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.