Double clubs, available as math.CT/0606733
and Tom Fiore:
14) Thomas M. Fiore, Pseudo algebras and pseudo double categories,
available as math.CT./0608760.
At this point I should admit that the terminology in this whole
field is a bit of a mess. I've made up simplified terminology
for the purposes of this article, but now I should explain how it
maps to the terminology most people use:
ME THEM
strict 2-category 2-category
weak 2-category bicategory
strict double category double category
weak double category double bicategory
horizontally weak double category pseudo double category
Verity used the term "double bicategory" to hint that his gadgets
have both squares and bigons, so they're like a blend of double
categories and bicategories. It's a slightly unfortunate term, since
experts know that a double category is a category object in Cat, but
Verity's double bicategories are not bicategory objects in BiCat.
Morton mainly uses Verity's double bicategories - but in the proof of
his big theorem, he also uses bicategory objects in BiCat.
There's a lot more to say, but I'll stop here and let you read the
rest in Jeff's paper!
----------------------------------------------------------------------
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