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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