From: Reinhard Brger, Prof. Pumpl n <Reinhard.Boerger@FernUni-Hagen.de> Since the set of maps from a non-empty set to the empty set is empty and since Ab x Ab\op has no empty hom-sets, there is no full embedding from Set to Ab x Ab\op. Does that answer your question? Not if I understand you, for two reasons. First, you said the word "hom-set" which isn't allowed in Ab---you have to say homgroup (or homab). There being no empty group, if Set embeds in Ab then the empty set must embed as a nonempty group. Emptiness in the sense definable via the forgetful functor U:Ab->Set is not a predicate known to be in Ab. Emptiness needs to be defined in terms of the Abelian group representing the empty set. You have to analyze any proposed embedding by treating both the objects and the homobjects between them as Abelian groups. The second problem is my fault. I noticed yesterday that my definition (repeated below, near the end) is too weak. As a simple counterexample, groups are well known to be definable in Set, but Grp certainly does not fully embed as a category in the closed self-dual category Set + Set\op (= Chu(Set,0)), which embeds Set as a closed category (that is, the embedding respects tensor products, not necessarily cartesian products). The objects are just sets and antisets, clearly nowhere to store Grp. I was basing my definition on two theorems of mine. 1. Every category of relational structures, and hence every full subcategory thereof definable by any wacky language you want, far wilder and less computable than first order logic, fully embeds in Chu(Set,2^k) where k is the total arity of the structure. To determine total arity k of a multisorted multirelation structure, form the disjoint union of all the carriers, add one unary predicate per carrier as a marker, and then form the "natural join" of all the relations. The embedding works for total arities up to any ordinal; for a proof look near the end of http://boole.stanford.edu/gamut.ps.gz 2. Every small concrete category embeds in Chu(Set,K) for some set K. Proof: Take K to be the disjoint union of the carriers of the objects, and proceed as in http://boole.stanford.edu/embed.ps.gz On the strength of these two theorems I speculated that any method of defining a category that confined its language to that of V would embed in Chu(V,k) for a suitable object of k. What does "confine language" mean? Terrible things! It might even entail giving up conventional equality in favor of weighted limits when V is not cartesian closed, as with Ab. Some of you might enjoy a world where equality didn't exist, not in the sense you have to settle for in Set where you have to convince skeptical students that equality doesn't exist, but in the more pragmatic sense that no one else in that world uses it either, because it really doesn't exist! What I forget to allow for in my definition was the variability of k. Larger k lets you define more. k functions as a simple but effective and pure notion of signature. Pure in the sense that the 256 colors on your monitor are the clean signature of pixel depth, purified of the grubby representational issues implied by talking instead of the 8 pixel planes as though each plane were a real thing. This representation is clearly meaningless for people whose computers have 6 pixel planes coding 729 colors in ternary. I had foolishly thought that the essence of the Chu construction could be purified to a statement about duality, but had clean forgotten to allow for the possibility that a definitional method might cunningly think to use language to express the definition! Just how little language can be present in a self-dual category embedding Set can be can be seen from the example above of Chu(Set,0), a self-dual category consisting *only* of sets and antisets (aka CABA's if you ignore concreteness issues). The sets are realized as the inconsistent Chu spaces (A,0), the antisets as the empty Chu spaces (0,A), with (0,0) as the one set that is also an antiset. (Obviously this doesn't work for what Dusko Pavlovic calls "little Chu," no repeated rows and columns.) Chu(Set,0) understood skeletally consists of just the "dynamic integers," thinking of (0,A) as minus (A,0). The only subcategories of this sparse category have as their objects some sets and some antisets. This does not even embed Set\op x Set = Chu(Set,1), which also is too small for anything much, e.g. no posets, which enter at Chu(Set,2). I see nothing for it for now but to be upfront about putting Chu in the definition. Old definition. A category C is *definable in* a closed category V when it embeds fully (as a category) in every self-dual closed category embedding V (as a closed category). New definition 1. A category C is *definable in* a closed category V when it embeds fully in the category Chu(V,k) for some k in ob(V). This definition steers between the Scylla of arbitrarily large self-dual categories and the Charibdis of self-dual categories barely bigger than V. It is also a tad shorter. But there is *still* a problem. In Set it is not hard to define Chu(Set,*) as a colimit of Chu(Set,K1) < Chu(Set,K2) < ... For example the category of hypergraphs described in @Article( DW80, Author="D{\"o}rfler, W. and Waller, D.A.", Title="A category-theoretical approach to hypergraphs", Journal="Archives of Mathematics", Volume=34, Number=2, Pages="185-192", Year=1980) does not to my knowledge embed in any Chu(Set,K) for any fixed K. (This is an open question, I imagine mainly because no one has looked hard at it.) But any *small* subcategory of it does, by the general result above that *every* small concrete category embeds in Chu(Set,K) for some K. This too is easily fixed. There is a *nonclosed* category Chu(Set,Set). The definition looks no different from the usual definition of Chu, except that in all small interactions (no proper class of participants), all Chu spaces involved are taken to be over the colimit of all their K's. The only failure of this category to be closed is its lack of tensor unit 1 and its dual _|_ (I've switched to Girard's notation because these days it makes a lot more sense to me). The rest of linear logic is fine; furthermore the absence of 1 and _|_ removes the odd-ball property 1@1 |- 1 that Lafont and Streicher noticed. With that and MIX gone, about all LL should need now is induction, and Chu(Set,Set) should then be a complete model of LL without multiplicative units. If there is any justice, all this should generalize to Chu(V,V) by arranging to embed the k's of any interaction of V-enriched Chu spaces in the appropriate colimit of them in V before letting them interact. This leads to the even shorter: New definition 2. A category C is *definable in* a closed category V when it embeds fully in the category Chu(V,V). In view of Dusko Pavlovic's paper on the couniversality of Chu, about the only improvement likely for this definition (barring further bugs, sigh) is to replace the explicit Chu construction by its abstract properties. The effect will be the same either way, though the definition is presumably not going to get any shorter that way! One residual question is how to do enriched category theory from scratch in V. Max, help! Vaughan Pratt