From cat-dist Sun Nov 2 14:54:49 1997 Received: by mailserv.mta.ca; id AA05915; Sun, 2 Nov 1997 14:53:52 -0400 Date: Sun, 2 Nov 1997 14:53:51 -0400 (AST) From: categories To: categories Subject: Abelian-topos (AT) categories Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Sun, 02 Nov 1997 06:52:02 -0800 From: Vaughan R. Pratt Surely a natural name for these "abelian-topos" categories would be abelian-topos categories, AT categories for short. (I barely grasp them, I only found out what abelian categories were a week ago, when someone on sci.math asked about them and I looked them up and answered him because I wanted to know too. It struck me as interesting that they had so much in common with toposes, whence my question about intersecting their respective theories. I made language a parameter because it seemed intuitively obvious that a strong enough language would make the models of that intersection the union of the respective classes. But as Peter points out this is already a triviality for any theory closed under classical disjunction, which I still can't believe I didn't know.) On the question of the right language for defining AT, I fully agree with Peter that universal Horn sentences are the appropriate logical strength of language. (I take back what I said before about "universal" not making a difference. Peter showed that for the Horn theory, i.e. allowing existential quantification, the models are representable as products AxT of an abelian category A with a pretopos T. Presumably the weaker universal Horn theory admits in addition the appropriate subcategories thereof.) So would it be correct to say that this makes AT a quasivariety in CAT with functors preserving the signature Peter is using? What I'm less clear about than the logical strength is the choice of signature. In particular what about the closed structure? Without the closed structure we have Peter's AxT representation theorem, with the theory finitely axiomatized to boot. Presumably this remains unchanged by the introduction of closed structure: we retain only those categories AxT such that A and T independently admit closed structure, and can finitely axiomatize the separate closed structure of each in terms of the associated retractions A and T (ugh, overloading), thereby axiomatizing the joint closed structure. For example the tensor unit I of AxT will be (Z,1) (Z the tensor unit of A), with AI = (Z,0) != I (except for pure abelian categories) and TI = (0,1) = 1 != I (except for pure pretoposes). But for objects a,b of A and t,u of T, the tensor product (a,t)@(b,u) will be (a@b,txu), with A(a@b,txu) = (a@b,0) = (a,0)@(b,0) = A(a,t)@A(b,u) and T(a@b,txu) = (0,txu) = (0,t)@(0,u) = T(a,t)@T(b,u). (So the retractions preserve tensor product but not tensor unit---TI just gives the terminator, but AI furnishes a new constant.) So should there perhaps be two classes, the AT categories and the closed AT categories? The latter would add I and @ to the signature (and presumably \aleph\lambda\rho, bless them). For the closed AT categories, TX can be neatly defined as 1@X, with the T-type objects identified as those having only one map to I (Mike Barr pointed out to me that strictness of 0, maps to 0 only from initial objects, would do this job), and with a topos defined as a closed AT category for which I is terminal (or 0 is strict). But although people seem comfortable working with toposes as opposed to pretoposes, what about abelian closed categories (if that's the right word order)? Despite Ab itself being a closed category, I don't see much discussion of closed structure for abelian categories. Why is this? Am I just reading the wrong stuff, or is the closed structure of abelian categories boring, or what? Or is FinAb (abelian but not abelian closed---no suitable tensor unit) too desirable to discard in this way? (What comparably interesting pretoposes are so lost? Not the topos FinSet.) Does the requirement of being closed kill off too many desirable abelian categories? I would have thought lack of closed structure would greatly impair the utility of a category, however beautiful its objects might be. I'm interested in these classes, especially those whose categories admit closed structure, because toposes sit at or near the left (geometric or discrete) end of what I've been calling the Stone gamut, while abelian categories sit near the middle. AT categories offer an entirely different approach to the Chu construction for mixing categories from strategic positions on the Stone gamut. The Chu construction Chu(V,k) mixes two closed categories, V and V\op, symmetrically positioned about the center of the Stone gamut, to yield all categories "in between" V and V\op (and "all"--certainly all small--categories period when V and V\op are at the outermost points, viz. Set and Set\op). In contrast AT categories mix categories from the far left (represented by Set) and the center (represented by Ab) to get a qualitatively different effect that I'm not sure how to relate to the Chu construction but which seems in some vague sense dual to it. One such sense is as follows. AT is the quasivariety generated just by Set and Ab alone. So in this sense at least it is the smallest quasivariety spanning the left half of the Stone gamut, assuming that the quasivariety generated by either Set or Ab alone does not span the gamut but crowds around their two respective positions on the gamut, left and middle. Include the duals of AT categories (not necessarily expanded to a quasivariety, see below) and now you cover the whole gamut in this minimal sense. On the other hand the various comprehensiveness properties I've been pointing out for the concrete subcategories of Chu(Set,K) for large enough K (including all small categories, even when concreteness is a requirement unlike the comprehensiveness results of the 1960's) make "sub-Chu" maximal over the Stone gamut. Vague question. The minimality of AT and the maximality of Chu is a very weak sort of duality, analogous to the minimal structure of sets vs. the maximal structure of Boolean algebras. Is there a more formal duality here, analogous to the duality of Set and CABA? That the retracts AX and TX seem to be dual notions, being respectively coreflective and reflective, gives them some of the flavor of Chu. But is AT itself dual to Chu in any categorical sense? More precise question. Is the quasivariety generated by all three of Set, Ab, and Set\op (aka CABA) finitely axiomatizable? And if not, does using Bool instead of CABA help or hinder? Vaughan From cat-dist Mon Nov 3 15:44:30 1997 Received: by mailserv.mta.ca; id AA09002; Mon, 3 Nov 1997 15:43:46 -0400 Date: Mon, 3 Nov 1997 15:43:46 -0400 (AST) From: categories To: categories Subject: local maps of toposes are always UIAO Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Mon, 03 Nov 1997 14:51:58 MEZ From: Thomas Streicher I'd like to know whether the following simple observation is well known. If F -| U : E -> S is a local map of toposes i.e. Gamma : Gl(F) -> Gl(Id_S) has a fibred right adjoint Nabla then U is full and faithful, i.e. one has the situation of a Unity and Identity of Adjoint Opposites in Lawvere's sense. Of course, UIAO entails that the geom. morph. is local. For the reverse direction the argument is as follows. If Gamma has a fibred right adjoint Nabla then Gamma preserves sums, i.e. is a cocartesian functor. Let FI ============ FI || | || | Ff be a cocartesian arrow in E / F || V FI -----------> FJ Ff then its image under Gamma is the left square in the diagram below i q I ----------> P ------>UFI || | | || | p pbk | UFf with q o i = eta_I || V V I ----------> J ------>UFJ f eta_J where i is an isomorphism as Gamma is cocartesian. That means eta_I I -------> UFI | | | | UFf is a pullback for all f : I -> J V V J -------> UFJ eta_J Choosing J = 1 we get that eta_I is an iso, i.e. eta is a natural iso. Thus, F and the right adjoint of U are both full and faithful. Of course, this argument doesn't go through when "local" is defined as U having an ordinary right adjoint. Thomas Streicher From cat-dist Mon Nov 3 15:44:30 1997 Received: by mailserv.mta.ca; id AA16380; Mon, 3 Nov 1997 15:42:50 -0400 Date: Mon, 3 Nov 1997 15:42:50 -0400 (AST) From: categories To: categories Subject: Workshop announcement Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Mon, 3 Nov 1997 12:01:30 +0000 (GMT) From: Ronnie Brown Open House for a Workshop on Global actions, groups and homotopy. Professor A Bak and two colleagues will be visiting Bangor under an ARC Scheme Dec 15-20, 1997, and a workshop will be held on the interactions of the above themes. Global actions: The algebraic counterpart of a topological space. Bak's theory introduces an algebraic concept of space with motion, which arose from K-theory considerations. A global action consists of a set X and a collection of groups acting on subsets of X. It yields a homotopy theory for algebraic structures which includes a natural, intuitive as well as rigorous concept of algebraic deformation of morphism. The aim of the workshop is to introduce these ideas and to relate them to methods of special interest at Bangor, for example algebraic models of homotopy types, homotopy coherence, higher dimensional algebra, computational aspects. All those interested are welcome, but no financial support is available. Information on accommodation in Bangor is available from the web site and participants are asked to make their own arrangements, although we will help if we can. Further information will be posted on http://www.bangor.ac.uk/ma/research/gagah/globact.htm Ronnie Brown Tony Bak Prof R. Brown, School of Mathematics, University of Wales, Bangor Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom Tel. direct:+44 1248 382474|office: 382475 fax: +44 1248 383663 World Wide Web: home page: http://www.bangor.ac.uk/~mas010/ New article: Higher dimensional group theory Symbolic Sculpture and Mathematics: http://www.bangor.ac.uk/SculMath/ Mathematics and Knots: http://www.bangor.ac.uk/ma/CPM/exhibit/welcome.htm From cat-dist Mon Nov 3 15:44:32 1997 Received: by mailserv.mta.ca; id AA19622; Mon, 3 Nov 1997 15:44:30 -0400 Date: Mon, 3 Nov 1997 15:44:30 -0400 (AST) From: categories To: categories Subject: Re: Abelian-topos (AT) categories Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Mon, 3 Nov 1997 10:00:59 -0500 (EST) From: Peter Freyd I've had trouble with starting a reply to Vaughan's most recent posting, the one headed, "Abelian-topos (AT) categories". It makes the wide gap that's come into being all too painful. Vaughan wrote, I don't see much discussion of closed structure for abelian categories. Why is this? Am I just reading the wrong stuff, or is the closed structure of abelian categories boring, or what? And that was after he had written, I only found out what abelian categories were a week ago. For a long time Category Theory