I guess that in the category of R-modules over a commutative ring R, a module M has a (good) dual iff it is finitely generated projective iff the endo-functor functor Hom(M,-) preserves all colimits (M is *compact* in a strong sense).
Obviously this is correct. But, on the other hand, Rel is a compact closed category (also: V-Prof, for suitable choice of V). So it is not necessarily the case that every object of a compact closed category is small/finite/compact.
Indeed, but in this case it is the objects of the category which are "compact," not the category itself. So if this is the argument, then a more natural term would be "locally compact" (clashing with "locally small," of course, but agreeing with "locally presentable" categories in which all objects are presentable).
Hmmm, even that last point is pretty tenuous... A locally presentable category may have the property that every object is presentable, but the converse is false. For example, Sup (the category of complete lattices and supremum-preserving maps) is not locally presentable; but it is monadic over Set and therefore has the property in question.
(I am *not* proposing to *actually* use "locally compact" -- I don't want to introduce yet another name for something that already has at least four names, even if none of the existing four are optimal.)
I disagree with this line of argument: if good terminology can be found, it will kill off its rivals PDQ. In fact, I have not been able to stop myself from thinking about this issue, and would like to propose "simply closed category" as a replacement for "autonomous category" (in the sense of: monoidal category in which every object has a left and a right dual). The point is that such a monoidal category is (both left and right) closed; moreover, it is one in which the "closed structure" (i.e. the pair of internal homs) admits an unusually simple description. One possible objection, aside from that which Mike has already made, is that the word "simple" already has an established mathematical meaning. My rebuttal to this is that there are precedents for using an adverb independently of the corresponding adjective. For example, I see no connection between the "completely" in "completely positive map" and any of the standard meanings of "complete". Cheers, Jeff. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]