Australian Category Seminar

A morphism $p \colon X \to Y$ in a category $\mathcal{C}$ with pullbacks is called powerful (or exponentiable) if the functor $p^* \colon \mathcal{C}/Y \to \mathcal{C}/X$, defined by pulling back along $p$, has a right adjoint. Giraud and Conduché characterized the powerful functors (i.e., the powerful morphisms in $\mathcal{C} = \mathbf{Cat}$) by explicit conditions. In this talk, I will present a proof of the equivalence of the powerfulness and the Giraud–Conduché condition for a functor, using simplicial sets.

(Based on discussion with Steve Lack.)