Interpreting category-valued presheaves
Dear categorists, I could not find an answer to the following, so I need your help: Given a category C we know that the category of set-valued presheaves on C may be interpreted as a free cocompletion of C. More precisely, if we denote PSh(C) the presheaf category, then colimit-preserving functors from PSh(C) to a cocomplete category D correspond to functors from C to D (via Yoneda extension). I am interested in an interpretation of category-valued presheaves. Is there some sort of 2-categorical version of the above fact? Best regards, Redi Haderi [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Redi Haderi Max Kelly's book ``Basic concepts of enriched category theory'' generalises this whole Kan business to V-categories for a decent base V. So, for the case V = Cat, you obtain the result for 2-categories. I say ``the result'', but of course, for 2-categories there are versions for weaker structures (bicategories, pseudofunctors, and so on). The Cat-enriched version is the strict case which you seem to want. Regards, Ross Street On 16 Feb 2020, at 6:11 AM, Redi Haderi <haderiredi@gmail.com<mailto:haderiredi@gmail.com>> wrote: Given a category C we know that the category of set-valued presheaves on C may be interpreted as a free cocompletion of C. More precisely, if we denote PSh(C) the presheaf category, then colimit-preserving functors from PSh(C) to a cocomplete category D correspond to functors from C to D (via Yoneda extension). [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Is not the result you seek (4.56) of Kelly's Basic Concepts of Enriched Category Theory, in the case where $\mathcal V = \mathbf Cat$ ? On Sun, Feb 16, 2020, 11:33 Redi Haderi <haderiredi@gmail.com> wrote:
Dear categorists,
I could not find an answer to the following, so I need your help:
Given a category C we know that the category of set-valued presheaves on C may be interpreted as a free cocompletion of C. More precisely, if we denote PSh(C) the presheaf category, then colimit-preserving functors from PSh(C) to a cocomplete category D correspond to functors from C to D (via Yoneda extension).
I am interested in an interpretation of category-valued presheaves. Is there some sort of 2-categorical version of the above fact?
Best regards, Redi Haderi
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
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Keith Harbaugh -
Redi Haderi -
Ross Street