Philippe Gaucher wrote:
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 ?
Thanks in advance for any help or any other pointer for DC. pg.
Th forgetful (dom functor) from DC to Cat preserves *lax* colimits (coproducts in Cat are such). Of course, this only makes sense if you regard DC as a 2-category. The reason for the preservation is naturally that the forgetful has a "lax right adjoint" R X |-> (pi' : X x C -> C) equipped with a lax transf \eta: 1 => R dom and a 2-natural transf \epsilon: dom R => 1 satisfying the triangle identities (these data gives a local adjunction, in the sense of Betti and Power). Here's a drafty little note with an application of the lax slice 2-cat DC to colimits http://maggie.cs.queensu.ca/chermida/papers/col-dec.ps Claudio 27-Nov-2002 12:21:48 -0400,2094;000000000000-00000000