June 2012 - Categories - categories.org.au

Partial functor
by David Leduc 16 Mar '15

16 Mar '15

Hi, A partial functor from C to D is given by a subcategory S of C and a functor from S to D. What is the appropriate notion of natural transformation between partial functors that would allow to turn small categories, partial functors and those "natural transformations" into a bicategory? The difficulty is that two partial functors from C to D might not have the same definition domain. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Research positions
by Maria Manuel Clementino 26 Feb '14

26 Feb '14

5-YEAR RESEARCH POSITIONS The Centre for Mathematics of the University of Coimbra (CMUC) invites applications for five 5-year research positions in all areas of Mathematics. Applicants should have a PhD in Mathematics and at least three years of post-doctoral experience, with an international-level publication record and demonstrated ability to perform independent research. Preference will be given to candidates working in one of the areas in which the members of CMUC are currently active. A very good command of English is required. The appointments will be made for three years, renewable for two more. Successful applicants are expected to develop research within the areas of interest of CMUC. They should also contribute to graduate studies, supervising students from the Department of Mathematics. The basic salary is equivalent to the index 195 of the Portuguese research career (in the order of 42 500 euros a year), in accordance with the Ciência 2008 guidelines http://alfa.fct.mctes.pt/apoios/contratacaodoutorados/edital2008 Benefits include private health insurance and social security. Applicants should send • cover sheet (available from www.mat.uc.pt/cmuc/coversheet.pdf) including names and contact information of three persons who can provide a letter of recommendation • curriculum vitae (publication list included) • a 5-year research proposal (2 pages max), emphasizing their own research goals as pdf files to cmuc(a)mat.uc.pt. The subject of the e-mail message must include the job reference C2008 FCTUC CMUC v15-v19. The deadline for applications is *September 10, 2008.* ------------------------------ More information can be found at www.mat.uc.pt

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separable locale
by Thomas Streicher 17 May '13

17 May '13

Recently rereading Fourman's "Continuous Truth" I came across the term "separable locale" but could nowhere find an explanation. Does it mean a cHa A for which there exists a countable subset B such that ever a in A is the supremum of those b in B with b leq a. This would be the point free account of "second countable", i.e. having a countable basis. Of course, second countable T_) spaces are separable, i.e. have a countable dense set. Is this reading the "usual" one? Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Two questions
by Michael Barr 15 Jul '12

15 Jul '12

Googling around, I have come on several claims that there are no non-trivial injectives in the category of groups (e.g., Mac Lane in the 1950 Duality for groups paper credits Baer with an elegant proof, but gives no hint of what it might be and Baer's earlier paper on injectives doesn't mention it). I have not come on any proof of this, however. Somewhere I have seen a proof that all monics in the category of groups are regular. I think it was in a paper by Eilenberg and ??? and it needed a special argument if there were elements of order 2. Can someone help me find this? Michael -- The United States has the best congress money can buy. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Computing with lambda-terms: a Festschrift dedicated to Corrado Bohm
by Stefano Guerrini 29 Jun '12

