Subject: categories: Question of Tom Leinster of 26 Nov: one-object closed categories

From: [email-masked] (Max Kelly)
Date: Thu, 10 Dec 1998 23:23:38 +1100 (EST)

Tom observed that an abelian monoid is a symmetric monoidal closed
category with one object, and asked whether anyone had studied categories
enriched in such a closed category.

Eilenberg and I, in our long article [Closed categories, in Proc. Conf. on
Categorical Algebra (La Jolla, 1965), Springer-Verlag 1966, 421 - 562]
remarked in our "Examples" section (Ch.4, Section3, page 553) that these
are, to within isomorphism, the only one-object s.m.closed categories. The
odd thing is that we don't seem to have looked at V-categories for such a
V - perhaps we did not want to give trivial-looking examples, although we
gave other little examples such as Heyting algebras.

The use we made of such a V arising from an abelian monoid M was to give an
interesting but unusual example of a monoidal functor. We observed that a
monoidal functor f: V --> Ab was the same thing as an M-algebra, commutative
precisely when the monoidal functor f is symmetric.

Anyway, I had a brief look at V-categories for such a V tonight, but with
too few details so far to say much about them before bedtime. Queer little
creatures, aren't they? A V-category A has objects a, b. c. and so on, but
each A(a,b) is the unique object * of V. All the action takes place at the
level of j: I --> A(a,a) and M: A(b,c) o A(a,b) --> A(a,c). Sorry to use the
traditional M for composition, when it was the monoid. Let the monoid be G.
When G is an abelian group, the M and j seem to be determined by elements
N_a,b depending on two objects of A. There is more meat in a V-functor. I
look forward to working through this - as I suppose Tom has done - and
looking especially at the cases where G is a two-element monoid. The
underlying ordinary category A_o of the V-category A seems to be an odd beast.

While I don't remember ever working through this, it still rings a bell.
Somewhere I have seen arrows decorated with something like numbers (elements
of G ?). Ross Street and his colleagues at Macquarie often use the word
"suspension" for the process of seeing an abelian as a one-object monoidal
category, a monoidal category as a one-object bicategory, and such things -
I don't know the full definition of "suspension"; but they and numerous
others are well aware of, for instance, the tricategory with one object,
whose 1-cells are commutative rings, whose 2-cells are two-sided algebras,
whose 3-cells are bimodules, and whose 4-cells are bimodule-homomorphisms.
Clearly I got that wrong; I think the objects should have been the 
commutative rings. Anyway, people doing this work are very likely to have
seen and used such one-object V.

Still, it looks like fun; and I've never seen an exposition of V-Cat for
the case where G is the two-element group. I'm taking it to bed.

Max Kelly.