Date: Tue, 6 Aug 1996 19:02:15 -0300 (ADT) Subject: extensive stuff Date: Tue, 6 Aug 1996 14:24:15 -0400 From: Peter Freyd The phrase DISTRIBUTIVE CATEGORY is established as referring to a category with finite products and coproducts wherein coproducts distribute with products. The phrase EXTENSIVE CATEGORY refers to a cartesian category (that is, one with finite limits) with finite coproducts wherein coproducts are preserved by pullbacks. (Bill, they tell me you gave it this name because measurements tend to be valued in such. Right?) An equivalent definition of an extensive category is a cartesian category that's _locally_ distributive, that is, every slice category is distributive. I've recently finally been able to find the right expansion of what I've called "cartesian logic" (the syntax of cartesian categories) to what I guess I will have to call "extensive logic" (the syntax of extensive categories). Cartesian logic can be sloganized as the logic of _unique_ existentials. For extensive logic add _exclusive_ disjunctions. (Yes, all you purists, "sloganized" is in the OED. Actually, the American Heritage Electronic Dictionary would seem to define it as the result of turning something into a Scottish war cry.) With cartesian logic we first obtain a completeness theorem with respect to the semantics of cartesian categories (by constructing the free cartesian category from a given theory and noting what rules of inference are needed to make the construction work) and then we obtain a completeness theorem with respect to the "elemental semantics" by using the fact that set-valued representations for any small cartesian category are collectively faithful. By the time we're done we need that the representations reflect not just equality and isomorphisms but non-split-epis into non-epis. (The Cayley representations, of course, do all this.) Similarly for extensive logic. Here we need the fact that the set-valued representations of any small extensive category are collectively faithful, indeed, collectively reflect (not just equality and isomorphisms but) split epis. By "representation" I mean a functor that preserves finite limits and finite coproducts. It would not be enough to preserve just finite products and coproducts for the semantics. That is, I must work in the context of extensive categories not distributive categories. (Cayley, of course, no longer suffices.) Do distributive categories arise in nature that aren't extensive? The quickest artificial example is the full subcategory, *A*, of (*S*)x(*S*), where *S* is the category of sets. A pair is in *A* if either both X and Y are non-empty or both are empty. *A* is coreflective, hence cartesian. It's closed under the formation of products and coproducts, hence distributive (indeed, it's closed under the formation of exponentials, hence an exponential category). *A* is not an extensive category. <{a,b},{a,b}> is the coproduct of <{a},{a}> and <{b},{b}>. The intersection of each of these subobjects with <{a},{b}> is <{},{}> hence pulling back along the inclusion map of <{a},{b}> does not preserve coproducts. But I seem to have stumbled across a more natural example. In Cats and Alligators a pair of idempotents e, e' are said to be "neighbors" if ee'e = e and e'ee' = e'. So, let's understand a SEMIGROUP OF NEIGHBORING IDEMPOTENTS to mean a semigroup satisfying the further equations: xx = x, xyx = x. Note that as a consequence: xyz = (xzx)yz = x(z(xy)z) = xz. The category of semigroups of neighboring idempotents is a distributive category because, even better, it's an exponential category. Construct A => B in the naive way, that is, as the set of homomorphisms from A to B. The semigroup structure on A => B is given pointwise: for homomorphisms f,g:A -> B, define fg = \a.(fa)(ga). The equation xyz = xz forces fg to be a homomorphism. A homomorphism h:X*A -> B curries to a homomorphism h':X ->(A => B) where the naive definition works for h', to wit, h' = \x.(\a.f). The equation xx = x is just what's needed to see that h'x is a homomorphism for each x, and then to see that h'(xy) = (h'x)(h'y). To see that the category of semigroups of neighboring idempotents is not an extensive category consider the four-element semigroup with multiplication given by: a b c d ________ a |a c c a b |d b b d c |a c c a d |d b b d It is a coproduct (in the category of semigroups of neighboring idempotents) of its one-element subsemigroups {a}, {b}. (In other words, it's the free semigroup of neighboring idempotents generated by a and b.) Now intersect these generating subsemigroups with the one-element subsemigroup {c} to see that pulling back along the inclusion map of {c} does not preserve coproducts. Still here? Guess what. The two categories are equivalent and the equivalence carries one example to the other. Given an object, , in *A* turn XxY into a semigroup by defining = . This is how the equivlence gets from *A* to the category of neighboring idempotents. Getting back is left to the reader. Now the real question: how much of all this is already in Johnstone? Date: Wed, 7 Aug 1996 12:55:55 -0300 (ADT) Subject: Peter Freyd's letter of 6 Aug Date: Wed, 7 Aug 1996 17:44:00 +1000 (EST) From: Max Kelly I refer to Peter Freyd's interesting letter of 6 Aug concerning distributive and extensive categories. Peter, you have not got quite right the current nomenclature for these: the definitive account of their interconnexions is in [Carboni, Lack, Walters, Introduction to extensive and distributive categories, JPAA 84 (1983), 145-158]. Their name for your "extensive" is "lextensive", which I think is due to Bill Lawvere; it means "lex and extensive", where "lex' is used to mean "having all finite limits". (By the way, I absolutely detest this usage of "lex"; a CATEGORY cannot be left exact!) Their "extensive" categories have only finite COPRODUCTS as part of the structure; but these are to be such that the canonical A/a x A/b --> A/(a+b) is to be an equivalence of categories. The point is important because a MORPHISM of extensive categories need preserve only finite coproducts, not finite limits (as a morphism of lextensive categories must). So the 2-categories involved are quite different, and this affects the notion of free category-with- -structure. One other thing: those semigroups you discuss are what I called "middle-ignoring semigroups" in my paper with Pultr [ On algebraic recognition of direct-product decompositions, JPAA 12(1978), 207-224], where I showed them equivalent to pairs of sets with neither empty or both - only as the simplest and most trivial example of our extension of Michael Barr's result on algebras for the "n-th power monad" sending A to A^n in any category. The funny thing is that I spoke on this as your guest in the colloquium at Philadelphia in 1977. Anyway, the more general situation of that paper may give more examples of distributive categories - I haven't yet had time to think about it. The category of middle-ignoring semigroups, by the way, and more generally the category of algebras for the n-th power monad P_n on the category of sets, is symmetric monoidal closed by Fred Linton's old result, since this monad is