September 2002 - Categories - categories.org.au

Presheaves etc. in a uniform way
by Vaughan Pratt 09 Sep '02

09 Sep '02

Presheaves etc. in a uniform way Set^C is normally produced directly from the small category C and the category Set as a functor category. Likewise Chu(Set,\Sigma) is produced from the set \Sigma and the category Set via the Chu construction. We give here a uniform way of producing all presheaf categories at once, or all Chu(Set,-) categories at once, without taking any of Set, C, or \Sigma as explicit parameters. This generalizes my note "Chu(Set,K) without sets" of a week ago, in the process answering and extending question (iii) of http://boole.stanford.edu/pub/couple.pdf Let G,K be two sets (of object symbols). Define a GK-category (C,g,k) to consist of a locally small category C and a pair of maps g:G->ob(C), k:K->ob(C) (interpreting object symbols as objects of C). Define a GK-functor F:(C,g,k)->(C',g',k') to be a functor F:C->C' between GK-categories satisfying Fg=g', Fk=k' and fully faithful on homsets from im(g) and homsets to im(k). Write GK-CAT for the 2-category of GK-categories and GK-functors. (GK-Cat is the same with every C small rather than merely locally small.) Associate to each (C,g,k) the full subcategories \G, \K of C having as objects those of im(g), im(k) respectively, together with a GxK matrix Q of cardinals giving the cardinality |C(g(i),k(j))| for i,j in G,K. These associated entities are all preserved up to isomorphism by GK-functors. GK-CAT thus partitions as a sum of connected components over all isomorphism classes of categories \G and \K with respectively |G| and |K| objects, and all GxK matrices of cardinals. It is a straightforward exercise to show that each such connected component of GK-CAT has a 2-final object, which we may call the locally 2-final objects of GK-CAT. These objects are equivalent to the following under the stated conditions. Set |G| = 1, |K| = 0, \G = 1 (the one-morphism category) Set\op |G| = 0, |K| = 1, \K = 1 Set x Set\op |G| = 1, |K| = 1, \G = 1, \K = 1, Q[0,0] = 1 Set^{M\op} (M-sets) |G| = 1, |K| = 0, M = \G (as a one-object category) (Set^M)\op |G| = 0, |K| = 1, M = \K Set^{C\op} (presheaves) |K| = 0, C = \G All presheaf categories |K| = 0 (Set^D)\op (dual preshvs) |G| = 0, D = \K Set^{C\op}x(Set^D)\op C = \G, D = \K, Q[i,j] = 1, i,j in G,K Chu(Set,\Sigma) |G| = |K| = 1, \G = \K = 1, Q[0,0] = \Sigma All Chu(Set,-) categories |G| = |K| = 1, \G = \K = 1 "Presheaf Chu" No restrictions The difference between Set^{C\op} x (Set^D)\op and Presheaf Chu is that the restriction Q[i,j] = 1 (meaning that homsets from im(g) to im(k) have exactly one morphism) of the former nullifies the effect of the Chu matrices (one matrix for the ordinary Chu construction). When this restriction is dropped the notion of continuity enters, with ordinary topological continuity at Q[0,0] = 2 and more general Chu continuity obtaining for larger cardinals in Q. It is plausible that a necessary and sufficient condition for a presheaf-Chu category to be *-autonomous is for \G and \K to be isomorphic, as a generalization of \G = \K = 1. Vaughan Pratt

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PSSL 77, 2nd and final announcement
by J=FCrgen Koslowski 08 Sep '02

