If has finite limits, then functors commuting with these finite limits are precisely what are called left exact functors. Sometimes they are called “finitely continuous.”
The archetypical example is the Set-valued hom-functor: its continuity in both arguments is indeed equivalent to the very definition of limit: for a diagram and , the covariant hom-functor satisfies by definition of limit
Topologists sometimes use “continuous functor” to mean a functor enriched over Top, since a functor between topologically enriched categories is enriched iff its actions on hom-spaces are continuous functions.
H. Bass in his treatment of K-theory uses the older term ‘right continuous functor’ for the dual notion of cocontinuous functor in a version which is additive. If the domain of an additive functor which commutes with direct sums is a cocomplete category, then the functor automatically has right adjoint. Following this fact, some people in ring theory and noncommutative geometry use the simple term ‘continuous functor’ for a functor with a right adjoint (even if the domain abelian category is not cocomplete). In general, of course, this is just a bit more than cocontinuous in the standard sense.
Left I could understand, but right? —Toby
The way I rewrote it explains it. It is unfortunate that the Eilenberg-Watts theorem treated in Bass was using only right adjoint functors so later they dropped word right. – Zoran
Revised on November 7, 2010 18:12:22
by Mike Shulman