Subject: categories: commutative monoids
Date: Thu, 26 Nov 1998 19:56:18 +0000 (GMT)
From: Tom Leinster
A commutative monoid is a strict monoidal category with one object, so in
this sense one can talk about a category enriched in a commutative
monoid. Has anyone looked very hard at such things?
Thanks,
Tom
Date: Thu, 10 Dec 1998 23:23:38 +1100 (EST)
From: maxk@maths.usyd.edu.au (Max Kelly)
Subject: categories: Question of Tom Leinster of 26 Nov: one-object closed categories
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.
Subject: categories: Re: one-object closed categories
Date: Thu, 10 Dec 1998 19:20:03 +0000 (GMT)
From: Tom Leinster
> From: maxk@maths.usyd.edu.au (Max Kelly)
>
> 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.
>
[...]
>
> 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).
[...]
Since I asked the question I've found a few examples; they've all got
the same flavour about them, so I'll just do my favourite.
If V is the commutative monoid, then a V-enriched category is a set A plus
two functions
[-,-,-]: A x A x A ---> V
[-]: A ---> V
satisfying
[a,c,d] + [a,b,c] = [a,b,d] + [b,c,d]
[a,a,b] + [a] = 0 = [a,b,b] + [b]
for all a, b, c, d.
The example: let A be a subset of the plane. Choose a smooth path P(a,b) from
a to b for each (a,b) in A x A, and define [a,b,c] to be the signed area
bounded by the loop
P(a,b) then P(b,c) then (P(a,c) run backwards);
also define [a] to be
-(area bounded by P(a,a)).
(There's meant to be an orientation on the plane, so that areas can be
negative.) Then the equations say obvious things about area - don't think
I'm up to that kind of ASCII art, though.
Tom
Date: Sat, 12 Dec 1998 23:31:20 +0200 (EET)
From: Mamuka Jibladze
Subject: categories: Re: one-object closed categories
Concerning categories enriched in monoidal categories with a single
object: another example is given by cocycles. It can be presented in
various ways. For example, a "\v Cech style" version: given an open cover
of X by Ui and a (suitably normalized) \v Cech 3-cocycle c of this cover
with values in M, this enrichs in the evident way the category whose
objects are the i's, with hom(i,j) either a singleton or empty according
to inhabitedness of the intersection of Ui and Uj. Other variations
suggest things like morphisms of simplicial sets to the nerve of M
considered as a 2-category with a single 1-cell. This is related to
K(M,2)-torsors, etc. Quite probably there are several publications
exploiting this. At least cocycles with values in monoids rather than
groups certainly have been considered.
What I certainly have not seen is a backwards generalization: has anybody
considered analogs of K(M,2)-torsors for general enrichments? Would be
very interested in a reference.
Happy holidays to all!
Mamuka
Date: Sat, 12 Dec 1998 19:31:36 +1100 (EST)
From: maxk@maths.usyd.edu.au (Max Kelly)
Subject: categories: V-categories where V "is" an abelian monoid.
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.
From: Peter Selinger
Subject: Re: categories: Re: one-object closed categories
Date: Fri, 11 Dec 1998 22:15:22 -0500 (EST)
> From Tom Leinster:
>
> Since I asked the question I've found a few examples; they've all got
> the same flavour about them, so I'll just do my favourite.
>
> If V is the commutative monoid, then a V-enriched category is a set A plus
> two functions
> [-,-,-]: A x A x A ---> V
> [-]: A ---> V
> satisfying
> [a,c,d] + [a,b,c] = [a,b,d] + [b,c,d]
> [a,a,b] + [a] = 0 = [a,b,b] + [b]
> for all a, b, c, d.
A few remarks: In the case where V is an abelian group, the first
axiom already implies the other two if we define [a] = -[a,a,a].
Namely, by letting a=b in the first axiom, it follows that [a,a,c] is
independent of c.
If V is an abelian group, then one can get an example of the above
structure from an arbitrary map {-,-} : A x A ---> V by letting
[a,b,c] = {a,b}+{b,c}-{a,c} and [a] = -{a,a}. Tom's "area" example is
of this form.
In fact, if V is an abelian group, then *any* example of a V-enriched
category is (non-uniquely) of the form described in the previous
paragraph: Fix some x in A (if any), and define {a,b} = [a,b,x].
What about the non-group case? In general, [a,b,c] need not always be
invertible in V. In fact, [a,b,a] need not be invertible. For a simple
example of this, let V be the natural numbers and define
[a] = 0,
[a,b,c] = 0, if a=b or b=c,
1, if a,b,c pairwise distinct,
2, otherwise (i.e., if a=c but a,b distinct).
This indeed works.
Best wishes,
-- Peter Selinger
> The example: let A be a subset of the plane. Choose a smooth path P(a,b) from
> a to b for each (a,b) in A x A, and define [a,b,c] to be the signed area
> bounded by the loop
> P(a,b) then P(b,c) then (P(a,c) run backwards);
> also define [a] to be
> -(area bounded by P(a,a)).
> (There's meant to be an orientation on the plane, so that areas can be
> negative.) Then the equations say obvious things about area - don't think
> I'm up to that kind of ASCII art, though.