Philippe Gaucher writes:
Hello,
Given a category C, I am interesting in the category DC defined as follows: an object is a functor F:I-->C where I is a small category, and a morphism from F:I-->C to G:J-->C is a functor from phi:I-->J together with a natural transformation from F to G o phi.
If C is locally small (resp. complete, cocomplete, cartesian closed), then so is DC (see CTGDC vol XVIII-4 (1977) "Decompositions et lax-completions", Guitart et Van Den Bril).
My question (for today) is : consider the "forgetful functor" from DC to Cat (the category of small categories) sending F:I-->C to I. Does this functor commute with colimits ? The paper above seems to claim that the commutativity holds if the colimit is a coproduct (see p376). What about general colimits ?
If C has a terminal object then the functor DC-->Cat has a right adjoint and so preserves any colimits which exist. However now many colimits do exist in DC (apart from coproducts). I don't have the paper of Guitart and Van Den Bril in front of me, but if I remember correctly, DC is the underlying category of a 2-category D'C, and D'C is the free completion of C (seen as a 2-category with no non-identity 2-cells) under lax colimits. And the 2-functor D'C-->Cat preserves lax colimits. Steve Lack. 28-Nov-2002 16:35:02 -0400,1567;000000000000-00000000