29 Jun '12

====================================================================== Computing with lambda-terms A special issue dedicated to Corrado Bohm for his 90th birthdays http://www.lipn.univ-paris13.fr/~guerrini/bohm90/ ====================================================================== In honor of Corrado Bohm at the occasion of his 90th birthday, a Festschrift will be published as a special issue of Mathematical Structures in Computer Science. The Festschrift will be presented during a special event in honor of Corrado Bohm at ETAPS 2013, 16-24 March 2013, Rome, Italy. SCOPE ===== We invite submissions on all the topics covered by Corrado Bohm's research and in particular new contributions on syntax and semantics of lambda-calculus, logic and type systems, functional programming. We also welcome surveys or position papers on current and future directions of research on the above topics. A more detailed, but not exclusive, list of topics of interest for submissions include: Lambda-calculus and higher-order rewriting systems: properties of reductions, reduction strategies, combinatory logic, Bohm-out and Bohm-trees, variants and extensions of lambda-calculus, term and graph rewriting systems, higher-order reduction. Semantics of lambda-calculus: Bohm-trees, models and theories of lambda-calculus, realizability. Logic and types: type systems, linear logic, types and logics for resources, implicit computational complexity. Functional programming: foundational aspects of functional programming, Curry-Howard isomorphism, applications and implementations of functional programming. New and alternative approaches to computing: quantum calculi, biologically inspired models and calculi. Submitted papers should stay within the scope of MSCS. SUBMISSION ========== The results must be unpublished and not submitted for publication elsewhere. Submissions will be refereed according to the usual high standards of MSCS. Submissions should not exceed 20 pages of content (excluding references and appendices). Submissions exceeding 25 pages (including references and appendices) will require a specific justification and the agreement of the editors. Further information and instructions about submission will be available at http://www.lipn.univ-paris13.fr/~guerrini/bohm90/ PLANNED SCHEDULE ================ Title and short abstract submission: 1 september 2012 Deadline for submission: 15 september 2012 (STRICT!) First referee report: 15 december 2012 Revised version: 15 january 2013 Notification: 1 march 2013 Final version: 1 june 2013 GUEST EDITORS ============= Henk Barendregt (henk at cs.ru.nl) Stefano Guerrini (guerrini at univ-paris13.fr) Adolfo Piperno (piperno at di.uniroma1.it) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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internal functional relations versus arrows
by Eduardo J. Dubuc 28 Jun '12

28 Jun '12

I am not an expert on elementary topos (meaning by this to work with the internal language in a Grothendiek topos) and I rather be told than struggle with the following: Consider an elementary topos EE, a locale G in EE, the unique locale morphism i^*: Omega --> G, and any arrow b: X x Y --> G: Consider the following formulae on "b": ed): Supremun_y b(x, y) = 1 uv): b(x, y) inf b(x, y') <_= i^*[[y = y']]. Let u: X x Y --> Map(X, Y) be the universal locale furnished with such an arrow (that is, forall b exists! t^* : Map(X, Y) --> G t^* u = b. Gavin Wraith (in "Localic Groups", Cahiers de Top. et Geom. Diff. Vol XXII-1 1981) defines an object Points(G) = LocalMorphisms(G, Omega) C Omega^G and says that it is clear that: a) Points(Map(X, Y)) = Y^X. From this taking global sections it follows that: b) There is a bijection between arrows X x Y --> Omega and arrows X --> Y (where the arrow on the right satisfy ed) and uv)) QUESTION 1] I ask for a convincing proof of a), or better, of the weaker ? b). Concerning b), consider the following conditions on a relation R C X x Y: exed) pi_1: R --> X is an epimorphism. exuv) The family y = pi_2 (x, y): Z --> R --> Y is a compatible family with respect to the family x = \pi_1 (x, y): Z --> R --> X (indexed by all (x, y): Z --> R) Then, using that epis are strict it follows using standard category theory: R satisfy exed) and exuv) iff exists! f: X --> Y such that R = Gamma_f (the Graph of f) Thus, b) will follow if we can prove : R satisfy exed) and exuv) iff cf_R satisfy ed) and uv) (cf_R = characteristic function). This is more related with the formula uv'): b(x, y) inf b(x', y') inf i^*[[x = x']] <_= i^*[[y = y']]. Also, by standard category theory it follows that "exed) + exuv)" are equivalent to "pi_1 is mono and epi". This may help ? NOTE: All the above is very clear in the topos of Sets, but my problem is that I am very uncomfortable when the experts write "we do as if the base topos is the topos of Sets". Does this slogan apply to the above ?. SUBQUESTION] Are uv) and uv') equivalent ? Now a more substantial question: QUESTION 2] Consider now a geometric morphism g: FF --> EE. We have a bijection between arrows X x Y --> g_* Omega_FF and arrows g^*X x g^*Y --> Omega__EE. I want to know if the arrows satisfying ed) and uv) correspond under this bijection. e.d. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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a note by Misha Gromov on category theory
by Micha Gavrilovich 28 Jun '12