commutative - at least I think it must be so, without stopping now to check it. About the nomenclature: do people agree that "lex" is really terrible? Peter Johnstone in Sussex recently called it something like a twice-dead metaphor - but now I forget what he wanted in its place - was it Peter Freyd's "cartesian" ? Max Kelly. Date: Wed, 7 Aug 1996 12:57:05 -0300 (ADT) Subject: Re: extensive stuff Date: Wed, 7 Aug 96 10:17 BST From: Dr. P.T. Johnstone >Now the real question: how much of all this is already in Johnstone? Not much of it, if you mean what is in Johnstone's published work, rather than in Johnstone's mind. But my paper "A syntactic approach to Diers' localizable categories" in Springer LNM 753 (the 1977 Durham Symposium proceedings) is relevant: in it I introduced what I then called "disjunctive logic", which is exactly what Peter now wants to call "extensive logic" (and I guess that's a better name). What I was doing there was to describe a class of theories whose model categories (in Sets) were just the "localizable categories" (aka multiply presentable categories) introduced by Yves Diers in his thesis: there is nothing about extensive categories in my paper, because I didn't know the concept at that time. However, I've known for some years now that extensive categories are exactly the class of categories in which this fragment of logic should be modelled -- but I haven't found a suitable opportunity to set this down in print. Incidentally, the corresponding class of sketches (those with arbitrary finite cones, but only discrete finite cocones) has been studied by many people -- see for example Barr & Wells (TTT), page 292. Peter Johnstone Date: Wed, 7 Aug 1996 12:57:56 -0300 (ADT) Subject: Extensive stuff Date: Wed, 7 Aug 96 10:41 BST From: Dr. P.T. Johnstone A quick PS: since "semigroups of neighbouring idempotents" satisfy the identity xyz = xz, they already have a name: they are what the semigroup-theorists call "rectangular bands". As such, they appear in my paper "Collapsed toposes and cartesian closed varieties" (J. Algebra 129 (1990); see top of p. 462), as the simplest nontrivial example of what I called a "commutative hyperaffine theory". The fact that they form a category equivalent to the two-valued collapse of Set x Set is in my paper (though it was known long before), as is the fact that this category is cartesian closed but not locally cartesian closed -- and from the proof that it's not lcc you can easily extract a proof that it's not extensive. Peter Johnstone Date: Wed, 7 Aug 1996 12:59:18 -0300 (ADT) Subject: Re: extensive stuff Date: Wed, 7 Aug 1996 09:40:48 -0300 From: RJ Wood Dear Peter Everybody keeps rediscovering your *A*. It probably belongs in Insights. Mike Barr's The Point of the Empty Set'' shows that the I-fold product functor, restricted to the full subcategory of *S*^I determined by the pure functors'', is monadic, in fact VTT. For I=2 the monad in question is TX=X^2 and a T-algebra is thus a set with a binary operation satisfying xx=x (xy)(zw)=xw (These are obviously equivalent to associativity and the equations you gave.) Anyway, this explains the equivalence of categories you mentioned. I certainly wasn't aware that *A* is distributive but not extensive. I'd just like to point out that *A* and its generalizations is also useful for showing that certain apparently multisorted algebraic categories are actually single sorted. For example, the obvious category of all modules over all rings is monadic over *A* and thus by the above also monadic over *S*. Others will have more to say on this. On a personal note, I first learned of *A* from my friend Kip Howlett who picked it up at some conference around 71. He realized that it was what I needed for my MSc thesis that involved M-sets with variable M and applications to automata theory. Anyway, sorting out *A* got me into category theory. Best regards RJ > The quickest artificial example is the full subcategory, *A*, of > (*S*)x(*S*), where *S* is the category of sets. A pair is > in *A* if either both X and Y are non-empty or both are empty. > *A* is coreflective, hence cartesian. It's closed under the formation > of products and coproducts, hence distributive (indeed, it's closed > under the formation of exponentials, hence an exponential category). > > *A* is not an extensive category. <{a,b},{a,b}> is the coproduct of > <{a},{a}> and <{b},{b}>. The intersection of each of these > subobjects with <{a},{b}> is <{},{}> hence pulling back along the > inclusion map of <{a},{b}> does not preserve coproducts. > > But I seem to have stumbled across a more natural example. In Cats and > Alligators a pair of idempotents e, e' are said to be "neighbors" > if ee'e = e and e'ee' = e'. So, let's understand a SEMIGROUP OF > NEIGHBORING IDEMPOTENTS to mean a semigroup satisfying the further > equations: > xx = x, > xyx = x. > > Note that as a consequence: xyz = (xzx)yz = x(z(xy)z) = xz. > > The category of semigroups of neighboring idempotents is a > distributive category because, even better, it's an exponential Date: Wed, 7 Aug 1996 13:01:16 -0300 (ADT) Subject: Re: extensive stuff Date: Wed, 7 Aug 1996 11:07:35 -0400 From: Peter Freyd Yikes! Paul Taylor has pointed out to me that I left out one of the conditions for distributive and extensive categories, to wit, the disjointness of coproducts. (Please note, however, that all the categories that I said were distributive are distributive.) He also points out that finite limits are not needed to say that coproducts are stable. Of course. But they're certainly needed for extensive logic. So: should the standard definition of extensive categories include finite limits? Or should I -- help! -- go around talking about "locally distributive logic"? (I included finite limits in the definition of locally distributive, but needn't have done so: the condition that the category and that every slice of the category be distributive easily implies finite limits.) Date: Wed, 7 Aug 1996 13:20:36 -0300 (ADT) Subject: Re: extensive stuff Date: Wed, 7 Aug 1996 09:13:45 -0700 From: james dolan -Still here? Guess what. The two categories are equivalent and the -equivalence carries one example to the other. - -Given an object, , in *A* turn XxY into a semigroup by -defining = . This is how the equivlence gets from -*A* to the category of neighboring idempotents. Getting back is left -to the reader. yes, the algebraic theory of semigroups retracts onto the initial theory in two ways (left projection and right projection), so we get a theory morphism to the square of the initial theory that's surjective for some reason or other. the square of the initial theory is also the natural structure theory of the right adjoint but very slightly non-monadic functor "binary cartesian product of sets". i think these funny semigroups are some variety of "bands". unfortunately i can't remember what a "band" is exactly. -Now the real question: how much of all this is already in Johnstone? don't remember seeing it discussed there, but at least the part about the funny semigroups being the algebras for the monad associated to the slightly non-monadic adjunction between sets and pairs