August 2012 - Categories - categories.org.au

Partial functor
by David Leduc 17 Mar '15

17 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/ ]

7 6

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

1 1

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/ ]

5 4

Re: [TYPES/announce] IAS program wiki
by Philip Scott 31 Aug '12

31 Aug '12

Hi Bob. Below is an announcement from Types, which may be interesting to others on the categories list: they are having a year on dependent types at the Institute of Advanced Study (run by Steve Awodey, Thierry Coquand, and Vladimir Voevodsky) , and many category theorists will be there (some even resident) during the course of the year. Here is the website: http://www.math.ias.edu/sp/univalent For now, though, the wiki started below is still very basic. Cheers, Phil Begin forwarded message: > From: Vladimir Voevodsky <vladimir(a)ias.edu> > Date: August 30, 2012 2:20:24 PM EDT > To: Types list list <types-announce(a)lists.seas.upenn.edu>, Coq Club <coq-club(a)inria.fr>, agda list <agda(a)lists.chalmers.se> > Subject: [TYPES/announce] IAS program wiki > > [ The Types Forum (announcements only), > http://lists.seas.upenn.edu/mailman/listinfo/types-announce ] > > Hello, > > I have created a wiki for the forthcoming special year on Univalent Foundations at IAS. The wiki is in a very rudimentary stage yet but it will eventually get more content. It is here: uf-ias-2012.wikispaces.com > > Vladimir. > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

1 0

Re: question on terminology
by claudio pisani 28 Aug '12

28 Aug '12

--- Sab 25/8/12, Fred E.J. Linton <fejlinton(a)usa.net> ha scritto: > Da: Fred E.J. Linton <fejlinton(a)usa.net> > Oggetto: Re: categories: question on terminology > A: "claudio pisani" <pisclau(a)yahoo.it>, categories(a)mta.ca > Data: Sabato 25 agosto 2012, 05:35 > Claudio Pisani asked, > > > Is there a standard name for those presheaves X on a > > category C such that Xf is a bijection for any f in C? > > Well, those presheaves are exactly the "restrictions to C" > of the > presheaves on the grouppoid reflection (the grouppoidal > 'quotient') of C > (by which I mean the category got by declaring invertible > every C-morphism). > > Does that suggest "grouppoidal action of C" might work? I > think I'd tend > to lobby against the use of the prefix "bi-" unless there > were *really* > compelling reasons in favor of it. > > Cheers, -- Fred > Dear Fred, thanks for the suggestion. It seems to me that its disadvantage is that "groupoidal action of C" may suggest that C itself is a groupoid, but probably the ambiguity disappears in the right context. By the way, I am actually interested in the (full and faithful, indexed) inclusion of presheaves on C' (where C' is the groupoid reflection of C) in presheaves on C and C^op (that is of groupoidal actions in left and in right actions). In fact it seems to provide a useful link between left and right actions. Claudio [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

3 2

Profunctors and spans of sets
by Mike Stay 27 Aug '12

27 Aug '12

A span of sets X <-f- S -g-> Y can be viewed as a profunctor between discrete categories by assigning to each pair (x,y) its preimage under (f,g) o Delta. Similarly, a map of spans of sets can be seen as a natural transformation between profunctors, while a span of spans of sets can be seen as a profunctor between the corresponding collages. Where has this been discussed in the literature? In the bicategory of categories, profunctors, and natural transformations, a profunctor from C to D with a right adjoint is (up to some details around Cauchy completion) just a functor from C to D. Is there a nice characterization of profunctors with right adjoints when the 2-cells are profunctors between collages? -- Mike Stay - metaweta(a)gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

2 1

question on terminology
by claudio pisani 25 Aug '12

25 Aug '12

Dear categorists, Is there a standard name for those presheaves X on a category C such that Xf is a bijection for any f in C? I have sometimes called them "biactions" since any such X (considered as, say, a left action of C) is paired with the obvious presheaf X' on C^op (a right action of C): X'f = (Xf)^-1. Of course, they correspond, as categories over C, to discrete bifibrations. I also know that the separable or decidable presheaves are those for which every Xf is injective. Claudio [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

3 2

TR : [ALGTOP-L] William Thurston has died
by Joyal, André 23 Aug '12

23 Aug '12

-------- Message d'origine-------- De: algtop-l-bounces+joyal.andre=uqam.ca(a)lists.lehigh.edu de la part de jim stasheff Date: mer. 22/08/2012 18:12 À: ALGTOP-L(a)lists.lehigh.edu Objet : [ALGTOP-L] William Thurston has died American Mathematical Society wrote: "William P. Thurston, whose geometric vision revolutionized topology, died August 21 at the age of 65. His honors include the Fields Medal (1982), the AMS Book Prize (2005), and the AMS Steele Prize for a Seminal Contribution to Research (2012). Feel free to share your reminiscences here." source: http://www.facebook.com/amermathsoc and see also: http://www.ams.org/news?news_id=1602 <http://www.facebook.com/l/NAQEJMW_R/www.ams.org/news?news_id=1602> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

1 0

Yoneda Lemma when there is a monad
by info＠christophertownsend.org 23 Aug '12

23 Aug '12

Hi Does anyone have any references for the following generalisation of the Yoneda lemma: Lemma: If (T,i,m) is a monad on a locally small category C then for any functor F:(C_T)^op->Set, contravariant from the Kleisli category to Set and for any T algebra (A,a) Nat[C^T(K_,(A,a)),F]=F(A,a) where K is the usual comparison functor from the Kleisli category, C_T, to the category of algebras of T, C^T. F(A,a) means the subset of F(A) consisting of elements x such that F(m_A)x=F(Ta)x. (And Nat[_] means the set of natural transformations.) // The usual Yoneda lemma is recovered by taking the trivial monad. The Lemma gives a generalised Yoneda embedding: C^T embeds in [(C_T)^op,Set]; i.e. any category of algebras embeds in the presheaf category of the Kleisli category. I wasn't aware of this quite trivial result and was hoping for some guidance as to where it is covered already in the literature? Thanks, Christopher [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

6 6

Graphs for Juncture
by Kosta Dosen 22 Aug '12

22 Aug '12

A new study entitled GRAPHS FOR JUNCTURE by Kosta Dosen and Zoran Petric (viii+128 pp.) is available at: http://www.mi.sanu.ac.rs/~kosta/jun.pdf _____________________________________ Abstract An alternative foundation for 2-categories is explored by studying graph-theoretically a partial operation on 2-cells named juncture, which can replace vertical and horizontal composition. Juncture is a generalized vertical composition of 2-cells that need not involve the whole target and the whole source; it may involve them only partially, provided the result is again a 2-cell. Since commuting diagrams of arrows of ordinary categories may be conceived as invertible 2-cells, this study concerns ordinary category theory too. The operation of juncture has a connection with proof theory, where it corresponds to a kind of cut rule on sequents, and it is related also to an operation on which the notion of operad can be based. The main achievement of the work is a detailed description of the specific planarity involved in juncture and graphs of 2-cells, comparable to the usual combinatorial characterizations of planarity in graph theory. ____________________________________ Subject Classification (MSC2010) Graph Theory: 05C10 (Planar graphs), 05C20 (Directed graphs), 05C62 (Graph representations), 05C76 (Graph operations) Category Theory: 18A10 (Graphs, diagram schemes), 18D05 (2-categories) ____________________________________ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

1 0