existed (say, in the Mathematical Reviews) as a subset of Homological Algebra -- which is a way of saying that category theory was abelian category theory. (I can't remember a new result in the theory of abelian categories in the last quarter-century. I do remember, alas, a bunch of new announcements of such results.) I have trouble with Vaughan's phrase "the closed structure" on an abelian category. There can be many. The category of finite- dimensional complex representations of a compact group has a distinguished symmetric monoidal closed structure. If they are viewed just as categories then for any two compact groups with the same number of conjugacy classes the categories are isomorphic. If viewed just as monoidal closed categories then the necessary and sufficient condition that they be isomorphic is that the groups have isomorphic character tables. On the other hand, if viewed as a _symmetric_ monoidal closed categories, one can recover the group from the category. If you want a specific example consider the two non-abelian groups of order eight, the dihedral and the quaternian. In each case the plain category of representations is the 5-fold cartesian power of the category of finite-dimensional complex vector spaces. As monoidal closed categories they are isomorphic (but not isomorphic with the 5-fold cartesian power of the closed monoidal category of finite- dimensional complex vector spaces). As symmetric monoidal closed categories they are different. Anyway, there's a whole body of material. A lot of it is now viewed as standard in a number of (non-categorical) subjects and as for all successfull branches of category theory the theory of abelian categories is no longer considered to be a branch of category theory. Back to pratt cats: if one wants to axiomatize those categories that are products of abelian cats and topoi and not worry about the intervening families the axioms can be made quite simple. After saying that it's a regular category with a coterminator contained in its terminator, I'd start with the P-E-l-r-/\ structure as in my last post, define TX as the image of rX and prove it to be the correflection of X into the full subcategory of type-T objects. 0xX -> X is easily seen to be the correflection of X into the full subcategory of type-A objects. Then the axiom that these two correflections yield a coproduct decomposition for each object allows one to prove quickly that the category is the cartesian product of the two correfletive subcategories. The type-T objects clearly form a topos. All that's needed now is a couple of axioms to make the type-A objects abelian. From cat-dist Tue Nov 4 08:18:01 1997 Received: by mailserv.mta.ca; id AA13863; Tue, 4 Nov 1997 08:17:20 -0400 Date: Tue, 4 Nov 1997 08:17:20 -0400 (AST) From: categories To: categories Subject: Re: Abelian-topos (AT) categories Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Mon, 03 Nov 1997 21:57:20 -0800 From: Vaughan R. Pratt I appreciate that there are people on the list with more years of experience with abelian categories than I have days. AC's don't seem to have penetrated much into computer science, and I have no idea whether they need to. But the finite axiomatizability of the quasivariety generated by Set and Ab definitely has my attention. And the fact that toposes and abelian categories, so far apart intuitively (sets vs. abelian groups?), are brought to within so short an axiom of each other by the definition of AT cats, has the potential to make abelian categories much more relevant to fans of toposes. Peter (and privately Fred Linton and Mike Barr) have answered my question about what I was naively calling "abelian closed". "Abelian" and "cartesian" are not interchangeable adjectives inasmuch as the latter describes the tensor product in the context of "cartesian closed" while the former names a quasivariety. While I was aware of the distinction, I was hoping that abelian categories as the models of the universal Horn theory of Ab, combined with Ab having closed structure, would somehow make the juxtaposition "abelian closed" meaningful, but the examples show this to be wishful thinking. And Peter's define TX as the image of rX removes any motivation to define TX as 1@X. (Meanwhile I've reconciled myself to TX as the pushout of the projections of 0xX.) >After saying that it's a regular category with a coterminator contained ^^^^^^^ [Is "effective" not needed? -v] >in its terminator, I'd start with the P-E-l-r-/\ structure as in my >last post, and prove it to be the correflection of X into the full >subcategory of type-T objects. 0xX -> X is easily seen to be the >correflection of X into the full subcategory of type-A objects. Then >the axiom that these two correflections yield a coproduct decomposition >for each object allows one to prove quickly that the category is the >cartesian product of the two correfletive subcategories. The type-T >objects clearly form a topos. All that's needed now is a couple of >axioms to make the type-A objects abelian. Not just finitely axiomatizable but beautifully so. Vaughan From cat-dist Tue Nov 4 08:18:08 1997 Received: by mailserv.mta.ca; id AA14936; Tue, 4 Nov 1997 08:18:07 -0400 Date: Tue, 4 Nov 1997 08:18:07 -0400 (AST) From: categories To: categories Subject: Re: local maps of toposes are always UIAO Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Tue, 4 Nov 1997 10:31:16 +0000 (GMT) From: Dr. P.T. Johnstone Thomas Streicher asked > I'd like to know whether the following simple observation is well known. > If F -| U : E -> S is a local map of toposes i.e. Gamma : Gl(F) -> Gl(Id_S) > has a fibred right adjoint Nabla then U is full and faithful, i.e. one has > the situation of a Unity and Identity of Adjoint Opposites in Lawvere's > sense. Yes, there is a simple proof of this fact in Proposition 1.4 of "Local maps of toposes" by Johnstone & Moerdijk (Proc. London Math. Soc. (3) 58 (1989), 281--305). From cat-dist Tue Nov 4 13:35:12 1997 Received: by mailserv.mta.ca; id AA30899; Tue, 4 Nov 1997 13:34:16 -0400 Date: Tue, 4 Nov 1997 13:34:16 -0400 (AST) From: categories To: categories Subject: Pratt slices Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Tue, 4 Nov 1997 16:55:54 +0000 From: Dr. P.T. Johnstone Just a thought about Vaughan's original question: the class of toposes (and that of pretoposes) is stable under slicing, as are all the `exactness properties' that they share with abelian categories. Slices of abelian categories aren't abelian; but, thanks to Aurelio Carboni, we know how to characterize them. So we ought surely to be looking for a common generalization, not of toposes and abelian categories, but of (pre)toposes and affine categories in Aurelio's sense. How about it, Peter? Peter J. From cat-dist Wed Nov 5 17:33:23 1997 Received: by mailserv.mta.ca; id AA10620; Wed, 5 Nov 1997 17:32:48 -0400 Date: Wed, 5 Nov 1997 17:32:47 -0400 (AST) From: categories To: categories Subject: CfP: ESSLLI-98 Workshop on Logical Abstract Machines Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Tue, 04 Nov 1997 17:44:46 +0000 From: Eike Ritter ESSLLI-98 Workshop on LOGICAL ABSTRACT MACHINES August 17 - 21, 1998 A workshop held as part of the 10th European Summer School in Logic, Language and Information (ESSLLI-98) August 17 - 28, 1998, Saarbrueken, Germany ** FIRST CALL FOR PAPERS ** ORGANISERS: Valeria de Paiva and Eike Ritter (University of Birmingham) Web site: http://www.dcs.warwick.ac.uk/~esslli98/workshops.html BACKGROUND: This workshop brings together recent work on the design of abstract machines for functional programming languages based on logical foundations. Abstract machines describe implementations of functional languages on a level of abstraction which is high enough to make it possible to reason about the implementation but low enough as to allow an easy coding of the abstract machine. The workshop is aimed at students and researchers with a basic understanding of functional programming and intuitionistic logic who want to work on the exciting field of programming with a solid logical basis. We focus the workshop along two main themes: explicit substitutions and abstract machines based on Linear Logic. Most of the more recent work on abstract machines is directed towards implementing and proving correct functional languages based on Linear Logic ideas. Linear Logic, being a resource logic, was deemed ideal to model resource control in functional languages. FORMAT OF THE WORKSHOP: The workshop consists of five sessions of 90 minutes and time will be allocated according to the quality of the submissions. We seek original papers on the full spectrum of abstract machines from theory to application. Among the topics of interest are: Theory Design and Implementation ------ ------------------------- formal semantics description of working systems explicit substitution calculi combinators type theory graph reduction techniques linear decorations run-time/memory management game theory for linear logic applications SUBMISSIONS: All researchers in the area, but especially Ph.D. students and young researchers, are encouraged to submit an extended abstract (up to 12 pages) and preferably in postscript A4 format by **February 15, 1998** via e-mail to E.Ritter@cs.bham.ac.uk or alternatively by post to Dr E. Ritter University of Birmingham School of Computer Science Edgbaston, Birmingham B15 2TT, England, UK Authors will be notified of acceptance or rejection by April 15. Final version of the accepted papers must be received in camera-ready form by June 1st, for inclusion in the informal proceedings. We are looking into formal publication of the proceedings. INVITED SPEAKERS (to be confirmed): Andrea Asperti, University of Bologna Pierre-Louis Curien, ENS, Paris Vincent Danos, Paris 7 Ian Mackie, Ecole Polytechnique Kristoffer Rose, ENS Lyon REGISTRATION: Workshop contributors will be required to register for ESSLLI-98, but they will be elligible for a reduced registration fee. IMPORTANT DATES: Feb 15, 98: Deadline for submissions Apr 15, 98: Notification of acceptance May 15, 98: Deadline for final copy Aug 17, 98: Start of workshop FURTHER INFORMATION: To obtain further information about ESSLLI-98 please visit the ESSLLI-98 home page at http://www.coli.uni-sb.de/esslli From cat-dist Wed Nov 5 17:33:38 1997 Received: by mailserv.mta.ca; id AA32402; Wed, 5 Nov 1997 17:32:21 -0400 Date: Wed, 5 Nov 1997 17:32:21 -0400 (AST) From: categories To: categories Subject: LL Workshop in Utrecht Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Tue, 28 Oct 1997 12:01:30 +0100 From: Workshop Problems Advances in the Semantics of Linear Logic ************************************************************** * * * WORKSHOP * * PROBLEMS AND ADVANCES IN THE SEMANTICS OF * * LINEAR LOGIC * * * ************************************************************** * Utrecht, 28 and 29 november 1997 * ************************************************************** On friday 28 and saturday 29 november there will be a workshop, hosted by the Mathematical Institute of Utrecht University, dedicated to recent work and work in progress in the semantics of linear logic: games, coherent and hyper coherent spaces, Chu spaces, phase semantics, bi-completion of categories, models for light linear logic et cetera. The meeting will take place in the center of the town of Utrecht. There will be a number of talks by invited speakers, including Jean Yves Girard, Samson Abramsky and Vaughan Pratt. We welcome your participation, as well as proposals for communications on work related to the subject of the workshop. For all information, please contact the organizers at: pasll@math.ruu.nl You can also look at URL: http://www.math.ruu.nl/pasll/ Vincent Danos Harold Schellinx Equipe de Logique Mathematique Mathematisch Instituut Universite Paris VII Universiteit Utrecht From cat-dist Wed Nov 5 17:33:53 1997 Received: by mailserv.mta.ca; id AA07787; Wed, 5 Nov 1997 17:33:47 -0400 Date: Wed, 5 Nov 1997 17:33:47 -0400 (AST) From: categories To: categories Subject: Re: Pratt slices Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Tue, 04 Nov 1997 11:15:46 -0800 From: Vaughan R. Pratt >From: Dr. P.T. Johnstone >So we ought surely to be looking for >a common generalization, not of toposes and abelian categories, but of >(pre)toposes and affine categories in Aurelio's sense. In that context let me rephrase my question about adding in Set\op as, is the common universal Horn theory of Set, Ab, and Set\op, along with their slices, finitely axiomatizable? Or (with or without the slices) is this nice link between Set and Ab confined to the geometric (discrete) half of mathematics? Vaughan From cat-dist Wed Nov 5 17:34:58 1997 Received: by mailserv.mta.ca; id AA17010; Wed, 5 Nov 1997 17:34:57 -0400 Date: Wed, 5 Nov 1997 17:34:57 -0400 (AST) From: categories To: categories Subject: abstract algebraic geometry Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Tue, 4 Nov 1997 14:17:40 -0800 From: Zhaohua Luo The following is the first part of "The language of analytic categories", which is a report on my paper CATEGORICAL GEOMETRY. The entire report (in LaTex) is available upon request. Again comments and suggestions are welcome. Z. Luo _________________________________________________ THE LANGUAGE OF ANALYTIC CATEGORIES By Zhaohua Luo (1997) Content 1. Analytic Categories 2. Distributive Properties 3. Coflat Maps 4. Analytic Monos 5. Reduced Objects 6. Integral Objects 7. Simple Objects 8. Local Objects 9. Analytic Geometries 10. Zariski Geometries References Appendix: Analytic Dictionary ------------------------------------------------------------- 1. Analytic Categories Consider a category with an initial object 0. Two maps u: U - -> X and v: V --> X are "disjoint" if 0 is the pullback of (u, v). Suppose X + Y is the sum of two objects with the injections (also called "direct monos") x: X --> X + Y and y: Y --> X + Y. Then X + Y is "disjoint" if the injections x and y are disjoint and monic. The sum X + Y is "stable" if for any map f: Z --> X + Y, the pullbacks Z_X --> Z and Z_Y --> Z of x and y along f exist, and the induced map Z_X + Z_Y --> Z is an isomorphism. Assume the category has pullbacks. A "strong mono" is a map (in fact, a mono) such that any of its pullbacks is not proper (i.e. non-isomorphic) epic. The category is "perfect" if any intersection of strong monos exist. If a map f is the composite m.e of an epi e followed by a strong mono m then the pair (e, m) is called an "epi-strong-mono factorization" of f; the codomain of e is called the "strong image" of f. In a perfect category any map has an epi-strong-mono factorization. An "analytic category" is a category satisfying the following axioms: (Axiom 1) Finite limits and finite sums exist. (Axiom 2) Finite sums are disjoint and stable. (Axiom 3) Any map has an epi-strong-mono factorization. Consider an analytic category. For any object X denote by R(X) the set of strong subobjects of X. Since finite limits exist, the poset R(X) has meets. Suppose u: U --> X and v: V --> X are two strong subobjects. Suppose T = U + V is the sum of U and V and t: T --> X is the map induced by u and v. Then the strong image t(T) of T in X is the join of U and V in R(X). It follows that R(X) has joins. Thus R(X) is a lattice, with 0_X: 0 --> X as zero and 1_X: X --> X as one. If the category is prefect then R(X) is a complete lattice. An object Z has exactly one strong subobject (i.e. 0_Z = 1_Z) iff it is initial. If u: U --> X is a mono, we denote by f^{-1}(u) the pullback of u along f. Then f^{-1}: R(X) --> R(Y) is a mapping preserving meets with f^{-1}(0_X) = 0_Y and f^{-1}(1_X) = 1_Y (i.e. f^{-1} is bounded). Also f^{-1} has a left adjoint f^{+1}: R(Y) --> R(X) sending each strong subobject v: V -- > Y to the strong image of the composite f. v: V --> X. If V = Y then f^{+1}(Y) is simply the strong image of f. The theories of analytic categories and Zariski geometries (including the notions of coflat maps and analytic monos) given below are based on the works of Diers (see [D] and [D1]). Note that we have only covered the most elementary part of the theory of Zariski geometries (up to the first three chapters of [D]). ----------------------------------------------------------- 2. Distributive Properties Let C be an analytic category. Recall that a "regular mono" is a map which can be written as the equalizer of some pair of maps. (2.1) The class of strong monos is closed under composition and stable under pullback; any intersection of strong monos is a strong mono. (2.2) An epi-strong-mono factorization of a map is unique up to isomorphism. (2.3) Any regular mono is a strong mono; any pullback of a regular mono is a regular mono; any direct mono is a regular mono; finite sums commute with equalizers. (2.4) Any proper (i.e. non-isomorphic) strong subobject is contained in a proper regular subobject; a map is not epic iff it factors through a proper regular (or strong) mono. (2.5) The initial object 0 is strict (i.e. any map X --> 0 is an isomorphism); any map 0 --> X is regular (thus is not epic unless X is initial). (2.6) If the terminal object 1 is strict (i.e. any map 1 --> X is an isomorphism) then the category is equivalent to the terminal category 1 (thus the opposite of an analytic category is not analytic unless it is a terminal category). (2.7) Let f_1: Y_1 --> X_1 and f_2: Y_2 --> X_2 be two maps. Then f_1 + f_2 is epic (resp. monic, resp. regular monic) iff f_1 and f_2 are so. --------------------------------------------------------------- 3. Coflat Maps A map f: Y --> X is "coflat" if the pullback functor C/X --> C/Y along f preserves epis. More generally a map f: Y --> X is called "precoflat" if the pullback of any epi to X along f is epic. A map is coflat iff it is "stable precoflat" (i.e. any of its pullback is precoflat). An analytic category is "coflat" if any map is coflat (or equivalently, any epi is stable). (3.1) Coflat maps (or monos) are closed under composition and stable under pullback; isomorphisms are coflat; any direct mono is coflat. (3.2) Finite products of coflat maps are coflat; a finite sum of maps is coflat iff each factor is coflat. (3.4) Suppose f: Y --> X is a mono and g: Z --> Y is a map. Then g is coflat if f.g is coflat. (3.5) For any object X, the codiagonal map X + X --> X is coflat. (3.6) Suppose {f_i: Y_i --> X} is a finite family of coflat maps. Then f = \sum (f_i): Y = \sum Y_i --> X is coflat. (3.7) Suppose f: Y --> X is a coflat bimorphisms. If g: Z --> Y is a map such that f.g is an epi, then g is an epi. (3.8) Suppose f: Y --> X is a coflat mono (bimorphisms) and g: Z --> Y is any map. Then g is a coflat mono (bimorphisms) iff f.g is a coflat mono (bimorphisms). (3.9) If x: X_1 --> X is a map which is disjoint with a coflat map f: Y --> X, then the strong image of x is disjoint with f. (3.10) If f: Y --> X is a coflat map, then f^{-1}: R(X) --> R(Y) is a morphism of lattice. (3.11) If f: Y --> X is a coflat mono, then f^{-1}f^{+1} is the identity R(Y) --> R(Y). (3.12) (Beck-Chevalley condition) Suppose f: Y --> X is a coflat map and g: S --> X is a map. Let (p: T --> Y, n: T --> S) be the pullback of (f, g). Then p^{+1}n^{-1} = f^{- 1}g^{+1}. ------------------------------------------------------------------ 4. Analytic Monos A mono u^c: U^c --> X is a "complement" of a mono u: U -- > X if u and u^c are disjoint, and any map v: T --> X such that u and v are disjoint factors through u^c (uniquely). The complement u^c of u, if exists, is uniquely determined up to isomorphism. A mono is "singular" if it is the complement of a strong mono. An "analytic mono" is a coflat singular mono. A mono is "disjunctable" if it has a coflat complement. An analytic category is "disjunctable" if any strong mono is disjunctable; it is "locally disjunctable" if any strong mono is an intersection of disjunctable strong monos. (4.1) Analytic monos are closed under composition and stable under pullbacks; isomorphisms are analytic monos; a mono is analytic iff it is a coflat complement of a mono; any direct mono is analytic. (4.2) The pullback of any disjunctable mono is disjunctable. (4.3) If u: U --> X and v: V --> X are two disjunctable strong subobjects of X, then u^c \cap v^c = (u \vee v)^c. (4.4) Finite intersections and finite sums of analytic monos are analytic monos. (4.5) Suppose any strong map is regular. Then C is disjunctable iff any object is disjunctable. It is locally disjunctable if there is a set of cogenerators consisting of disjunctable objects. ------------------------------------------------------------------ 5. Reduced Objects A map is "unipotent" if any of its pullback is not proper initial. A map (in fact, a mono) is "normal" if any of its pullback is not proper unipotent. A "reduced object" is an object such that any unipotent map to it is epic. A unipotent reduced strong subobject of an object X is called the "radical" of X, denoted by rad(X). A "reduced model" of an object X is the largest reduced strong subobject of X, denoted by red(X). An analytic category is "reduced" if any object is reduced. An analytic category is "reducible" if any non-initial object has a non-initial reduced strong subobject. If f: Y --> X is an epi we simply say that X is a "quotient" of Y. A "locally direct mono" is a mono which is an intersection of direct monos. An analytic category is "decidable" (resp. "locally decidable") if any strong mono is a direct (resp. locally direct) mono. (5.1) An object is reduced iff any unipotent strong mono to it is an isomorphism (i.e. any object has no proper unipotent strong subobject). (5.2) Any stable epi is unipotent; conversely any unipotent map in a reduced analytic category is a stable epi. (5.3) A unipotent strong subobject contains each reduced subobject. (5.4) A radical is