Transfer of limits across functors

Ross Street – 4 December 1996

This was a talk on elementary category theory. By examining and slightly extending a little-known result of John Isbell appearing in ["Structure of categories" Bull. AMS 72 (1966) 619-655], we deduced a well-known result of Bob Paré and discussed an unpublished (I think) result of Peter Freyd. The main result was that, if we have functors i : A --> X, r : X --> A, a natural transformation eta : 1 --> r i, and, if idempotents split in A, then A has a terminal object provided X does. Notice that eta does not need to be invertible. This gives the usual results about transfer of limits and colimits to a full reflective subcategory. It also gives the result of Paré that only one of the adjunction triangles implies the construction of an adjoint via splitting an idempotent. The result of Freyd was the Most (?) General Adjoint Functor Theorem involving "uniform continuity" of a functor.