Subject: categories: V-categories where V "is" an abelian monoid.

From: [email-masked] (Max Kelly)
Date: Sat, 12 Dec 1998 19:31:36 +1100 (EST)

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.