the largest reduced and the smallest unipotent strong subobject (therefore is unique). (5.5) Any quotient of a reduced object is reduced; if f: Y --> X is a map and U is a reduced strong subobject of Y then its strong image f^{+1}(U) in X is reduced. (5.6) Any reduced subobject is contained in a reduced strong subobject. (5.7) The join of a set of reduced strong subobjects of an object (in the lattice of strong subobjects) is reduced. (5.8) Any analytic subobject of a reduced object is reduced. (5.9) An analytic category is reduced iff every strong mono is normal. (5.10) Any object in a perfect analytic category has a reduced model. (5.11) If X has a reduced model red(X) then any map from a reduced object to X factors uniquely through the mono r(X) --> X. (5.12) In a perfect analytic category the full subcategory of reduced subobjects is a coreflective subcategory. (5.13) The radical of an object X is the reduced model of X. (5.14) In a reducible analytic category the reduced model of an object is unipotent (thus is the radical); any object in a perfect reducible analytic category has a radical. (5.15) Any decidable or locally decidable analytic category is reduced. ----------------------------------------------------------------- 6. Integral Objects A non-initial reduced object is "integral" if any non-initial coflat map to it is epic. An integral strong subobject of an object X is called a "prime" of X. Denote by Spec(X) the set of primes of X. An analytic category is "spatial" if any non- initial object has a non-initial prime. (6.1) Any quotient of an integral object is integral; if f: Y --> X is a map and U a prime of Y, then f^{+1}(U) is a prime of X. (6.2) If U and V are two non-initial coflat (resp. analytic) subobjects of an integral object, then the intersection of U and V is non-initial. (6.3) Any non-initial analytic subobject of an integral object is integral. (6.4) In a locally disjunctable analytic category the following are equivalent for a non-initial reduced object X: (a) X is integral; (b) Any non-initial analytic mono is epic; (c) X is not the join of two proper strong subobjects in R(X). ------------------------------------------------------------------ THE END OF THE FIRST PART From cat-dist Wed Nov 5 17:35:31 1997 Received: by mailserv.mta.ca; id AA15826; Wed, 5 Nov 1997 17:35:26 -0400 Date: Wed, 5 Nov 1997 17:35:26 -0400 (AST) From: categories To: categories Subject: Re: Abelian-topos (AT) categories Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Wed, 5 Nov 1997 11:11:52 +0000 From: Steven Vickers Under one view, there is a mismatch in the comparison between toposes and Abelian categories. Consider enriched category theory over Set and Ab[elian groups]. Over Set: enriched category A = small category, A-action = functor from A to Set (covariant or contra- for right or left action), cat of A-actions = Set^A or Set^A^op, wlog a presheaf topos, "quotient" (by Grothendieck topology) = general Grothendieck topos. Over Ab: enriched category A = ringoid ("ring with several objects"), A-action = right or left module over A, cat of A-actions = Mod-A or A-Mod, "quotient" (by Gabriel topology, a.k.a. hereditary torsion theory) = Grothendieck category, i.e. cocomplete Abelian category in which direct limits are exact and there is a generator. By the Lubkin-Heron-Freyd-Mitchell theorems, Abelian categories embed fully faithfully in Grothendieck categories but are more general. Assuming this parallel Grothendieck toposes || Grothendieck categories is a good one, is there a natural parallel of Abelian categories on the Set-enriched side? Steve Vickers. From cat-dist Thu Nov 6 16:36:54 1997 Received: by mailserv.mta.ca; id AA16745; Thu, 6 Nov 1997 16:36:17 -0400 Date: Thu, 6 Nov 1997 16:36:17 -0400 (AST) From: categories To: categories Subject: Re: Abelian-topos (AT) categories Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Thu, 6 Nov 1997 10:39:35 GMT From: Michael Barr In reply to Steve Vickers' post, I have a few comments. First off, it was not Lubkin-Heron-Freyd-Mitchell who proved the full embedding theorem. The first three proved only a faithful functor into set (and subject to some smallness condition), while Mitchell showed the full embedding theorem. And not merely into a Grothendieck abelian category, but one with a small projective generator. The analogue would be an embedding into a set-valued functor category. Unfortunately, a beautiful observation of Makkai's shows that that is impossible. Makkai pointed out that under a full embedding that preserves finite limits, finite sums and epis the boolean algebra of complemented subobjects, which is classified by maps into 1 + 1, would have to be preserved. But in a functor category that lattice is complete and atomic (that is, completely distributive), so that fact, which is not true for toposes in general, becomes a necessary condition for the existence of an embedding. (Is it sufficient?) There is, of course, a full embedding theorem for small exact categories, but that is a lot less than a topos. But is true that additive + exact = abelian, so maybe that is also a good analogy. Michael From cat-dist Fri Nov 7 14:47:03 1997 Received: by mailserv.mta.ca; id AA29834; Fri, 7 Nov 1997 14:45:36 -0400 Date: Fri, 7 Nov 1997 14:45:36 -0400 (AST) From: categories To: categories Subject: Re: Abelian-topos (AT) categories Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Thu, 6 Nov 1997 16:38:20 -0500 (EST) From: Peter Freyd The result that Steve Vickers cited -- that every small abelian cateogory can be fully embedded into a Grothendieck category -- actually must have come before anything proved by Lubkin, Heron, Freyd or Mitchell. I'm sure that Grothendieck knew about the canonical representation of a small abelian category into its category of abelian pre-canonical sheaves. From cat-dist Sun Nov 9 14:22:11 1997 Received: by mailserv.mta.ca; id AA04553; Sun, 9 Nov 1997 14:20:51 -0400 Date: Sun, 9 Nov 1997 14:20:51 -0400 (AST) From: categories To: categories Subject: higher-dimensional multicategories and weak n-categories Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Fri, 7 Nov 1997 17:45:34 +0000 (GMT) From: Claudio Hermida Dear all, I've just finished scanning the slides of my talks at CT97 (Vancouver) and the AMS meeting at Montreal on Higher-dimensional multicategories aimed as a set up for weak n-categories in the Baez/Dolan sense, ie. using universally defined composites to avoid coherence conditions. The slides are a series of .JPG files accessible through my web page http://www.math.mcgill.ca/~hermida Claudio Hermida From cat-dist Mon Nov 10 17:10:27 1997 Received: by mailserv.mta.ca; id AA28086; Mon, 10 Nov 1997 17:09:27 -0400 Date: Mon, 10 Nov 1997 17:09:26 -0400 (AST) From: categories To: categories Subject: Automated Software Engineering 97 Panel on Categories for Software Engineering Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Sun, 9 Nov 1997 12:52:12 -0800 (PST) From: Joseph Goguen Hi folks! I had to prepare a report on this meeting for another purpose, and thought that a slight rewrite of it might be interesting and encouraging for this group. --Joseph *********************************************************************** Joseph Goguen, Dept. Computer Science & Engineering, University of California at San Diego, 9500 Gilman Drive, La Jolla CA 92093-0114 USA email: goguen@cs.ucsd.edu www: http://www.cs.ucsd.edu/users/goguen/ phone: (619) 534-4197 [my office]; -1246 [dept office]; -7029 [dept fax]; (619) 822-0702 [secy: Lisa Bodecker] office: 3131 Applied Physics and Math Building ****************************************************************************** >From 3 to 5 November, I attended a meeting of ASE97 (Automated Software Engineering) at Lake Tahoe Nevada; this is a major conference in the software engineering community, attended this year by about 400 people. The interest in category theory at this meeting really amazed me; I dont think such a thing could have happened even two years ago. Two tutorials (out of 6) were actually based on category theory, and both were sold out; 3 papers out of 32 heavily used category theory, and 3 more used tools or methods that involve category theory. The most surprising and hopeful event was a panel on the role of category theory in software engineering on Tuesday morning (4 Nov), organized and moderated by Mike Lowry (NASA), cochair of the conference program committee; the panel was very well attended (maybe 200 people) and was very positive. The panelists were Capt. Mark Gerken (USAF), myself (UCSD), Richard Jullig (ArrowLogics, a small categorical startup), and Doug Smith (Kestrel); we also tried to get Mike Healey (Boeing), but he didnt attend the conference. Lowry introduced the panelists and the questions they were asked to address. He then sketched the early history of category theory and some basic definitions. Gerken outlined a substantial research program, much of which involved applying a large and successful system called SpecWare from Kestrel; several applications involved generating code for complex scheduling algorithms from their (algebraic) specifications. Colimits are the main bit of category theory in SpecWare, along the lines pioneered by my work on categorical general system theory in the late 60s, and the languages Clear and OBJ that I helped develop. Gerken considers software architecture and reuse very promising areas for future research. Goguen described several past applications, noting how they used category theory in different ways, e.g., for generality (in general systems theory and institution-based specification languages), for deeper conceptual understanding (axioms for categories of fuzzy sets), and to state results more sharply (abstract data types as initial algebras, and minimal realizations as adjoints). He also described how initiality and colimits entered computer science (the former via Lawvere's axiom of infinity). Goguen suggested sheaf theory and enriched categories as promising future areas, the first especially for concurrency and the second for semiotics. Jullig described some principles behind SpecWare and the new system that his company is building, saying why colimit based systems were superior to more traditional language mechanisms, like those found in Z, ML, etc. He also suggested fibrations and categorical logic as promising future topics. Smith went deeper into principles of SpecWare, emphasizing the role of theory morphisms, e.g., in supporting structured specifications, refinement and parameterized specifications, all of which are useful for algorithm design and code generation, among other things. Smith also noted the role of the theory of institutions for avoiding commitment to any particular logic, and described several applications done with SpecWare; he was particularly happy with a classification scheme for algorithms. Members of the audience