of sets is probably well-known, i'd guess. Date: Sat, 10 Aug 1996 23:52:22 -0300 (ADT) Subject: Re: extensive stuff Date: Thu, 8 Aug 1996 11:43:51 +1000 (EST) From: stevel@maths.su.oz.au Dear Peter, I too rediscovered'' your *A*, precisely in looking for an example of a distributive category which failed to be locally distributive, but never realized that it had such an illustrious history. As for distributive categories which fail to be extensive, there is another important and natural class of examples. Any distributive lattice, thought of as a preorder, is a distributive category, indeed a locally distributive category but has coproducts which are very far from being disjoint and so fails to be extensive. In a distributive category admits a subdirect decomposition into a distributive and extensive category, and a distributive preorder, in the following way. Given a distributive category D, one can form the preorder reflection D_pr, and this is a distributive preorder and the projection D ---> D_pr a distributive functor. On the other hand, one can form the extensive reflection'' D_ext of D (this is the image of D under the left biadjoint to the inclusion of the 2-category of distributive and extensive cats in the 2-category of distributive cats), and the projection D ---> D_ext is also a distributive functor, and moreover the induced functor D ---> D_ext x D_pr is fully faithful. This is similar to a result of Cockett which fully embeds a _locally_ distributive category in the product of a lextensive category and a distributive preorder. The category D_ext has a very simple construction. An object of D_ext is an arrow a:A-->1+1 in D. An arrow from a:A-->1+1 to b:B-->1+1 is an arrow f:A-->B+1 in D satisfying the condition f A ----> B+1 | | a | | b+1 | | v v 1+1---> 1+1+1 inj_13 Steve. Date: Sat, 10 Aug 1996 23:53:19 -0300 (ADT) Subject: Re: extensive stuff Date: Thu, 8 Aug 1996 11:59:20 +1000 (EST) From: stevel@maths.su.oz.au > > He also points out that finite limits are not needed to say that > coproducts are stable. Of course. But they're certainly needed for > extensive logic. So: should the standard definition of extensive > categories include finite limits? Or should I -- help! -- go around > talking about "locally distributive logic"? (I included finite limits > in the definition of locally distributive, but needn't have done so: > the condition that the category and that every slice of the category > be distributive easily implies finite limits.) > > As Max Kelly pointed out, an extensive category with finite limits has been called a lextensive category. The problem still remains what one should call an extensive category with finite products (such a category being necessarily distributive). Extensive and distributive category'' is a bit of a mouthful, and prextensive is obviously unacceptable. Other possibilities that have been suggested include 2-rig'' and arithmetic category''. Steve. Date: Sat, 10 Aug 1996 23:54:20 -0300 (ADT) Subject: Re: extensive stuff Date: Thu, 8 Aug 96 09:55 BST From: Dr. P.T. Johnstone Max asks: >About the nomenclature: do people agree that "lex" is really terrible? Peter Johnstone in Sussex recently called it something like a twice-dead metaphor - but now I forget what he wanted in its place - was it Peter Freyd's "cartesian" ? and Peter asks: should the standard definition of extensive categories include finite limits? Or should I -- help! -- go around talking about "locally distributive logic"? Yes, I've been a convert for some time now to the Freydian use of "cartesian" for categories having (or functors preserving) all finite limits. I know this annoys the computer-science people who want to use it for (finite products but not equalizers), but I can't think of a better term. Actually, as applied to categories, "left exact" is a thrice-dead metaphor (twice-dead as applied to functors, since "exact sequence" is a dead metaphor for "exact differential", and "left exact" as applied to functors is a dead metaphor for "preserving the left- hand ends of exact sequences"). What status that gives to the term "lextensive" for "extensive plus all finite limits", I shudder to think. Should the standard definition of extensive categories include finite limits? Obviously Max (and Bob Walters, and Steve Lack, and probably Bill Lawvere) would say "no". But if you want to develop a syntax for these categories, you're not going to get very far without the finite limits; so we do need a name for "extensive plus finite limits". Perhaps, after all, we should revive the term "disjunctive" from my SLNM 753 paper, and call them "disjunctive categories". What do other people think? Peter Johnstone P.S. - If anyone was confused by the two messages I contributed to this discussion yesterday, please note that they were sent out in the reverse of the order in which I sent them in. Not that it matters very much. Date: Sat, 10 Aug 1996 23:55:28 -0300 (ADT) Subject: Re: Peter Freyd's letter of 6 Aug Date: Thu, 8 Aug 1996 11:47:50 +0200 From: Dr. Reinhard B/rger (Prof. Dr. Pumpl^nn) Just for completeness, I think that the category of middle-ignoring semigroups is equivalent to the category of pairs of sets which are either both empty or both non-empty. Another example of a category which is distributive but not extensive is the dual of the category of unital rings; note that the dual of the category of unital commutative rings is even extensive. Greetings Reinhard Date: Wed, 14 Aug 1996 23:04:48 -0300 (ADT) Subject: disjunctive stuff Date: Sun, 11 Aug 1996 16:11:09 -0400 From: Peter Freyd Regarding some comments of Peter Johnstone: I haven't succeeded in interpreting disjunction in arbitrary wcartesian extensive categories and would therefore hesitate in calling them "disjunctive categories." Computer scientists used to use the phrase "concrete" to mean "well- pointed" but seemed to have stoped as they became aware of the clash with existing category terminology. (By the way, I can't think at the moment of many people in CS, other than Robin C and friends, who use "cartesian" just to mean products.) Date: Wed, 14 Aug 1996 23:06:33 -0300 (ADT) Subject: alternating stuff Date: Sun, 11 Aug 1996 17:15:58 -0400 From: Peter Freyd How about "alternating categories"? Alternatives, in ordinary language, are usually understood to be mutually exclusive. So an extensive cartesian category (i.e. a locally distributive category with terminator) would be called an "alternating category" and the corresponding syntax, "alternating logic". A major problem: it would be hard to keep others -- since I find it hard to keep myself -- from corrupting this to "alternative logic". (On the other hand, whenever one is an environment where "linear logic" is sure to be totally misinterpreted, one could claim also to be studying "alternative logic".) Let me go on record here for the syntax. As in cartesian logic conjunction is the only connective and the only terms are conjunctions of primitive predicates each followed by an appropriate sequence of variables. Cartesian logic has just one primitive assertion, written A ue> B, where A and B are terms. Given an elemental interpretation of the primitive predicates, A ue> B is satisfied in the elemental cartesian semantics if for every instantiation of the variables of A such that A holds there is a unique instantiation of the remaining variables of B such that B holds. In alternating logic the primitive assertions are written A ue> B1|B2|...