Dear all, In section B1.1 of the Elephant, weighted limits in a 2-category are defined up to equivalence by pseudo cones. Let me say that strict weighted limits exist if we can replace pseudo cones by genuine cones. These limits are then defined up to a unique isomorphism. In the category CAT of categories (fill in universes if you don't want to run into meta categories) such strict limits exist. Let me say that a 2-category is strictly finitely 2-complete if such strict weighted limits exist for all finite (in an obvious sense) weighted categories. QUESTION: Let S be a category with finite limits. I denote by Cat(S) the 2-category of internal categories of S. Under which conditions on S is Cat(S) strictly finitely 2-complete? Best to all, Jean [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear all, Let me first thank John Powers and David Roberts for their answers. However they both say that if S is a category with finite limits the 2-category Cat(S) is what I called strictly finitely 2-complete. I doubt it. Could they, or anybody else, tell me how yo prove Cat(S) has strict inverters? The construction in the Elephant gives strict identifiers, and they do exist in Cat(S). But they don't coincide with inverters if we insist on strict notions. Look for example at what they are when S is the category of sets, where both constructions are possible but distinct. Best regards, Jean
Dear Jean,
Cat(S) admits powers by the arrow category {0 --> 1}, as one can form the internal arrow category by hand, for any internal category in finitely complete S. Cat(S) also admits products, pullbacks and a terminal object, so I believe under your standing assumption on S (finitely complete) Cat(S) is strictly finitely 2-complete.
Best regards, David
On 14 February 2017 at 01:11, Jean Benabou <jean.benabou@wanadoo.fr> wrote: Dear all,
In section B1.1 of the Elephant, weighted limits in a 2-category are defined up to equivalence by pseudo cones. Let me say that strict weighted limits exist if we can replace pseudo cones by genuine cones. These limits are then defined up to a unique isomorphism. In the category CAT of categories (fill in universes if you don't want to run into meta categories) such strict limits exist. Let me say that a 2-category is strictly finitely 2-complete if such strict weighted limits exist for all finite (in an obvious sense) weighted categories.
QUESTION: Let S be a category with finite limits. I denote by Cat(S) the 2-category of internal categories of S. Under which conditions on S is Cat(S) strictly finitely 2-complete?
Best to all,
Jean
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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Jean Benabou