then initited a robust discussion of educational issues, for example, how category theory could be learned by software engineers, whether it is too hard or could be considered fun, what alternative formalisms exist, what to read, etc. Of course, as a participant I was at a disadvantage in taking notes, and there are no doubt many inaccuracies of ascription and omission in the above, but I still hope it may convey the flavor of the event. From cat-dist Tue Nov 11 17:18:05 1997 Received: by mailserv.mta.ca; id AA23336; Tue, 11 Nov 1997 17:16:48 -0400 Date: Tue, 11 Nov 1997 17:16:48 -0400 (AST) From: categories To: categories Subject: BRICS Int. PhD School: Call for Admission and Grant Applications Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Tue, 11 Nov 1997 01:38:58 +0100 (MET) From: Uffe Henrik Engberg [Please accept our apologies if you receive this more than once] B R I C S International PhD School in Computer Science University of Aarhus Denmark Call for Admission and Grant Applications This is a call for admission and grant applications from students to BRICS International PhD School in Computer Science at University of Aarhus, Denmark. The call is aimed at students starting August 1998, with application deadline January 1st, 1998. BRICS International PhD School is an integrated part of the BRICS (Basic Research in Computer Science) Research Centre, and both are funded by the Danish National Research Foundation. The school admits 10-15 students (Danish and foreign) annually, and it provides a substantial number of student grants. The core areas of the PhD School are: Semantics of Computation; Logic in Computer Science; Computational Complexity; Design and Analysis of Algorithms; Programming Languages; Distributed Computing; Verification; and Data Security and Cryptology. The PhD school will provide its students with a solid background in the theoretical foundation centred around BRICS activities. From this foundation, the students may either continue in one of the core areas or venture into areas of a more applied or experimental nature as possible areas of thesis specialisation. The PhD School wishes to recruit PhD students of the highest international standards. It provides an excellent research environment and scientific training facilities, and aims at making its PhD graduates attractive for a wide spectrum of employers - in private and public research and development institutions, both in Denmark and abroad. So, if you are a student with at least four years of full-time study by the summer of 1998, highly motivated and well prepared for a PhD study in computer science within a truly international environment, please send us an application following the instructions below. For more details, please visit http://www.brics.dk, or contact us by e-mail at phdschool@brics.dk. Also, we would appreciate your passing on this information to interested students and colleagues at your university or research institute. BRICS The Research Centre BRICS (Basic Research In Computer Science) was founded in 1994 by the Danish National Research Foundation at the Universities of Aarhus and Aalborg. BRICS is a centre of basic research in Algorithmics, Logic and Semantics, and has a scientific staff of 30 (permanent and long term visitors), and around 100 short term visitors annually. Admission Prerequisites and Study Structure Admission is based on knowledge corresponding to four years of full-time studies, including basic courses in programming and programming languages, computer systems, algorithms and data structures, computability and mathematics (this list will be interpreted in a flexible way) The time allocated for the PhD studies is a further four years, where the first two years include some mandatory course work and introductory research, concluded by a qualifying examination, whereas the last two years are dedicated to the writing of the thesis, finishing with a defence. All students admitted to the school will enter this study structure. We may, however, take into account merits from previous study. Students are normally admitted for the semester start September 1st, - however, we advocate arrival to Aarhus from August 1st so that there will be ample time for the students to settle in and get used to the city and the university environment - but admission may take place throughout the year. PhD Student Grants The school offers student grants of different types: 1. tuition waiver and full studentship 2. tuition waiver and partial studentship 3. tuition waiver Only a limited amount of type 1. grants are available. How to apply 1. Fill in the application form on http://www.brics.dk/PhDSchool/Application.html Alternatively, send an e-mail to phdschool@brics.dk with subject "PhD Application" including: - your full name, personal address and phone number, college / university, URL of home page (if applicable), and e-mail address, - in case you apply for grants from the PhD School, the type of your application (see above - for details see http://www.brics.dk). 2. Send by ordinary mail to the address below - a covering letter, including the information from your application form/e-mail, - a short curriculum vitae, - complete official transcripts from colleges or universities, documenting minimally four years of full time study, - names of three people whom we may ask for letters of recommendation, - an indication of your motivation for a PhD study, and particular research area of initial interest (max two pages). Applications for admission August 1st, 1998 should be sent as soon as possible and before January 1st, 1998. Decisions will be announced in March, 1998. Postal Address BRICS International PhD School Department of Computer Science University of Aarhus Ny Munkegade, Bldg. 540 DK-8000 Aarhus C Denmark Phone: +45 8942 3264 Fax: +45 8942 3255 From cat-dist Tue Nov 11 17:18:43 1997 Received: by mailserv.mta.ca; id AA22341; Tue, 11 Nov 1997 17:18:33 -0400 Date: Tue, 11 Nov 1997 17:18:33 -0400 (AST) From: categories To: categories Subject: On a remark of Barr Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Tue, 11 Nov 1997 11:35:10 -0500 (EST) From: Peter Freyd Mike Barr on the question: "is every small topos fully representable into a set-valued functor category?": Unfortunately, a beautiful observation of Makkai's shows that that is impossible. Makkai pointed out that under a full embedding that preserves finite limits, finite sums and epis the boolean algebra of complemented subobjects, which is classified by maps into 1 + 1, would have to be preserved. But in a functor category that lattice is complete and atomic (that is, completely distributive), so that fact, which is not true for toposes in general, becomes a necessary condition for the existence of an embedding. (Is it sufficient?) One can go further. The embedding will necessarily preserve all finite colimits and all infinite coproducts that happen to exist. Extend the Makkai observation as follows. For any pair of objects A and B note that the lattice of partial maps with complemented domains can be constructed as the set of maps from A to 1+B (give 1+B the "flat ordering" of CS: start with B with the trivial partial ordering and adjoin a bottom). The fact that it's a complete lattice is enough to show that arbitrary disjoint unions are coproducts. There is an argument for the case that the topos does not have a natural numbers object but I'll assume here that it does have such. The standard points of the natural numbers object are complemented, hence their union must exist; that union is a coproduct; it must therefore be the entire natural numbers object. It is preserved by the embedding. In Aspects of Topoi there's a lemma entitled "one coequalizer for all" that now suffices to show that the representation preserves all coequalizers. From cat-dist Thu Nov 13 15:58:18 1997 Received: by mailserv.mta.ca; id AA00103; Thu, 13 Nov 1997 15:56:11 -0400 Date: Thu, 13 Nov 1997 15:56:11 -0400 (AST) From: categories To: categories Subject: preprints available Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Wed, 12 Nov 1997 14:16:30 +0100 (MET) From: Anders Kock The following preprints are available: Differential Forms as Infinitesimal Cochains This is essentially my contribution at the Vancvouver Category Theory Meeting in July. It proves that the simplicial complex given by the first neighbourhood of the diagonal of a manifold (in a well adapted model for SDG) has de Rham cohomology of the manifold as its R-dual. Extension Theory for Local Groupoids We relate Extension Theory for (non-abelian) groups (a la Eilenberg-Mac Lane) with the theory of Connections (a la Ehresmann), via a notion of local groupoid. In particular, we give in this setting a kind of converse to the statement "the curvature 2-form of a connection satisfies Bianchi identity". Both these preprints are accessible via my home page: http://www.mi.aau.dk/~kock/ or directly at ftp://ftp.mi.aau.dk/pub/kock/Cochains.ps (respectively ../locg.ps) Anders Kock From cat-dist Thu Nov 13 15:58:28 1997 Received: by mailserv.mta.ca; id AA14329; Thu, 13 Nov 1997 15:56:57 -0400 Date: Thu, 13 Nov 1997 15:56:57 -0400 (AST) From: categories To: categories Subject: Limits in double categories, preprint Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Wed, 12 Nov 1997 19:34:32 +0100 From: Marco Grandis The following preprint is available: Limits in double categories by Marco Grandis and Robert Pare Abstract. We define the notion of (horizontal) double limit for a double functor F: I -> A between double categories, and we give a construction theorem for such limits, from double products, double equalisers and tabulators (the double limits of vertical arrows). Double limits can describe important tools; for instance, the Grothendieck construction of a profunctor is its tabulator, in the "double category" of categories, functors and profunctors. If A is a 2-category, the previous result reduces to Street's construction theorem of weighted limits, by ordinary limits and cotensors 2*X (the tabulator of the vertical identity of the object X). Marco Grandis Dipartimento di Matematica Universita' di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it tel: +39.10.353 6805 fax: +39.10.353 6752 home page: http://www.dima.unige.it/STAFF/GRANDIS/ From cat-dist Mon Nov 17 13:14:09 1997 Received: by mailserv.mta.ca; id AA08791; Mon, 17 Nov 1997 13:12:08 -0400 Date: Mon, 17 Nov 1997 13:12:08 -0400 (AST) From: categories To: categories Subject: last cfp MPC'98: Mathematics of Program Construction Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Content-Transfer-Encoding: QUOTED-PRINTABLE Status: O X-Status: Date: Mon, 17 Nov 1997 18:00:04 +0100 (MET) From: Johan Jeuring MPC '98 Fourth International Conference on=20 MATHEMATICS OF PROGRAM CONSTRUCTION ----------------------------------- http://www.md.chalmers.se/Conf/MPC98/ =20 June 15 - 17, 1998 