08 Sep '02

PERIPATETIC SEMINAR ON SHEAVES AND LOGIC 77th meeting - second and final announcement http://www.iti.cs.tu-bs.de/TI-INFO/koslowj/PSSL77.html News since the first announcement: Visitors from foreign countries who intend to arrive early should note that Thursday, October 3, is a holiday in Germany. The situation concerning accommodation seems to be less problematic than I made it sound in some postings about a month ago. This was due to my (then) imminent departure for CTCS. Please don't hesitate to visit us in Braunschweig on account of that premature alarm! Reminder of the first announcement: The 77th meeting of the seminar will be held at the Department of Theoretical Computer Science of the Technical University in Braunschweig, Germany, over the weekend of October 5/6, 2002. As usual, the seminar welcomes talks using or addressing category theory or logic, either explicitly or implicitly, in the study of any aspect of mathematics, computer science or science in general. Braunschweig is located about 60 km East of Hannover and can easily be reached by car or train. The closest airport is in Hannover. Since Expo 2000, there is a fast train connection from that airport to the Hannover main train station. Trains to Braunschweig leave at least once every hour and take about 40 minutes or less. For train travel within Germany and from neighboring countries you also may consult =09http://reiseauskunft.bahn.de/bin/query.exe/e Your destination should be entered as "Braunschweig Hbf". We will send further information on the location of the seminar, along with details on local travel and accommodation to those who register, at least a week before the meeting. Electronic registration is available on the Conference Page =09http://www.iti.cs.tu-bs.de/TI-INFO/koslowj/PSSL77.html. We are looking forward to welcoming you to our new quarters in the Informatikzentrum of the Technical University of Braunschweig! Jiri Adamek Juergen Koslowski Stefan Milius Jiri Velebil --=20 Juergen Koslowski If I don't see you no more on this world ITI, TU Braunschweig I'll meet you on the next one koslowj(a)iti.cs.tu-bs.de and don't be late! http://www.iti.cs.tu-bs.de/~koslowj Jimi Hendrix (Voodoo Child, SR)

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Topos question
by Galchin Vasili 07 Sep '02

07 Sep '02

Hello CT Community, I am continuing my study of topoi by reading Goldblatt. In an arbitrary topos E, can we demonstrate that Hom (1, omega) contains any arrows/morphisms other than "true"? I suspect the answer is yes because Hom (1, omega) is a Heyting algebra. Hence, it must have a least element ("true" is the greatest element). I just don't know how to to prove that Hom (1, omega) contains more elements than "true". Can somebody point me in the right direction? Thank you, Bill Halchin

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NEW! FMCO: Symposium on Objects and Components
by F.S.de.Boer＠cwi.nl 06 Sep '02

06 Sep '02

We apologize for the reception of multiple copies. **************************CALL FOR PARTICIPATION***************************= **** First International Symposium on FORMAL METHODS FOR OBJECTS AND COMPONENTS (FMCO 2002) http://fmco.liacs.nl/fmco02.html IMPORTANT DATES - The symposium will be held November 5-8, 2002. - Deadline for REGISTRATION is September 15, 2002 NOTE The deadline is APPROACHING!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! PLACE Leiden, The Netherlands INVITED SPEAKERS Tutorials Manfred Broy (Technical University of Munchen) Werner Damm (University of Oldenburg) David Harel (The Weizmann Institute of Science) Bertrand Meyer (ETH Zurich and ISE) Perdita Stevens (Edinburgh University) Clemens Szyperski (Microsoft Research) Jos Warmer (Klasse Objecten) Technical presentations Erika Abraham-Mumm (CAU) Farhad Arbab (CWI) David Garlan (Carnegie Mellon University) John Hatcliff (Kansas State University) Jozef Hooman (University of Nijmegen) Bart Jacobs (University of Nijmegen) Paul Klint (CWI) Leslie Lamport (Microsoft Research) Gary Leavens (Iowa State University) Rustan Leino (Microsoft Research) Ugo Montanari (Pisa University) Oscar Nierstrasz (University of Berne) Ernst-Ruediger Olderog (University of Oldenburg) Amir Pnueli (The Weizmann Institute of Science) Bernhard Rumpe (Technical University of Munchen) Emil Sekerinski (McMaster University) Kaisa Sere (=C5bo University) Joseph Sifakis (Verimag) Martin Wirsing (Ludwig-Maximilians-Universit=E4t M=FCnchen) THEME Large and complex software systems provide the necessary infrastucture in all industries today. In order to construct such large systems in a systematic manner, the focus in the development methodologies has switched in the last two decades from functional issues to structural issues: both data and functions are encapsulated into software units which are integrated into large systems by means of various techniques supporting reusability and modifiability. This encapsulation principle is essential to both the object-oriented and the more recent component-based sofware engineering paradigms. Formal methods have been applied successfully to the verification of medium-sized programs in protocol and hardware design. However, their application to the development of large systems requires more emphasis on specification, modeling and validation techniques supporting the concepts of reusability and modifiability, and their implementation in new extensions of existing programming languages like Java. The objective of this symposium is to bring together researchers and practioners in the areas of software engineering and formal methods to discuss the concepts of reusability and modifiability in component-based and object-oriented software systems. The following issues related to modifiability and reusability in components and objects will be discussed by leading experts in the fields of Formal Methods and Software Engineering: - Specification - Verification - Validation - Modeling techniques - Architectures - Programming languages - Semantics - Type theory - Design methodology - Compositionality - Refinement - Theorem-proving - Model checking - Tools FORMAT The Symposium is a four days event in the style of the former REX workshops, organized to provide an atmosphere that fosters collaborative work, discussions and interaction. Lectures are given by the invited speakers listed above. Participation is limited to about 80 people. The program consists of invited tutorials and more technical presentations, and contains an exquisite social event. Both tutorial and technical contributions will be published after the workshop in a proceedings of Lecture Notes in Computer Science by Springer-Verlag. ORGANISING COMMITTEE F.S. de Boer (CWI) M.M. Bonsangue (LIACS-Leiden University) S. Graf (Verimag) W.-P. de Roever (CAU) *****************************PARTICIPATION*********************************= ** For more information about participation and registration see the FMCO site http://fmco.liacs.nl/fmco02.html For further information consult F.S. de Boer (frb(a)cwi.nl) or M.M. Bonsangue (marcello(a)liacs.nl)