|Bn where A, B1, B2,...,Bn are terms. Such an assertion is said to be satisfied in the elemental alternating semantics if for every instantiation of the variables of A such that A holds there is a unique index, i, such that the remaining variables of Bi can be instantiated so that Bi holds and, further, there is just one such instantiation of the remaining variables of Bi. E.G.: for (decidable) fields, add to the cartesian theory of unital rings the alternating axiom x=x ue> (x=0)|(xy=1). As for the categorical semantics, given an alternative category in which each primitive predicate has been interpreted, extend the interpretation to terms -- just as for cartesian logic -- using finite limits. (For example: if A has variables x and y, and B has variables y and z then, using brackets to designate the interpretations, the interpretation of A^B is characterized by [A^B] l/ \r [A] [B] / \ / \ [x] [y] [z] where the rhombus is a pullback.) Note that the interpretation of a conjunction comes equipped with the two maps, l and r. In cartesian logic the key definition: A ue> B is satisfied iff l:[A^B] -> [A] is an isomorphism. In alternating logic: A ue> B1|B2|...|Bn is satisfied iff the l's combine to give an isomorphism [A^B1] + [A^B2] +...+ [A^Bn] -> [A]. te: Sun, 18 Aug 1996 11:29:00 -0300 (ADT) Subject: alternating stuff Date: Thu, 15 Aug 1996 00:22:45 -0400 From: James Otto Date: Sun, 11 Aug 1996 17:15:58 -0400 From: Peter Freyd How about "alternating categories"? Alternatives, in ordinary language, are usually understood to be mutually exclusive. So an Alternating (Turing) machines are fundamental to complexity and logic programming. The alternation is of bounded quantifiers. The exclusive, inclusive distinction is captured by the orthogonal, injective distinction, which is a strong model, weak model distinction. extensive cartesian category (i.e. a locally distributive category with terminator) would be called an "alternating category" and the corresponding syntax, "alternating logic". ... Let me go on record here for the syntax. As in cartesian logic .... In cartesian logic the key definition: A ue> B is satisfied iff l:[A^B] -> [A] is an isomorphism. On the models side of dualities, this is precisely being orthogonal to a map. In alternating logic: A ue> B1|B2|...|Bn is satisfied iff the l's combine to give an isomorphism [A^B1] + [A^B2] +...+ [A^Bn] -> [A]. On the models side of dualities, this is precisely being orthogonal to a small cone with discrete base. E.g. see Ad\'amek and Rosick\'y's book including its historical remarks e.g. on Y. Diers. I like P. Johnstones's multi-' for the map, cone distinction. On the theory side of dualities, P. Johnstones's remarks may help me. Just out of curiousity, if one drops the closed' (which I do not) from cartesian closed' or locally cartesian closed', what would one expect to have? Regards, Jim Otto Date: Sun, 18 Aug 1996 11:29:59 -0300 (ADT) Subject: Re: alternating stuff Date: Thu, 15 Aug 1996 17:01:33 +1000 (EST) From: stevel@maths.su.oz.au > Date: Wed, 14 Aug 1996 23:06:24 -0300 (ADT) > From: categories > > Date: Sun, 11 Aug 1996 17:15:58 -0400 > From: Peter Freyd > > How about "alternating categories"? Alternatives, in ordinary > language, are usually understood to be mutually exclusive. So an > extensive cartesian category (i.e. a locally distributive category > with terminator) would be called an "alternating category" and the > corresponding syntax, "alternating logic". > An extensive cartesian category is _not_ the same as a locally distributive category with terminator: as has already been pointed out, extensive categories are also required to have disjoint coproducts, which locally distributive categories need not, as the example of distributive lattices shows. Although I entirely agree that calling a category lex'' if it has finite limits is a bad thing, and am not thrilled about the name lextensive for an extensive category with finite limits, I do think that the name does have some advantages. In particular it makes clear that the two notions are closely linked. As various people have pointed out, for many uses the extensive categories won't be much use unless you have the finite limits as well; the reason, at least from my point of view, for isolating the extensive categories is to emphasize the fact that they are in fact categories with finite coproducts satisfying a certain _property_ which does not depend on the _structure_ of having all finite limits. As an excuse'' for the name lextensive, one can think of it as standing for (finite )l(imits +)extensive. Steve. Date: Sun, 18 Aug 1996 11:31:13 -0300 (ADT) Subject: Re: disjunctive stuff Date: Thu, 15 Aug 96 10:27 BST From: Dr. P.T. Johnstone I'm not sure about alternating logic' and alternating categories'; perhaps I just need time to get used to them. I think the trouble is that in everyday speech alternating' means something dynamic (switching back and forth between two states), and the logic doesn't have any such feature. Alternative' doesn't have the same connotation, but I agree that alternative logic' is impossible. The original reason for disjunctive' was a pun: it stands for both disjunction' and disjoint'. The way I formulated the syntax (essentially, an extension of Michel Coste's version of cartesian logic), you are allowed to write down a disjunction of formulae if you can prove that it's disjoint (just as you can use an existential quantifier if you can prove that the thing being quantified is unique). Peter Johnstone Date: Sun, 18 Aug 1996 11:33:04 -0300 (ADT) Subject: Re: extensive stuff Date: Thu, 15 Aug 1996 13:31:16 +0000 From: Steve Vickers If "lex" fills a need in our vocabulary - "having finite limits" when applied to structures, "preserving finite limits" when applied to transformations -, can't we forgive it the deadness of its metaphorical origins? I suspect there are other metaphorically dead words in mathematics - what about ring or field? The fundamental problem with "Cartesian" now is that it is ambiguous. Whenever it's used we have to investigate whether non-product limits are required. Steve Vickers. Date: Sun, 18 Aug 1996 11:34:04 -0300 (ADT) Subject: Re: disjunctive stuff Date: Thu, 15 Aug 1996 15:09:14 -0400 From: Michael Barr A propos the discussion of "cartesian", I might add that I think Descartes also invented the idea of the graph of a function and that is an equalizer. FWIW. Michael Date: Sun, 18 Aug 1996 11:37:23 -0300 (ADT) Subject: strongly (or exclusive) disjunctive logic: Hu, Tholen Date: Sat, 17 Aug 1996 11:22:07 -0400 From: James Otto Dear People, Perhaps I already said more about strongly (or exclusive) disjunctive logic than I wished. (So this is 2 of 2.) But I should note H. Hu, W. Tholen, Limits in free coproduct completions, JPAA 105 ('95) Rather than a duality as in P. Gabriel, F. Ulmer, Springer LNM 221 ('71) M. Makkai, A. Pitts, TAMS 299 ('87) but somewhat as in P. Johnstone, In Springer LNM 753 ('79) they construct a dual and a double dual: small with multi-limits of finite diagrams | flat functors to set v finitely accessible with connected limits | functors to set preserving filtered colimits and connected limits v having finite limits and stable disjoint small coproducts with net image the coprimes. By the way, for accessible categories one could see J. Ad\'amek, J. Rosick\'y, Cambridge ('94) F. Borceux, Volumes 1-2, Cambridge ('94) for alternating Turing machines D. Bovet, P. Crescenzi, Prentice Hall ('94) and for quantifiers, games, and interactive proofs (a book which I am less familiar with than the previous 4) J. K\"obler, U. Sch\"oning, J. Tor\'an, Birkh\"auser ('93) Regards, Jim Otto Date: Thu, 22 Aug 1996 09:27:03 -0300 (ADT) Subject: Re: cartesian/(L)extensive/... stuff Date: Tue, 20 Aug 1996 15:38:11 -0600 (MDT) From: Robin Cockett (1) First, in response to Peter's earlier comments about the term "cartesian": I would certainly like to consider Peter as a friend! Some names for mathematical concepts are like old jeans: the more threadbare and holes they have the more comfortable they feel. I suspect the term cartesian might be one such. I tend to think of a "cartesian tensor" as being a product - as saying a bit of structure is cartesian usually means it arises from limits - and have abused terminology by calling what I should have probably called a "cartesian tensor category" (or a "cartesian monoidal category") a "cartesian category" -- yes this is a category with products. I am usually careful, however, to make this usage explicit as I am aware that equalizers are often assumed. (2) Another article of comfy clothing is the term extensive! I agree with Steve Lack that it is unfortunate that extensive categories are not assumed to have a final object (as these creatures are the more common) I merrily suggest abusing terminology to avoid that hiccup. I would be less happy, however, to let the LEXness - or should I say cartesianness - be carried by the context. My reason for this hesitation is as follows: If you start with a distributive category and form its extensive completion, as explained by Lack, then all datatypes (natural numbers, lists, etc.) lift into that completion. These datatypes are shapely in the sense of Barry Jay (naturallity square are cartesian etc.). However, if you, taking the construction from another angle, freely add equalizers to a distributive category with datatypes (i.e. finitely complete it) all bets are off: certainly datatypes which do lift need not be shapely but I conjecture that there is, in fact, no guarantee that they lift at all ... Conjecture: There is a distributive category X with (strong) NNO such that ========== E(X) is its equalizer completion does not have a (strong) NNO. I do not have a proof that this is so ...or not! Coproducts appear to lift as the equalizer completion 2-functor preserves products. However, clearly (consider a distributive lattice) the resulting coproducts need not be extensive -- although, of course, the category E(X) is distributive (another source of non-extensive distributive categories). If the distributive category is separated (or decidable) in the appropriate sense then certainly datatypes lift (as the extensive completion and equalizer completion then coincide). This is the reason why the initial distributive category with (strong) datatypes may be finitely completed to preserve all datatypes. This makes me sensitive about the passage to Lextensive even from extensive. To obtain a completion in a RIGHT SENSE may be a little more delicate (or brutish ... depending on your approach). The point is there are some outstanding issues here .. (3) Lastly a remark on the connection between distributive categories and categorical proof theory: One motivation for developing the theory of weakly distributive categories (wdc) was to provide a unification of the proof theory of "classical" and linear logic. In particular, we supposed that the "and--or" fragment of classical logic has as a proof theory the free distributive category on its propositions. Accordingly, in that original paper, we sketched a proof that distributive categories are cartesian wdc's (i.e. wdc's in which the tensor is a product and the cotensor a coproduct). Subsequently we never revisited the result. Over the summer while studying the nucleus of these categories we realized something was amiss. Re-examing the proof we realized that one of the "obvious" coherence conditions was obviously false. In fact, so badly does it fail for distributive categories that the revision of the result states: Prop. A cartesian wdc which is simultaneously a distributive category is necessarily a preorder. It should be mentioned that cartesian wdc's abound: pointed sets, vector spaces, semi-lattices, ... are examples. Thus cartesian wdc's definitely do not collapse. (Back to names!!!! This does remove one of our motivations for the name "weakly distributive categories": Barr suggested the term "linearly distributive". However, to us the original name is now one of those comfy bits of clothing (even if somewhat frayed) ... In fact, if we had NOT made this oversight it is likely we would have been altogether more hesitant in the development of weakly distributive categories! Mathematics moves in mysterious ways.) There is philosophical significance to the correct result: classical semantic settings separate at an earlier stage than we had suspected from (categorical) proof theoretic settings. In particular, distributive categories (a core fragment of classical setting) do not permit the process of cut elimination (unless they are preorders). -robin (Robin Cockett) (p.s. Revised papers are available under Seely's home page: ftp://triples.math.mcgill.ca/pub/rags/ragstriples.html) Date: Thu, 22 Aug 1996 22:46:00 -0300 (ADT) Subject: Re: cartesian/(L)extensive/... stuff Date: Fri, 23 Aug 1996 11:02:53 +1000 (EST) From: Steve Lack > > (2) Another article of comfy clothing is the term extensive! I agree > with Steve Lack that it is unfortunate that extensive categories are not > assumed to have a final object (as these creatures are the more common) I can't quite imagine in what context I might have said that; it is certainly true that life becomes easier when the extensive category in question has a terminal object, but the whole point is that the notion of extensivity is a property of finite coproducts. (So in particular an extensive functor is one that preserves finite coproducts only, although of course it then follows that pullbacks along coproduct injections are also preserved.) > > If you start with a distributive category and form its extensive completion, > as explained by Lack, then all datatypes (natural numbers, lists, etc.) lift > into that completion. These datatypes are shapely in the sense of > Barry Jay (naturallity square are cartesian etc.). However, if you, > taking the construction from another angle, freely add equalizers to a > distributive category with datatypes (i.e. finitely complete it) all bets > are off: certainly datatypes which do lift