presentable vs. locally presentable

presentable vs. locally presentable

[Michael Shulman]

Dear all,

I wonder how the following 2-categories are related?

1. The 2-category of locally presentable categories and left adjoints.

2. The full sub-2-category of the 2-category of cocomplete categories
and cocontinuous functors, consisting of those categories which are
small bicolimits (in that 2-category) of diagrams of presheaf
categories on small categories.

3. Like (2), but consisting only of those categories which are
codescent objects of diagrams of presheaf categories on small
categories.

Since every locally presentable category is a small-orthogonality
class in a presheaf category, I think it follows that it is a
coinverter  (in the 2-category of cocomplete categories) of a
transformation between two presheaf categories.  Thus (1) is a
subcategory of (2), and a full subcategory by the adjoint functor
theorem.  It is of course clear that (3) is a full subcategory of (2),
and I think they should be the same, since bicolimits can be
constructed from coproducts, copowers, and codescent objects, while
small coproducts and copowers of presheaf categories are again
presheaf categories.  It seems likely to me that (1) and (2) are also
the same; has anyone studied this question?

I am wondering about this because some people have recently started
using "presentable category" as a synonym for "locally presentable
category," with (as far as I understand) the intuition that the above
description of a locally presentable category as a coinverter is a
"presentation" of it -- in contrast with the meaning of "locally
presentable category" that it is the *objects* of the category which
are presentable (in the sense that homming out of them preserves
sufficiently highly filtered colimits).  I would find the most
intuitive sort of "presentation" for an object of a 2-category to be a
codescent object of a diagram of free objects, rather than a
coinverter; for instance, that is the sort of presentation that seems
to arise in pseudo-monadicity theorems.  So I wondered whether locally
presentable categories are also the categories that can be "presented"
as codescent objects of diagrams of presheaf categories on small
categories (the latter being, of course, the free cocomplete
categories on small categories), and the step to all small colimits is
natural.

Regards,
Mike


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presentable vs. locally presentable

[Joyal, André]

Dear Michael,

You are obviously raising interesting questions.
I may have and answer but I dont know!
Excuse my ignorance, but I do not understand your language.

1)What is a bicolimit?

2)What is a codescent object?

3)What are the pseudo-monadicity theorems?

Best,
André


-------- Message d'origine--------
De: categories@mta.ca de la part de Michael Shulman
Date: lun. 24/05/2010 14:42
À: categories
Objet : categories: presentable vs. locally presentable
 
Dear all,

I wonder how the following 2-categories are related?

1. The 2-category of locally presentable categories and left adjoints.

2. The full sub-2-category of the 2-category of cocomplete categories
and cocontinuous functors, consisting of those categories which are
small bicolimits (in that 2-category) of diagrams of presheaf
categories on small categories.

3. Like (2), but consisting only of those categories which are
codescent objects of diagrams of presheaf categories on small
categories.


...


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RE: presentable vs. locally presentable

[Marta Bunge]

Dear Mike,
Very relevant to your questions is the paper 
Marta Bunge and Aurelio Carboni,The symmetric toposJ. Pure and Applied Algebra 105 (1995) 233-249
If you still have questions after looking at it, please ask. I cannot answer extensively now.
Best wishes, Marta

************************************************
Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics 
McGill UniversityBurnside Hall, Office 1005
805 Sherbrooke St. West
Montreal, QC, Canada H3A 2K6
Office: (514) 398-3810/3800  
Home: (514) 935-3618
marta.bunge@mcgill.ca 
http://www.math.mcgill.ca/~bunge/
************************************************



> Date: Mon, 24 May 2010 14:42:37 -0400
> Subject: categories: presentable vs. locally presentable
> From: shulman@math.uchicago.edu
> To: categories@mta.ca
> 
> Dear all,
> 
> I wonder how the following 2-categories are related?
> 
> 1. The 2-category of locally presentable categories and left adjoints.
> 
> 2. The full sub-2-category of the 2-category of cocomplete categories
> and cocontinuous functors, consisting of those categories which are
> small bicolimits (in that 2-category) of diagrams of presheaf
> categories on small categories.
> 
> 3. Like (2), but consisting only of those categories which are
> codescent objects of diagrams of presheaf categories on small
> categories.
> 

...


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