28 Jun '12

Dear All, I thought the list members may be interested in the following recent note of Misha Gromov which *very* carefully uses ideas of category theory to talk about entropy. Particuarly, his point of view on category theory may be appealing. In a Search for a Structure, Part 1: On Entropy. June 19, 2012 http://www.ihes.fr/~gromov/PDF/structres-entropy-june-2012.pdf At the end of the paper, he says: Yet, I hope that I managed to convey the message: the mathematical language d eveloped by the end of the 20th century by far exceeds in its expressive power anything, even imaginable, say, before 1960. Any *meaningful* idea coming from science can be fully developed in this language. In the paper, he also says: To see this, you just need to realize that ”something” of a physicist, is a covariant functor from a suitable ”category of protocols” to the category of sets – outcomes of exper- iments; all you have to do afterwards is to follow the guidelines prescribed by the syntax of category theory. (Arguably, the category language, some call it ”abstract”, reflects mental undercurrents that surface as our ”intuitive reasoning”; a comprehensive math- ematical description of this ”reasoning”, will be, probably, even farther removed from the ”real world” than categories and functors.) Sincerely, Misha [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Coproducts of monads
by Jiri Adamek 27 Jun '12

27 Jun '12

This is to announce an extended abstract of our paper Coproducts of Monads in Set that will be presented at the conference LICS 2012. In the category of monads on Set we give a necessary and sufficient condition for a coproduct to exist, and a concrete formula for the coproduct. If one of the monads is inconsistent, i.e. a submonad of the (trivial) terminal monad, the coproduct is inconsistent. Another case where the coproduct always exists is the exception monad SX = X+E: the coproduct with another monad T is given by T(X+E). Theorem. Two consistent monads on Set have a coproduct iff they have arbitrarily large joint fixed points, or one is a submonad of the exception monad. The coproduct injections are monomorphisms. Example. A monad S has a coproduct with the power-set monad iff it has coproducts with all monads. This is the case precisely when S i a submonad of the exception monad, or is inconsistent. The extended abstract can be downloaded at http://www.iti.cs.tu-bs.de/~milius/publications/papers.html J. Adamek, N. Bowler, P. Levy and S. Milius [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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advertising a postdoc at CS in Oxford
by Mehrnoosh Sadrzadeh 26 Jun '12

26 Jun '12

Hi, would it be possible to advertise this post, if you see appropriate? It is a logic-based postdoc but, with applications in mind Many thanks. Postdoctoral Research Assistant - Logic for Autonomous Systems Department of Computer Science, University of Oxford Salary in range £29,249-£35,938 per annum (grade 7) The Department of Computer Science of University of Oxford seeks a postdoctoral researcher to work on a 3-year EPSRC project entitled `New Foundational Structures for Engineering Multi-UAV Protocols'. This interdisciplinary project aims to verify autonomous Unmanned Aerial Vehicles (UAV's) that not only navigate, but also communicate with each other. The project aims to span the full verification lifecycle: from high-level logical specifications, to oracle-based test-set reasoning through a bounded model checker, to low-level code on our four Pelican UAVs. The position requires a researcher with a proven track record of high-quality research in logic and its applications - of particular relevance are formalisms such as: - Modal and Epistemic Logics - Dynamic Program Logics - Probabilistic and Fuzzy logics Preference will be given to researchers with experience in modelling and reasoning about epistemic properties of protocols of communicative Multi-Agent Systems. The researcher will work closely with Dr Mehrnoosh Sadrzadeh and Professor Michael Wooldridge, and will also be expected to interact with Professor Daniel Kroening and his team working on the UAV and verification aspects of the project in order to derive requirements and case studies. Informal enquiries are welcome, and should be directed to Dr Sadrzadeh (mailto:Mehrnoosh.Sadrzadeh@cs.ox.ac.uk). This post is a full-time appointment, and is available for up to 3 years. Closing date for applications: **12 noon on Monday 30 July 2012** [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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The method of mathematics
by Venkata Rayudu Posina 26 Jun '12

26 Jun '12

Dear All, Is the method of mathematics a discipline in the Kantian sense of containing deviation [with ‘do not’] from rule (in addition to being confined to rule with definition)? For example, in studying the definition of function I, in addition to drawing an internal diagram with an arrow from every dot in domain to a dot in codomain, do not draw more than one arrow from any dot in the domain. But the problem with attending to ‘do not’ is a problem of enumerating error(s). I’d appreciate very much any corrections and clarifications that you, your time permitting, could provide. Thank you, posina [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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