Marstrand, Sweden Post-conference workshops:=20 * Workshop on Generic Programming, WGP'98 http://www.cse.ogi.edu/PacSoft/conf/wgp/ * International Workshop on Constructive Methods for=20 Parallel Programming, CMPP'98 http://brahms.fmi.uni-passau.de/cl/cmpp98/index.html * Formal Techniques for Hardware and Hardware-like=20 Systems, FTH'98 http://www.cs.chalmers.se/~ms/FTH98/ CALL FOR PAPERS The general theme of this series of conferences is the use of crisp, clear mathematics in the discovery and design of algorithms and in the development of corresponding software or hardware. The conference theme reflects the growing interest in formal, mathematically based methods for the construction of software and hardware. The goal of the MPC conferences is to report on and significantly advance the state of the art in this area. Previous conferences were held in 1989 at Twente, The Netherlands, organised by the Rijksuniversiteit Groningen, in 1992 at Oxford, United Kingdom, and in 1995 at Kloster Irsee, Germany, organised by Augsburg University. SUBMISSION =20 Full papers should be submitted in Postscript format by e-mail to reach Johan Jeuring by December 15, 1997. The details of the submission procedure can be found at http://www.md.chalmers.se/Conf/MPC98/how_to_submit.html=20 Although there is no page limit, submissions should strive for brevity. Simultaneous submission to the conference and a post-conference workshop is allowed. TOPICS The emphasis is on the combination of c o n c i s e n e s s and=20 p r e c i s i o n in c a l c u l a t i o n a l t e c h n i q u e s=20 for program construction. We solicit high quality papers on original research, typically in one of the following areas: - formal specification of sequential and concurrent programs; - constructing implementations to meet specifications; in particular, - program transformation; - program analysis; - program verification; - convincing case studies. While this list is not exclusive it is intended to show the focus of the conference. We expect to publish the proceedings as a Springer LNCS, ready at the conference. VENUE Marstrand is a small island on the beautiful westcoast of Sweden, 40 km from G=F6teborg. The charming old houses, the fortress, the walking paths, and the absence of cars make this island a very pleasant resort. There are direct flights to G=F6teborg Landvetter from most European main cities, and busses from G=F6teborg to Marstrand. PROGRAMME COMMITTEE=20 Ralph-Johan Back Finland =20 Roland Backhouse The Netherlands =20 Richard Bird UK =20 Eerke Boiten UK =20 Dave Carrington Australia =20 Robin Cockett Canada =20 David Gries USA =20 Lindsay Groves New Zealand=20 Wim Hesselink The Netherlands=20 Zhenjiang Hu Japan=20 Barry Jay Australia=20 Johan Jeuring Sweden (Chair)=20 Dick Kieburtz USA =20 Christian Lengauer Germany =20 Lambert Meertens The Netherlands =20 Sigurd Meldal Norway =20 Bernhard M=F6ller Germany Chris Okasaki USA=20 Jose Oliveira Portugal Ross Paterson UK =20 Mary Sheeran Sweden =20 Doug Smith USA =20 LOCAL ORGANISATION MPC '98 is organised by the Computing Science department of Chalmers University of Technology and University of G=F6teborg. The organisation committee consists of the following people: Patrik Jansson Johan Jeuring Marie Larsson Mary Sheeran IMPORTANT DATES Submission December 15, 1997 Notification February 9, 1998 Final version due March 30, 1998 POST-CONFERENCE WORKSHOPS The following one-day workshops are being organised in conjunction with=20 MPC '98 and will take place after the main conference. * International Workshop on Generic Programming.=20 http://www.cse.ogi.edu/PacSoft/conf/wgp/ * International Workshop on Constructive Methods for=20 Parallel Programming, CMPP'98: http://brahms.fmi.uni-passau.de/cl/cmpp98/index.html * Formal Techniques for Hardware and Hardware-like Systems, FTH'98: http://www.cs.chalmers.se/~ms/FTH98/ CORRESPONDENCE Johan Jeuring (MPC '98) Department of Computing Science Chalmers University of Technology S-412 96 G=F6teborg Sweden E-mail: mpc98@cs.chalmers.se Fax: +46 31 165655 From cat-dist Mon Nov 17 13:14:13 1997 Received: by mailserv.mta.ca; id AA00357; Mon, 17 Nov 1997 13:12:50 -0400 Date: Mon, 17 Nov 1997 13:12:50 -0400 (AST) From: categories To: categories Subject: Gentzen and Hilbert formulations of Linear logic Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Sun, 16 Nov 1997 14:01:09 -0400 (AST) From: Moneesha Mehta I am looking for ( wihtin the next week or two) a Gentzen formulation of linear logic with no empty sequence at the left. Girard formulates his with an empty sequence at the left (1995) and Retore (1997) only does the multiplicative part. I would also like a Hilbert-style formulation if one has ever been written. Does anyone have any reference(s) or ideas? Thanks, M. Mehta From cat-dist Tue Nov 18 21:10:26 1997 Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id VAA21655; Tue, 18 Nov 1997 21:09:27 -0400 (AST) Date: Tue, 18 Nov 1997 21:09:27 -0400 (AST) From: categories To: categories Subject: PhD Studentship Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Mon, 17 Nov 1997 17:59:19 GMT From: Barney Hilken Please could you bring this to the attention of any suitable candidates? My apologies to those who receive multiple copies. Barney. ----------------------------------------------------------------------- **** Research studentship leading to a PhD **** University of Manchester, Department of Computer Science. A new research project, `Topological Duality for Modal, Temporal and Program Logics', has recently been awarded funding by the Engineering and Physical Sciences Research Council. A three-year PhD studentship is on offer on this project. This is an opportunity to join an active research group working in applied logic and theoretical Computer Science. The student will gain training in research skills and learn how to apply mathematics to fundamental problems in Computer Science. The Computer Science Department of Manchester University is a large and active department, with research activities in many aspects of Computer Science. The project aims to use new results in modal logic to describe the behaviour of computing systems. The student will work on applications to dynamic logic (a logic of system states), and through this work will gain a thorough training in modal logic, category theory, semantics and theoretical computer science. Prerequisites are a good degree, at least half of which is mathematics, including knowledge of algebra, topology and logic. Some knowledge of Computer Science, though not essential, would be useful. The funding covers fees and subsistence for UK citizens, or fees only for EU citizens. The start date is as soon as possible, and not later than 1st October 1998. Investigators: Dr. David Rydeheard and Dr. Harold Simmons, with Dr. Barnaby Hilken. For further details contact: Dr. David E. Rydeheard, email: david@cs.man.ac.uk, tel: +161 275 6164 For application forms contact: Mrs. Janet Boyd, The Postgraduate Office, Department of Computer Science, The University, Oxford Road Mancheter, M13 9PL. Quoting: GR/L85756 Or email: janetb@cs.man.ac.uk From cat-dist Tue Nov 18 21:10:31 1997 Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id VAA12414; Tue, 18 Nov 1997 21:10:20 -0400 (AST) Date: Tue, 18 Nov 1997 21:10:20 -0400 (AST) From: categories To: categories Subject: cfp: Workshop on generic programming, WGP'98 Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Tue, 18 Nov 1997 15:09:21 +0100 (MET) From: Johan Jeuring Workshop on Generic Programming June 18th 1998 Marstrand, Sweden http://www.cse.ogi.edu/PacSoft/conf/wgp/ Call for papers Generic programming is about making programs more adaptable by making them more general. Generic programs often embody non-traditional kinds of polymorphism; ordinary programs are obtained from them by suitably instantiating their parameters. In contrast with normal programs, the parameters of a generic programs are often quite rich in structure. For example they may be other programs, types or type constructors, or even programming paradigms. Generic programming techniques have always been of interest, both to practitioners and theoreticians, but to date have rarely been a specific focus of research. Recent developments in functional and object-oriented programming lead the organizers of this workshop to believe that there is sufficient interest to warrant the organisation of a one-day workshop on the theme of generic programming. The workshop will be on June 18th, 1998, directly following the Mathematics of Program Construction conference to be held in Marstrand Sweden. We cordially invite all those with an active interest in this important new area to submit a short position paper on their work to one of the organizers. Deadline for submission to the workshop is Feb. 16, 1997. Notification of acceptance will be March 7, 1997 The organizers are as follows: Roland Backhouse (Cochair) Netherlands Tim Sheard (Cochair) USA Johan Jeuring Oege de Moor Bernhard Moeller Jose Oliveira Barry Jay Robin Cocket who is director of the Charity Project Karl Lieberherr Fritz Ruehr From cat-dist Tue Nov 18 21:11:25 1997 Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id VAA00022; Tue, 18 Nov 1997 21:11:24 -0400 (AST) Date: Tue, 18 Nov 1997 21:11:24 -0400 (AST) From: categories To: categories Subject: "weakened" operads Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Mon, 17 Nov 1997 16:07:01 -0800 (PST) From: john baez In their book "Homotopy invariant structures on topological structures", Boardman and Vogt construct from any topological operad O a new one WO, which can be thought of as a "weakened" version of O in which all the laws of O now hold only up to homotopy in a coherent way. Has there been any subsequent work clarifying this construction? Here I'm not interested so much in all the *other* approaches to infinite loop space machines, as in the notion of "weakening" a topological operad. For example, Boardman and Vogt point out that their construction can be used to discuss homotopy colimits, which makes me wonder if the construction of WO from O could be done slickly using homotopy colimits. Has anyone discussed this? Best, John Baez From cat-dist Thu Nov 20 13:59:48 1997 Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id NAA18334; Thu, 20 Nov 1997 13:58:08 -0400 (AST) Date: Thu, 20 Nov 1997 13:58:07 -0400 (AST) From: categories To: categories Subject: PSSL - Announcement Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Thu, 20 Nov 1997 11:52:40 +0100 (MET) From: Jiri Rosicky PRELIMINARY ANNOUNCEMENT The Federated Conference Mathematical Foundations of Computer Science and Computer Science Logic will take place in Brno (Czech Republic) during August 23-28, 1998. In this connection, I am going to hold the meeting of the Peripatetic Seminar on Sheaves and Logic in Brno over the weekend of August 29-30, 1998. As