Since I wrote some first thoughts the other night about Tom Leinster's question on the above, I've had some second thoughts which are perhaps a little more sensible, and may remove some of the mystery from these strange critters (which may be quite beautiful - I've just seen Tom's example of 10 Dec.). To get the notation straight, let G be an abelian monoid (perhaps a group), and V_o the category with one object * and V_o(*,*) = G. Then V_o underlies a symmetric monoidal closed category V with one object *, with tensor product o given on objects by *o* = * and on maps by fog = fg, with unit object I = *, and with internal-hom given on objects by [*,*] = *, and on maps by [f,g] = fg. So we can speak of V-categories, V-functors, and V-natural transformations at the level of my old paper with Eilenberg, incloding the ordinary category A_o underlying a V-category A. However all the richer theory of V-categories etc., as in my book on the subject, needs completeness of V_o for the definition of functor-categories, and hence for limit- and colimit-notions, Kan extension, and so on; as well as cocompleteness too, for these to work well. So, to this extent, V is a lousy closed category, being neither complete nor cocomplete. However there is a cure for incompleteness, called "completion"; although "cocompletion" works more smoothly. So let us embed V_o by Yoneda in its free cocompletion, the ordinary functor-category [V_o^op, Set]. We don't need the "op" here, since G is abelian. This functor-category is nothing but the presheaf category of G-sets (sets with an action of G, with the usual axioms (fg)x = f(gx) and 1x = x). This has a cartesian-closed structure, but forget that; it also has a symmetric monoidal closed structure arising from that on V using Day's convolution process. This is nothing but the Linton- -type s.m.closed structure where the tensor-product A o B represents the bi-homomorphisms out of A x B, and the internal-hom [A,B] is the G-set of all homomorphisms of G-sets from A to B. Explicitly, A o B is the quotient of A x B by the relation (fx,y) = (x,fy). Let us call THIS s.m.closed category W. Then V is embedded in W by Yoneda, and the image in W of * is the G-set G itself, seen as a G-set using its own multiplication - the "regular representation". So we may see V as this "part" of W. Now a V-category is nothing but a W-category whose hom-objects all happen to lie in V (which is a sub-monoidal category of W). Such a category A, with objects a,b, c, and so on, no longer need be said to have each A(a,b) equal to *, but instead to have A(a,b) = G. So the V-categories are nothing but these very special W-categories, and W is a highly-respectable s.m.closed category, first cousin to R-Mod for a commurative ring R. In fact, there is a "free" W-category F(B) on any ordinary category B; it has the same objects as B, and (F(B))(a,b) is the free G-set on the set B(a,b). The V-categories are just those W-categories of the form F(B) where the ordinary category B is CHAOTIC; that is to say, each B(a,b) is a singleton. So at least these nice new objects have a kind of legitimate origin. Max Kelly.