unital weak functor?

unital weak functor?

[John Baez]

On Thu, Sep 20, 2007 at 07:03:07AM +1000, Stephen Lack wrote:

> These are often called normal lax functors
> (or normal morphisms of bicategories).

I suggested "normalized", but "normal" is clearly better:
you normalize something to make it normal.

On a wholly different note - I hope people take a look at the
new videos by the Catsters.  They're using YouTube in an
interesting new way: to explain monads, adjunctions and the like.

For more info:

http://golem.ph.utexas.edu/category/2007/09/the_catsters_latest_hit_adjunc.html

Best,
jb

Re: unital weak functor?

[Joachim Kock]

>Josh Nichols-Barrer wrote:
>
>>Is there a name for a weak functor between bicategories which takes
>>identity 1-morphisms to identity 1-morphisms?

The term 'normalised' (already mentioned in the replies) goes back at
least to Grothendieck, SGA1, Exp VI: he calls a cleavage for a fibred
category E -> B normalised if the cartesian lift of each identity arrow
is an identity arrow -- this is exactly the condition for the
correponding pseudo-functor B^op -> Cat to preserve identity arrows
strictly.

Joachim.

unital weak functor?

[John Baez]

Josh Nichols-Barrer wrote:

> Hi everyone,
>
> Is there a name for a weak functor between bicategories which takes
> identity 1-morphisms to identity 1-morphisms?  "Unital weak functor"
> would
> seem an apt name, but if there is another with more precedent I'll just
> use that instead.

I don't think there's a standard term.

But, I like the term "normalized", since in certain circumstances weak
functors between
bicategories are described by cocycles in group cohomology, and the
cocycle is then
said to be "normalized" when the weak functor preserves identity
1-morphisms. It's an old
fact that every cocycle is equivalent to a normalized one, and this is
related to the fact
that every weak functor is isomorphic to a normalized one.

For more information on this, see:

Andre Joyal and Ross Street, Braided monoidal categories,
Macquarie Mathematics Report No. 860081, November 1986.
Also available at http://rutherglen.ics.mq.edu.au/~street/JS86.pdf

or for a pedagogical treatment, try section 8.3, "Classifying 2-groups
using
group cohomology", of this:

John Baez and Aaron Lauda, Higher-dimensional algebra V: 2-Groups,
Theory and Applications of Categories 12 (2004), 423-491.
Also available at http://arxiv.org/abs/math.QA/0307200

Best,
jb

RE: unital weak functor?

[Stephen Lack]

Dear Josh,

These are often called normal lax functors (or normal morphisms of bicategories).

Regards,

Steve.

-----Original Message-----
From: cat-dist@mta.ca on behalf of Josh Nichols-Barrer
Sent: Wed 9/19/2007 10:58 AM
To: categories@mta.ca
Subject: categories: unital weak functor?
 
Hi everyone,

Is there a name for a weak functor between bicategories which takes
identity 1-morphisms to identity 1-morphisms?  "Unital weak functor" would
seem an apt name, but if there is another with more precedent I'll just
use that instead.

Best,
Josh

Re: unital weak functor?

[Robin Houston]

On Tue, Sep 18, 2007 at 08:58:46PM -0400, Josh Nichols-Barrer wrote:
> Is there a name for a weak functor between bicategories which takes
> identity 1-morphisms to identity 1-morphisms?

They are usually called 'normal', at least by the Sydney school.

Robin

unital weak functor?

[Josh Nichols-Barrer]

Hi everyone,

Is there a name for a weak functor between bicategories which takes
identity 1-morphisms to identity 1-morphisms?  "Unital weak functor" would
seem an apt name, but if there is another with more precedent I'll just
use that instead.

Best,
Josh