usual, there will be welcomed talks on category theory, sheaves, logic and related areas (theoretical computer science in particular). Jiri Rosicky From cat-dist Thu Nov 20 15:51:38 1997 Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id PAA23337; Thu, 20 Nov 1997 15:51:23 -0400 (AST) Date: Thu, 20 Nov 1997 15:51:23 -0400 (AST) From: categories To: categories Subject: Re: "weakened" operads Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Thu, 20 Nov 1997 14:02:49 -0500 (EST) From: James Stasheff perhaps relevant is the recent developments of bar constructions for operads cf the special case of going fromt he dg Lie operad to the L_\infty operad ************************************************************ Until August 10, 1998, I am on leave from UNC and am at the University of Pennsylvania Jim Stasheff jds@math.upenn.edu 146 Woodland Dr Lansdale PA 19446 (215)822-6707 Jim Stasheff jds@math.unc.edu Math-UNC (919)-962-9607 Chapel Hill NC FAX:(919)-962-2568 27599-3250 On Tue, 18 Nov 1997, categories wrote: > Date: Mon, 17 Nov 1997 16:07:01 -0800 (PST) > From: john baez > > In their book "Homotopy invariant structures on topological structures", > Boardman and Vogt construct from any topological operad O a new one WO, > which can be thought of as a "weakened" version of O in which all the > laws of O now hold only up to homotopy in a coherent way. > > Has there been any subsequent work clarifying this construction? Here > I'm not interested so much in all the *other* approaches to infinite > loop space machines, as in the notion of "weakening" a topological > operad. For example, Boardman and Vogt point out that their construction > can be used to discuss homotopy colimits, which makes me wonder if the > construction of WO from O could be done slickly using homotopy colimits. > Has anyone discussed this? > > Best, > John Baez > > > From cat-dist Sat Nov 22 08:56:22 1997 Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id IAA17212; Sat, 22 Nov 1997 08:55:34 -0400 (AST) Date: Sat, 22 Nov 1997 08:55:34 -0400 (AST) From: categories To: categories Subject: Request Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Sat, 15 Nov 1997 01:01:43 +0500 From: SXCC Please let me know if an electronic version of an introductory material is available on Elementary Toposes. I am also available at the e-mail address : sxclg@giascl01.vsnl.net.in Partha Pratim Ghosh From cat-dist Sat Nov 22 08:56:46 1997 Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id IAA19385; Sat, 22 Nov 1997 08:56:46 -0400 (AST) Date: Sat, 22 Nov 1997 08:56:46 -0400 (AST) From: categories To: categories Subject: abstract algebraic geometry Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Thu, 20 Nov 1997 12:40:02 -0800 From: Zhaohua Luo The following is the second part of "The language of analytic categories", which is a report on my paper CATEGORICAL GEOMETRY. Please note that Section 6 on integral objects (which was included in the first part) has been modified in order to conform with the notion of a primary object by Diers. The fact is that there are several ways to introduce a primary object in a general analytic category, and the one give by Diers (for a Zariski category) happens to be the weakest one. The new definition of an integral object given below (being a reduced primary object) is therefore weaker than the old one given in the first part of this note, but the basic properties remain the same (see (6.1) - (6.3)). On the other hand, Diers's definition of an integral object in a Zariski category (being a quotient of a simple object) is the strongest one. In practice these definitions agree in most cases (for instance, see (6.4) and (6.5) below). Z. Luo ---------------------------------------------------------------- The opposite RING^op of the category RING of commutative rings (with unit) is an analytic category, which is equivalent to the category of affine schemes. Following Diers we have the following list: RING^op RING simple field integral integral domain reduced without non-null nilpotent elements radical the residue class ring with respect to its radical pseudo-simple exactly one prime ideal quasi-primary ab = 0 => (a or b is nilpotent) primary any zero divisor is nilpotent analytically closed total ring of quotients irreducible the ideal {0} is irreducible with respect to intersection regular von Neumann regular ring local local ring generic residue quotient field ---------------------------------------------------------------- THE LANGUAGE OF ANALYTIC CATEGORIES By Zhaohua Luo (1997) Content 1. Analytic Categories 2. Distributive Properties 3. Coflat Maps 4. Analytic Monos 5. Reduced Objects 6. Integral Objects 7. Simple Objects 8. Local Objects 9. Analytic Geometries 10. Zariski Geometries References Appendix: Analytic Dictionary ---------------------------------------------------------------------- SECOND PART ------------------------------------------------------------------------ 6. Integral Objects Let C be an analytic category (i.e. a lextensive category with epi-strong-mono factorizations). A non-initial object is "primary" if any non-initial analytic subobject is epic. A non-initial object is "quasi-primary" if any two non-initial analytic subobjects has a non-initial intersection. An "integral" object is a reduced primary object. A "prime" of an object is an integral strong subobject. A non-initial object is "irreducible" if it is not the join of two proper strong subobjects. For any object X denote by Spec(X) the set of primes of X. If U is any analytic subobject of X we denote by X(U) the set of primes of X which is not disjoint with U, called an "affine subset" of X. Using (4.3) one can show that the class of affine subsets is closed under intersection. Thus affine subsets form a base for a topology on Spec(X). The resulting topological space Spec(X) is called the "spectrum" of X. Since the pullback of an analytic mono is analytic, it follows from (6.2) below that Spec is naturally a functor from C to the (meta)category of topological spaces. For instance, if C is the category of affine schemes or affine varieties then Spec coincides with the classical Zariski topology. (6.1) Any quotient of a primary object is primary; any primary object is quasi-primary. (6.2) Any quotient of an integral object is integral; if f: Y --> X is a map and U a prime of Y, then f^{+1}(U) is a prime of X. (6.3) Any non-initial analytic subobject of a primary object is primary; any non-initial analytic subobject of an integral object is integral. (6.4) Suppose C is locally disjunctable. The following are equivalent for a non-initial reduced object X: (a) Any non-initial coflat map to X is epic. (b) X is primary. (c) X is quasi-primary. (d) X is irreducible. (6.5) Suppose C is locally disjunctable. Then (a) An object is integral iff it is reduced and quasi-primary. (b) An object is integral iff it is reduced and irreducible. 7. Simple Objects A mono (or subobject) is called a "fraction" if it is coflat normal. A map to an object X is called "local" (resp. "generic") if it is not disjoint with any non-initial strong subobject (resp. analytic subobject). A map to an object X is called "quasi-local" if it does not factor through any proper fraction to X. A map to an object X is called "prelocal" if it does not factor through any proper analytic mono to X. A non-initial object is called "simple" (resp. "extremal simple", resp. "unisimple", resp. "pseudo-simple", resp. "quasi- simple", resp. "presimple") if any non-initial map to it is epic (resp. extremal epic, resp. unipotent, resp. local, resp. quasi- local, resp. prelocal). (7.1) The class of fractions is closed under composition and stable under pullback. (7.2) Any local map is quasi-local; any quasi-local map is prelocal; the class of local (resp. generic, resp. quasi-local, resp. prelocal) maps is closed under composition; a quasi- local fraction (resp. prelocal analytic mono) is an isomorphism. (7.3) Any unipotent map is both local and generic; any epi is generic. (7.4) An object X is simple (resp. extremal simple, resp. unisimple, resp. quasi-simple, resp. presimple) iff it has exactly two strong subobjects (resp. subobjects, resp. normal sieves, resp. fractions, resp. analytic subobjects). (7.5) Any simple object is integral; any extremal simple object and any reduced unisimple object is simple. (7.6) A non-initial object is pseudo-simple iff any non-initial strong subobject is unipotent; any simple object, extremal simple object, and unisimple object is pseudo-simple; any pseudo-simple object is quasi-simple; any quasi-simple object is presimple; any presimple object is primary. (7.7) Any reduced pseudo-simple object is simple; the radical of any pseudo-simple object is simple. (7.8) Suppose C is locally disjunctable reducible. The following are equivalent for an object X: (a) X is pseudo-simple. (b) X is quasi-simple. (c) X is presimple. (d) The radical of X is simple. (7.9) Suppose any coflat unipotent map is regular epic and any map to a simple object is coflat. Then (a) Any coflat mono is normal. (b) Any simple object is extremal simple and unisimple. 8. Local Objects A non-initial object X is called "local" if non-initial strong subobjects of X has a non-initial intersection M. An epic simple fraction of an integral object X is called a "generic residue" of X. A mono (or subobject) p: P --> X is called a "residue" of X if P is a generic residue of a prime of X. An object is called "regular" if any disjunctable strong mono to it is direct. An object is "analytically closed" if any epic analytic mono to it is an isomorphism. (8.1) Suppose X is a local object with the strong subobject M as above. Then M is the unique simple prime of X; any proper fraction U of X is disjoint with M; M --> X is a local map. (8.2) Any integral object has at most one generic residue, which is the intersection of all the non-initial fractions; any generic residue is a generic subobject. (8.3) Any simple fraction and any simple prime is a residue; any residue of an object is a maximal simple subobject (i.e. it is not contained in any other simple subobject); any two distinct residues of an object are disjoint with each other. (8.4) Suppose p: P --> U is a residue and u: U --> X is a fraction (resp. strong mono). Then u.p: P --> X is a residue of X. (8.5) Suppose f: P --> Z is a local map with P simple. Then Z is local and f^{+1}(P) is the simple prime of Z. (8.6) Suppose f: X --> Z is a local map and X is local. Then Z is local. (8.7) Suppose f: P --> X is a map and P is simple. Then (a) f is a local epi iff X is simple. (b) f is a local strong mono iff X is local with the simple prime P. (c) f is an epic fraction iff X is integral with the generic residue P. (8.8) Suppose C is locally disjunctable reducible. (a) Suppose f: P --> Z is a prelocal map with P simple. Then f is a local map; Z is a local object with f^{+1}(P) as the