Australian Category Seminar

Suppose $\partial$ : A - C is a fibration, A, C are finitely complete and $\partial$ is left exact. Various categories over C are defined: Mon_C(A), Hom_C(A, B), Idem_C(A). Write A_gpd for the category with the same objects as A and cartesian arrows. Call A wellpowered when each Mon_C(A)_gpd has a terminal object. Say A has small homs when each Hom_C(A,B) has a terminal object. Say A has small idempotency when each Idem_C(A) has a terminal object. Theorems: (i) Wellpowered implies Mon_C(A) is essentially small. (ii) Small homs iff small idempotency.