need not be shapely but I conjecture > that there is, in fact, no guarantee that they lift at all ... > > Conjecture: There is a distributive category X with (strong) NNO such that > ========== E(X) is its equalizer completion does not have a (strong) NNO. > > I do not have a proof that this is so ...or not! > > Coproducts appear to lift as the equalizer completion 2-functor > preserves products. However, clearly (consider a distributive lattice) the > resulting coproducts need not be extensive -- although, of course, the category > E(X) is distributive (another source of non-extensive distributive categories). In fact, (1) If X is distributive then E(X) is locally distributive but (2) If X is distributive and E(X) is extensive then X is equivalent to the trivial category 1. To (freely) pass from a distributive category to a lextensive one, you first form E(X) and then the slice category p/E(X) where p is the equalizer i ---> p ---> 1 ---> 1+1 j of the coproduct injections i,j:1--->1+1. The passage from extensive categories to lextensive categories, is, as Robin says, more delicate. Steve Lack. Date: Sat, 24 Aug 1996 11:44:46 -0300 (ADT) Subject: Re: cartesian/(L)extensive/... stuff Date: Fri, 23 Aug 1996 11:53:12 -0600 (MDT) From: Robin Cockett I did mean extensive + products and mistyped "final object." The point being that these are common beasties which deserve a snappy name! Further, Steve Lack is correct about the fact that if E(X) is extensive and X is distributive then X = 1. So I should correct my comments about when the equalizer completion and extensive completion coincide! ... and I have to chuckle at this point as I have tripped on my own snag ... The problem is the initial object. In a distributive category this object is only one way connected to the rest of the category so that the poset collapse of a distributive category is always non-trivial while the category itself is. This means the added equalizer i ---> p ---> 1 ---> 1+1 j of the coproduct injections i,j:1--->1+1 in E(X) is never isomorphic to the initial object. Hence Steve Lack's comment. Some time ago (when I was in Oz, in fact) I sensitized the community there to exactly these issues. In fact, I became a bit of a heretic by considering distributive categories without an initial object (I called these predistributive). It is the initial of these gadgets (with "non-empty" inductive datatypes) which has E(X) extensive ... and the initial object is provided precisely by the added equalizer p, above. However, it really is infinitely better to talk about LEXT(X), the lextensive completion, not the raw E(X) where some preinitial baggage may still be present. The conjecture still stands but is better expressed in the form: Conjecture: There is a distributive category X with (strong) NNO such that ========== LEXT(X) is its lextensive completion does not have a (strong) NNO. -robin Date: Tue, 27 Aug 1996 10:19:53 -0300 (ADT) Subject: terminology Date: Mon, 26 Aug 1996 11:07:26 +1000 (EST) From: Steve Lack Regarding the debate on cartesian/extensive, we'd like to suggest the name "arithmetic category" for an extensive category with products. Certainly such categories seem well structured enough to do some arithmetic, and free such are closely enough related to the free analogies in the algebraic context (ie rigs) to be thought of as behaving "arithmetically" in some sense. Robbie Gates Steve Lack -- ---------------------------------------------------------------------- robbie gates | apprentice algebraist | http://cat.maths.usyd.edu.au/~robbie pgp key available | Date: Wed, 28 Aug 1996 10:42:44 -0300 (ADT) Subject: Re: terminology Date: Tue, 27 Aug 1996 14:45:24 +0000 From: Steve Vickers >Regarding the debate on cartesian/extensive, we'd like to suggest >the name "arithmetic category" for an extensive category with products. > >Certainly such categories seem well structured enough to do some >arithmetic, and free such are closely enough related to the free >analogies in the algebraic context (ie rigs) to be thought of as >behaving "arithmetically" in some sense. > >Robbie Gates >Steve Lack The phrase "Arithmetic Category" with this sense doesn't sit comfortably next to Joyal's "Arithmetic Universes", in which the word "arithmetic" also conveys recursive structure. Steve Vickers. Date: Thu, 29 Aug 1996 13:19:42 -0300 (ADT) Subject: Re: alternating stuff Date: Thu, 29 Aug 1996 14:59:54 +0200 (MET DST) From: Jiri Rosicky In a recent paper (An algebraic description of locally multipresentable categories, TAC 2 (1996), 40-54), we have introduced essentially multialgebraic theories and showed that they correspond to locally finitely multipresentable categories in the same way as essentially algebraic theories correspond to locally finitely presentable categories. J.Adamek, J.Rosicky Date: Thu, 29 Aug 1996 13:19:00 -0300 (ADT) Subject: Re: terminology Date: Thu, 29 Aug 1996 03:00:41 -0400 (EDT) From: F William Lawvere The term EXTENSIVE was applied to certain categories in 1991 by Carboni, Lawvere, and Walters on the basis of the following considerations. For centuries, mathematical philosophy has distinguished between extensive quantities and intensive quantities, for example in thermodynamics of inhomogeneous bodies, volume, mass, energy, and entropy on the one hand are distinguished from pressure, density, temperature on the other. Lawvere in 1982 (SLNM 1174) and in 1990 (Categories of Space and of Quantity) had proposed to explain these as modes of variation of quantity. Quantities vary over a domain of variation in both cases. ( A domain of variation is a "space" , which in turn is an object in a category of space, which will be rather more determinate than "category" in general). An extensive quantity type is a covariant functor from a category of space, preserving finite coproducts, to a linear category (= one in which coproducts and products coincide). A distribution, a measure, a current, a homology class on a sum domain is given by a tuple of such , one on each summand ; thus all these are elements of extensive quantity types. An intensive quantity type is a contravariant functor from a category of space which also takes coproducts to products ; but more: intensive quantities usually act linearly on extensive quantities, lending them (not only a linear but also a) multiplicative structure which is also preserved by the contravariant functorality. For example, functions act as densities on distributions and measures, similarly differential forms act on currents, and cohomology classes act on homology classes. Often intensive quantity types are representable and related extensive types are definable as linear duals ( Riesz, Pontrjagin, Eilenberg- Mac Lane etc),but these are not the only possibilities. A fundamental example of a linear "category" is the 2-category of all categories with coproducts; it has an obvious abstraction functor to the linear category of commutative monoids, by taking isomorphism- classes. Part of the idea of K-theory and of K-homology and of the "non-linear K-theory" which I with Schanuel and others have been pursuing under the name of "objective number theory" is that it is useful to "objectify" quantities by lifting their type across this abstraction functor. The most fundamental measure of a thing is the thing itself. But measures can be pushed-forward ( a common colloquial expression for the covariant functorality of extensive quantity).However pullback is more familiar (already in 1844 Grassmann complained that intensives were more familiar than Ausdehnungen) : On any category with pullbacks, there is the contravariant functor to cat which takes the "slice" categories of each object.This functor is commonly viewed as consisting if "functions", namely functions whose values are the fibers. Indeed if the category satisfies suitable conditions , this will be an intensive quantity type. But what extensives will it act on ? On any category at all the slice categories constitute an even simpler covariant cat-valued functor, simply composing the transforming map following the structural map to define the new structural map. It is often appropriate to consider that the structural map distributes the total in the base ( though distributions usually do not have values at points, they do often have totals ) and that the mentioned composing pushes the distribution forward. Indeed if the category has coproducts, this naive pushing forward is automatically linear. Thus a category with coproducts defines an extensive quantity type on itself if and only if it is an extensive category. If we call lextensive any extensive category with (finite) limits, then also by pullback, the intensives act on the extensives in a way that satisfies all reasonable functoralities, including the crucial CCR or "projection formula". More exactly, note that there will typically be lots of subcategories of such a "category of space" which are extensive but do not themselves have products or even a terminal object; it suffices to be closed under sums and summands. Namely the empty space together with all spaces of dimension exactly 7. Any such subcategory A defines the extensive quantity type "distributions of A-dimensional spaces " in each base space X, namely the subslice category. Given A and B two extensive subcategories, there is an objective intensive quantity type which acts roughly as " B-A dimensional cohomology" namely for any X consider the category of all those spaces over X such that for any space over X whose total is in A, the pullback has total in B ; this pulls back along any change of X. Of course many extensive subcategories of a (lextensive) category of space will have products, for example (as Joyal pointed out to me in 1984) if A=B is the "compact" objects , thus defining the extensive notion of distributing a compact object in a base space, the corresponding intensive quantities which act on these are the proper fiberings . Date: Fri, 30 Aug 1996 13:57:50 -0300 (ADT) Subject: Re: terminology Date: Fri, 30 Aug 1996 12:08:55 -0400 (EDT) From: F William Lawvere The following is an improved version of the text sent Thurs. Any suggestions for further improvements will be welcome . Note that the distinction between the objective extensive and the objective intensive is essentially "sampling/sorting" distinction discussed in Conceptual Mathematics. Bill The term EXTENSIVE was applied to certain categories in 1991 by Carboni, Lawvere, and Walters on the basis of the following considerations. For centuries, mathematical philosophy has distinguished between extensive quantities and intensive quantities, for example in thermodynamics of inhomogeneous bodies, volume, mass, energy, and entropy on the one hand are distinguished from pressure, density, temperature on the other. Lawvere in 1982 (SLNM 1174) and in 1990 (Categories of Space and of Quantity) had proposed to explain these as modes of variation of quantity. Quantities vary over a domain of variation in both cases. ( A domain of variation is a "space" , which in turn is an object in a category of space, which will be rather more determinate than "category" in general). An extensive quantity type is a covariant functor from a category of space, preserving finite coproducts, to a linear category (= one in which coproducts and products coincide). A distribution, a measure, a current, a homology class on a sum domain is given by a tuple of such , one on each summand ; thus all these are elements of extensive quantity types. An intensive quantity type is a contravariant functor from a category of space which also takes coproducts to products ; but more: intensive quantities usually act linearly on extensive quantities, lending them (not only a linear but also a) multiplicative structure which is also preserved by the contravariant functorality. For example, functions act as densities on distributions and measures, similarly differential forms act on currents, and cohomology classes act on homology classes. Often intensive quantity types are representable and related extensive types are definable as linear duals ( Riesz, Pontrjagin, Eilenberg- Mac Lane etc),but these are not the only possibilities. A fundamental example of a linear "category" is the 2-category of all categories with coproducts; it has an obvious abstraction functor to the linear category of commutative monoids, by taking isomorphism- classes. Part of the idea of K-theory and of K-homology and of the "non-linear K-theory" which I with Schanuel and others have been pursuing under the name of "objective number theory" is that it is useful to "objectify" quantities by lifting their type across this abstraction functor. The most fundamental measure of a thing is the thing itself. But measures can be pushed-forward ( a common colloquial expression for the covariant functorality of extensive quantity).However pullback is more familiar (already in 1844 Grassmann complained that intensives were more familiar than Ausdehnungen) : On any category with pullbacks, there is the contravariant functor to cat which takes the "slice" categories of each object.This functor is commonly viewed as consisting of "functions", namely functions whose values are the fibers. Indeed if the category satisfies suitable conditions , this will be an intensive quantity type. But what extensives will it act on ? On any category at all the slice categories constitute an even simpler COVARIANT cat-valued functor, simply composing the transforming map following the structural map to define the new structural map. It is often appropriate to consider that the structural map distributes the total in the base ( though distributions usually do not have values at points, they do often have totals ) and that the mentioned composing pushes the distribution forward. Indeed if the category has coproducts, this naive pushing forward is automatically linear.