Colimits in 2-dimensional slices

We generalize to dimension 2 the well-known fact that a colimit in a 1-dimensional slice is precisely the map from the colimit of the domains which is induced by the universal property. We show two different approaches to this; a more intuitive one, based on the reduction of the weighted 2-colimits to oplax normal conical ones, and a more abstract one, based on an original concept of colim-fibration and on an extension of the Grothendieck construction. We find the need to consider lax slices, and prove results of preservation, reflection and lifting of 2-colimits for the domain 2-functor from a lax slice. The preservation result is shown by proving a general theorem of F-category theory, which states that a lax left adjoint preserves appropriate colimits if the adjunction is strict on one side and is suitably F-categorical. Finally, we apply this theorem of preservation of 2-colimits to the 2-functor of change of base along a split Grothendieck opfibration between lax slices, after showing that it is such a left adjoint by laxifying the proof that Conduché functors are exponentiable. We conclude extending the result of preservation of 2-colimits for the change of base 2-functor to any finitely complete 2-category with a dense generator.

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