On Fri, Apr 17, 2009 at 8:06 PM, Andrew Salch <asalch@math.jhu.edu> wrote:
In a recent paper of Connes and Consani, they consider the following "pasting along an adjunction":
[...]
I would like to know if this "pasting along an adjunction" is a special case of some more general construction already known to category theory, and if basic properties of pasting along an adjunction have already been worked out and written down somewhere.
In section 2.3.1 of "Higher Topos Theory" http://arxiv.org/abs/math.CT/0608040 Jacob Lurie motivates the notion of "inner fibrations" and of Cartesian fibrations of (oo,1)-categories as a generalization of this "pasting" construction. "Pasting" along any bifunctor C^op x D --> Set is the same as having an inner fibration over the interval (which is an arbitrary functor for 1-categories), and the particular "pasting" that you mention, over hom_D(F(-),-) coming from a functor F : C \to D, gives a Cartesian fibration over the interval (top of p. 88, leading over to section 2.4). Best, Urs