( But even though the slice categories have terminal objects, these are not preservedby the relevant extensive functorality.) THUS A CATEGORY WITH COPRODUCTS DEFINES AN EXTENSIVE QUANTITY TYPE ON ITSELF IF AND ONLY IF IT IS AN EXTENSIVE CATEGORY. If we call lextensive any extensive category with (finite) limits, then also by pullback, the intensives act on the extensives in a way that satisfies all reasonable functoralities, including the crucial CCR or "projection formula". More exactly, note that there will typically be lots of subcategories of such a "category of space" which are extensive but do not themselves have products or even a terminal object ; it suffices to be closed under sums and summands. Namely the empty space together with all spaces of dimension exactly 7. Any such subcategory A defines the extensive quantity type "distributions of A-dimensional spaces " in each base space X, namely the subslice category. Given A and B two extensive subcategories, there is an objective intensive quantity type which acts roughly as " B-A dimensional cohomology" namely for any X consider the category of all those spaces over X such that for any space over X whose total is in A, the pullback has total in B ; this pulls back along any change of X. Of course many extensive subcategories of a (lextensive) category of space will have products, for example (as Joyal pointed out to me in 1984) if A=B is the "compact" objects , thus defining the extensive notion of distributing a compact object in a base space, the corresponding intensive quantities which act on these are the PROPER fiberings . Date: Thu, 5 Sep 1996 14:11:33 -0300 (ADT) Subject: Correction to paper - distributive is not weakly distributive Date: Tue, 3 Sep 1996 16:13:35 -0400 From: Robert A. G. Seely The following notice and discussion amplifies some recent remarks made by Robin Cockett on the CATEGORIES list. We wish to announce a correction to a statement in the paper Weakly distributive categories by J.R.B. Cockett and R.A.G. Seely An error in Proposition 3.1, where we claimed that distributive categories are weakly distributive, was found in proof. The result is totally incorrect: a distributive category is a cartesian weakly distributive category if and only if it a preorder. (Note: a weakly distributive category may be cartesian - by which we just mean the tensor and cotensor ("par") are cartesian product and coproduct respectively - without being a preorder; it is the distributivity that causes the collapse.) In particular, any distributive category which satisfies equation (13): \delta^R_R (A+B)x(C+D) ------------> A+(Bx(C+D)) | | \delta^L_L | | v | ((A+B)xC)+D | 1 + \delta^L_L | | \delta^R_R + 1 | | v a v (A+(BxC))+D ------------> A+((BxC)+D) (where we write x for the tensor, + for the cotensor (par), and 1 for identity) for the choice of weak distributions described in the paper is immediately a preorder. This because in that diagram if A=D=1 and B=C=0 then, up to equivalence, we obtain for the two sides of diagram the coproduct embeddings of 1 into 1+1. This suffices to cause collapse. The argument can be modified to show that in any distributive category which is simultaneously weakly distributive (no matter how the weak distributions are defined), Boolean negation must have a fixed point. This also suffices to cause collapse. A consequence of this observation is that the categorical proof theory of not-necessarily-intuitionist AND/OR logic is somewhat subtle. In the absence of any connective for implication, there is no apparent a priori reason not to have multiple-conclusion sequents; let's see what this yields. We start with the premise that a good semantics for AND/OR logic ought to be a polycategory; in particular, that the morphisms interpreting the following two derivations must be equal. (That these are equal is a consequence of the polycategory definition, but you can judge them on their own merits if you like. This type of permutation of cuts is pretty standard, and categorical cut elimination then would demand that they be equal.) (Notation: I use -> for the sequent turnstile, and x and + for AND and OR. The interpretation of the commas is, as is usual in such logics, AND on the left and OR on the right, so there are evident identity maps representing A,B -> AxB and A+B -> A,B. All deduction steps are cuts. The cut rule is XX,A -> YY and WW -> A,UU entail XX,WW -> YY,UU and variants via exchange.) B,C -> BxC A+B -> A,B ------------------------- A+B,C -> A,BxC C+D -> C,D -------------------------------- A+B,C+D -> A,BxC,D B,C -> BxC C+D -> C,D ------------------------- = B,C+D -> BxC,D A+B -> A,B -------------------------------- A+B,C+D -> A,BxC,D But here's the catch - with the obvious interpretation, these come out different in SETS: think of the image of a pair in (A+B)x(C+D), where a \in A and d \in D. For the top map, this is mapped to a, whereas for the bottom map it is mapped to d. This is just our equation (13) again, so the point of our initial comment is that in any distributive category, with any interpretation, these two maps are equal iff the category is a preorder. This is a pretty "stripped down" example - it seems that categorical cut elimination is inconsistent with using distributive categories for AND/OR logic and general sequents. This problem is averted of course if one restricts oneself to "intuitionist" sequents (with the right of the turnstile restricted to single formulas), but then this result may be seen as indicating how the folkloric result concerning the collapse of categorical proof theory for classical logic (Joyal) doesn't really depend on very much structure - note that we have assumed no structure rules beyond cut, and the linear versions of the AND/OR sequent rules; the collapse just needs multiple-conclusion sequents and distributivity. It is interesting to note, however, that by carefully choosing the weak distributions one can construct a cartesian weakly distributive category from an elementary distributive category by simply passing to the Kleisli category of the `exception monad'' E(X) = X+1. So, for example, although SETS is not weakly distributive itself, POINTED_SETS is. The error means, of course, that all discussion in the paper of non-posetal distributive categories as examples of weakly distributive categories must be discounted. This mainly affects the Introduction and Section 3, where Proposition 3.1 must be restated as indicated above, and the surrounding text must take this restatement into account. In particular, Theorem 3.3, although still correct, ought to be stengthened to state that a cartesian weakly distributive category is a preorder if and only if it has a strict initial object. A version of this paper which contains a rewritten Introduction and Section 3 may be found on rags' WWW home page at this URL: . These comments will also appear in the published version of the paper (to appear in JPAA). Finally, the inevitable controversy about terminology: we have decided to continue calling these categories "weakly distributive", since we have done so for so long and in so many places. Besides, Hyland and dePaiva had arrived at the same name for the "weak distributivities", independently, and at the same time. But we keep an open mind about these matters: if another name seems to have near-universal approval, we will adopt it too. The most promising seems to be Barr's suggestion of "linearly distributive". Indeed, had that suggestion been made in 1991, we might have adopted it then (it certainly beats "dissociative categories"!) Robin Cockett Robert Seely (for ftp help: rags@math.mcgill.ca)