simple prime of Z. (b) Suppose f: X --> Z is a prelocal map and X is local. Then f is a local map and Z is a local object. (8.9) Any sum of regular objects is regular; any extremal quotient of a regular object is regular; any regular and presimple object is analytically closed. (8.10) Suppose C is a complete and cocomplete, well- powered and co-well-powered analytic category. Then (a) The union of any family of subobjects consisting of regular objects is regular. (b) The full subcategory of regular objects is a coreflective subcategory. (8.11) Suppose C is a locally disjunctable analytic category. Then (a) Any regular object is reduced. (b) A regular object is integral iff it is simple. ------------------------------------------------------------------ END OF SECOND PART From cat-dist Tue Nov 25 09:19:11 1997 Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id JAA26446; Tue, 25 Nov 1997 09:17:23 -0400 (AST) Date: Tue, 25 Nov 1997 09:17:23 -0400 (AST) From: categories To: categories Subject: Adjoining intedeterminates to cartesian categories Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Tue, 25 Nov 1997 10:51:06 +0000 From: Benedict R. Gaster Hi, I have a simple question concerning the adjoining of intedeterminates to cartesian (closed) categories. Before asking the question I describe the two results I am interested in then state my question. It is possible to adjoin an intedeterminate arrow X: 1 -> C to a cartesian (closed) category as follows: Proposition 1: If C is a cartesian (closed) category, X: 1 -> C an intedeterminate, and I: C -> C[X] is cartesian (closed) functor, there exists a unique cartesian functor G: C[X] -> D, such that G(X) = d and the following diagram commutes: I C -------> C[X] \ | \ | \ | \ | \ | F \ | G \ | \ | \ | \ | \ | D, where D is a cartesian category, F: C -> D is a cartesian functor, and d: 1 -> F(C). The proof of this proposition can be given directly or via an alternative definition in terms of the reader comonad and the corresponding Klesli construction (personally I find the later simply and more intuitive). As far as I am aware these constructions were originally given by Lambek [lam74], proofing the corresponding deduction theorem for cartesian closed categories, and revised by Lambek and Scott [lam&sco86]. Although the above result is well known it seems that the result of adjoining an intedeterminate object to a cartesian closed category, in particular it's proof, is not so well known. This result may be stated as follows: Proposition 2: If C is a cartesian (closed) category, X an interterminate, and I: C -> C[X] is a functor, there exists a unique cartesian (closed) functor G: C[X] -> D, such that G(X) = D, and the following diagram commutes: I C -------> C[X] \ | \ | \ | \ | \ | F \ | G \ | \ | \ | \ | \ | D, where D is a cartesian (closed) category, F: C -> D is a cartesian closed functor, and D is in Obj(D). The problem is that it is not clear how to proof Proposition 2. I believe that it is possible to use results from the work of Kelly and various colleagues in the field of Two-dimensional universal algebra [back89]. However, I was hoping that there may be a more direct route to proving this result, which does not rely on the Two-dimensional monad theory. My question is simply this: Is there a more direct, and hopefully simpler, proof of Proposition 2? Thank you for any help you can provide on this subject. Best wishes Ben -------- Benedict R. Gaster. Languages and Programming Group, University of Nottingham. A thing of beauty is a joy forever. -- John Keats (1795--1821). --------------------------------------------------------------------------- References: [back89] R. Backwell and G.M. Kelly and A.J. Power Two-Dimensional Monad Theory Journal of Pure and Applied Algebra year = 1989 [lambek74] J. Lambek Functional Completeness of Cartesian Categories Annals of Mathematical Logic} 1974 [lam&sco86] J. Lambek and P. J. Scott Introduction to higher order categorical logic Cambridge University Press Cambridge studies in advanced mathematics 7 1986 From cat-dist Tue Nov 25 14:18:36 1997 Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id OAA13231; Tue, 25 Nov 1997 14:18:14 -0400 (AST) Date: Tue, 25 Nov 1997 14:18:14 -0400 (AST) From: categories To: categories Subject: Re: Adjoining intedeterminates to cartesian categories Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Tue, 25 Nov 1997 10:16:26 -0500 (EST) From: Peter Freyd The existence is a quick consequence of the standard adjoint functor theorems. From cat-dist Wed Nov 26 13:43:22 1997 Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id NAA13738; Wed, 26 Nov 1997 13:42:30 -0400 (AST) Date: Wed, 26 Nov 1997 13:42:29 -0400 (AST) From: categories To: categories Subject: Cocompletions of Categories Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Wed, 26 Nov 97 10:19:37 +0100 From: Jiri Velebil Dear Colleagues, I wonder whether the following description of a free F-conservative completion of any category under C-colimits (where F and C are classes of small categories such that F is a subclass of C and "F-conservative" means "preserving existing F-colimits"). The description of the cocompletion is as follows: Suppose C is a class of small categories and F is a subclass of C. Let X be any category. Denote by [X^op,Set] the quasicategory of all functors and all natural transformations between them. Denote by F^op-[X^op,Set] the quasicategory of all functors which preserve F^op-limits, i.e. limits of functors d : D -> X^op with D^op in F. Claim 1. F^op-[X^op,Set] is reflective in [X^op,Set] (The proof uses the fact that the above Claim holds for the case when X is small - Korollar 8.14 in Gabriel, Ulmer: Lokal pr"asentierbare Kategorien.) By Claim 1., F^op-[X^op,Set] has all small colimits. Denote by D(X) the closure of X (embedded by Yoneda) in F^op-[X^op,Set] under C-colimits. Then one can prove that D(X) is a legitimate category. The codomain-restriction I: X -> D(X) of the Yoneda embedding fulfills the following: 1. D(X) has C-colimits. 2. I preserves F-colimits. 3. D(X) has the following universal property: for any functor H : X -> Y which preserves F-colimits and the category Y has C-colimits there is a unique (up to an isomorphism) functor H* : D(X) -> Y such that H* preserves C-colimits and H*.I = H. In fact, this gives a 2-adjunction between C-CAT_C : the 2-quasicategory of all categories having C-colimits, all functors preserving C-colimits and all natural transformations and CAT_F : the 2-quasicategory of all categories, all functors preserving F-colimits and all natural transformations. The result also holds for V-categories, instead of a class C of small categories one has to work with a class of small indexing types. Thank you, Jiri Velebil velebil@math.feld.cvut.cz Department of Mathematics FEL CVUT Technicka 2 Praha 6 Czech Republic From cat-dist Wed Nov 26 13:43:26 1997 Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id NAA14209; Wed, 26 Nov 1997 13:43:12 -0400 (AST) Date: Wed, 26 Nov 1997 13:43:12 -0400 (AST) From: categories To: categories Subject: Algebraic Theories/Operads Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Wed, 26 Nov 1997 08:57:36 -0600 (CST) From: David Metzler I'm looking for a good reference on the relation between algebraic theories (a la Lawvere) and operads. David Metzler From cat-dist Thu Nov 27 16:06:21 1997 Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id QAA05940; Thu, 27 Nov 1997 16:01:50 -0400 (AST) Date: Thu, 27 Nov 1997 16:01:49 -0400 (AST) From: categories To: categories Subject: Big Omega available Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from QUOTED-PRINTABLE to 8bit by mailserv.mta.ca id QAA05940 Status: RO X-Status: Date: Thu, 27 Nov 1997 10:31:13 +1100 From: Ross Street Dear Colleagues This is to announce the availability at http://www-math.mpce.mq.edu.au/~mbatanin/Bigomega.ps of a short 10 page note which begins as follows: ********************************************************************** "The universal property of the multitude of trees" Michael Batanin and Ross Street Macquarie University, N S W 2109 AUSTRALIA Email and November 1997 Lawvere [BL] essentially pointed out that the category Delta, whose objects are finite ordinals and whose arrows are order-preserving functions, is the generic monoidal category containing a monoid. Let Mon be the category of monoids in the category Set of sets. Bénabou [Be] pointed out that the (simplicial) nerve of the category Delta is the standard resolution [BB] of the terminal monoid via the comonad generated by the underlying functor Mon --> Set and its left adjoint. Let Omcat denote the category of omega-categories and let Glob denote the category of globular sets. In this note we announce a generic property of the category BigOmega whose nerve is the standard resolution of the terminal omega-category via the comonad generated by the underlying functor Omcat --> Glob and its left adjoint. We also give a concrete model for BigOmega in terms of trees. Furthermore, we make connections with the recent work of Joyal [J]. Full proofs of our claims will appear elsewhere. ************************************************************************** Regards, Michael Batanin and Ross Street From cat-dist Thu Nov 27 16:06:21 1997 Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id QAA06096; Thu, 27 Nov 1997 16:03:05 -0400 (AST) Date: Thu, 27 Nov 1997 16:03:05 -0400 (AST) From: categories To: categories Subject: Re: Algebraic Theories/Operads Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Wed, 26 Nov 1997 16:22:14 -0800 (PST) From: john baez David Metzler writes: > I'm looking for a good reference on the relation between > algebraic theories (a la Lawvere) and operads. Me too! The closest thing I know is Boardman and Vogt's book on homotopy invariant algebraic structures --- which uses the framework of PROPs rather than operads. From cat-dist Thu Nov 27 16:06:21 1997 Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id QAA06255; Thu, 27 Nov 1997 16:04:05 -0400 (AST) Date: Thu, 27 Nov 1997 16:04:05 -0400 (AST) From: categories To: categories Subject: Re: Algebraic Theories/Operads Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Thu, 27 Nov 1997 10:40 +0530 From: CAYLEY@tifrvax.tifr.res.in The volume 202 of Contemprory mathematics series of the AMS titled Operads: Proc. of Renaissance conferences ed by J.Loday et al is a possible source. P.S.Subramanian, Tata Institute. From cat-dist Thu Nov 27 16:07:06 1997 Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id QAA06369; Thu, 27 Nov 1997 16:04:51 -0400 (AST) Date: Thu, 27 Nov 1997 16:04:50 -0400 (AST) From: categories To: categories Subject: RE: Algebraic Theories/Operads Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Thu, 27 Nov 1997 16:10:31 MET From: Heinrich.Kleisli@unifr.ch May, J. P.: Operads, algebras and modules. Contemp. Math. 202, 15-31 (1997).