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preprint: Computads and slices of operads
by Michael Batanin 04 Sep '02

04 Sep '02

Dear colleagues, my new preprint is now available in Category Theory section http://au.arxiv.org/abs/math.CT/0209035 You also can download it from my home page http://www.math.mq.edu.au/~mbatanin/papers.html Below is the abstract ------------------------------------- \\

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None
by Michael A.Batanin 04 Sep '02

04 Sep '02

Title: Computads and slices of operads Authors: M.A.Batanin Comments: 18 pages Subj-class: Category Theory MSC-class: 18C20, 18D05 \\ For a given $\omega$-operad $A$ on globular sets we introduce a sequence of symmetric operads on $Set$ called slices of $A$ and show how the connected limit preserving properties of slices are related to the property of the category of $n$-computads of $A$ being a presheaf topos. \\

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preprint: On the representation theory of Galois and Atomic Topoi
by Eduardo J. Dubuc 03 Sep '02

03 Sep '02

This is to announce the posting at the los alamos site of the following paper: to download the paper: http://arXiv.org/abs/math.CT/0208222 Title: On the representation theory of Galois and Atomic Topoi Authors: Eduardo J. Dubuc Comments: 34 pages Subj-class: Category Theory We elaborate on the representation theorems of topoi as topoi of discrete actions of various kinds of localic groups and groupoids. We introduce the concept of "proessential point" and use it to give a new characterization of pointed Galois topoi. We comment and develop on Grothendieck's galois theory and its generalization by Joyal-Tierney, and related work on these theories within the contex of topos theory by other authors. We have the hierarchy (for connected topoi): 1. essentially pointed Atomic = locally simply connected. 2. proessentially pointed Atomic = pointed Galois. 3. pointed Atomic The corresponding group of automorphisms of the point in the fundamental theorem are: 1. discrete group. 2. prodiscrete localic group. 3. localic group. We analyze also the respective groupoid versions (allways for connected topoi) and show that the groupoid in the fundamental theorem can allways be interpreted as the groupoid of all points. 1. connected discrete groupoid. 2. connected groupoid with discrete space of objects and prodiscrete localic spaces of hom-sets. 3. connected groupoid with discrete space of objects and localic spaces of hom-sets. We analyze also the unpoited version, and show that for a Galois topos, may be pointless, the groupoid can also be considered as the groupoid of "points". 1. allways with points. 2. connected (may be pointless) prodiscrete localic groupoids. 3. we do not know how to define the groupoid of points for a unpointed (may be pointless) connected atomic topos. 30-Sep-2002 14:23:08 -0300,869;000000000000-00000000

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TACReprints #1: Metric spaces, generalized logic and closed categories
by TAC 02 Sep '02

02 Sep '02

This is to announce the first in the TAC Reprints series: Metric spaces, generalized logic and closed categories F. W. Lawvere Originally published as: Rendiconti del Seminario Matematico e Fisico di Milano, , XLIII (1973), 135-166. Keywords: Metric spaces, enriched categories, logic 2000 MSC: 18D20 Republished in: Reprints in Theory and Applications of Categories, No. 1 (2002) pp 1-37 http://www.tac.mta.ca/tac/reprints/articles/1/tr1.pdf

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