September 2011 - Categories - categories.org.au

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: Simplicial versus (cubical with connections)
by Ronnie Brown 27 Oct '11

27 Oct '11

Dear Colleagues, I have found the `very good' properties of cubical sets essential for many aspects of my work, and are used in our new book `Nonabelian algebraic topology', (downloadable pdf from my web pages) which puts in one place the work with Higgins et al; with regard to the monoidal closed structures this is especially seen in Chapter 15, whose Introduction p. 513 mentions a crucial natural transformation \eta: \rho (X_*) \otimes \rho(Y_*) \to \rho(X_* \otimes Y_*) where X_*, Y_* are filtered spaces, and \rho gives the cubical higher homotopy groupoid, which is almost `obvious' in this cubical context, and so can be translated to other contexts. Whether these `very good properties' are `excellent' in a formal sense I do not know, but it would be very surprising if that failed! I do not find analogous properties simplicially. I got the impression from John Moore that although Dan Kan's initial work on combinatorial homotopy was cubical (see his first PNAS Note) the problems of the homotopy type of the realisation of the cartesian product discouraged any further cubical work, as the simplicial method went smoothly, although often technical. So nobody tried to fix the cubical theory. Our `fix', i.e. the connections, was forced on us for other reasons, initially to relate crossed modules and double groupoids; they were crucial for the 2-d van Kampen theorem (all done in the 1970s) and then for higher analogues. I doubt if much of this local-to-global stuff could have been conjectured, let alone proved, simplicially. On the other hand, there are two papers on classifying spaces for equivariant crossed complexes with Golasinski, Porter and Tonks where simplicial methods are used, because they link nicely with work on coherence of Cordier and Porter. A recent paper using cubical methods which could be relevant for Dmitry is Faria~Martins, J. and Picken, R. Surface {H}olonomy for {N}on-{A}belian 2-{B}undles via {D}ouble {G}roupoids}.{Adv. Math.} 226(2011) 3309–3366. Ronnie On 15/09/2011 12:56, Dmitry Roytenberg wrote: > Dear colleagues, > > This raises the following question. In homotopy theory one almost > always considers categories enriched in simplicial sets, endowed with > the classical Kan-Quillen model structure, but are cubical sets (with > or without connections) equally good for that purpose? In his "Higher > Topos Theory" Lurie considers monoidal model categories S enjoying > certain additional properties and proves that for each such "excellent > model category" S the category Cat(S) of small S-enriched categories > admits a model structure with a particularly nice description of > fibrations and weak equivalences (that S = simplicial sets with the > classical model structure and the Cartesian product is excellent is a > result of Dwyer and Kan; the corresponding model structure on Cat(S) > is due to Bergner). > > So my question is, is it known whether the category of cubical sets, > with or without connections, admits an excellent model structure? The > reason I am asking is because in geometric applications I have in mind > cubes are often easier to work with than simplices (e.g. it is easier > to compute integrals of differential forms over cubes). > > Thank you in advance. > > Dmitry > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

8 10

Natural Functorial Categorical Intuition
by Ellis D. Cooper 26 Oct '11

26 Oct '11

My motto has been "Rigor cleans the window through which intuition shines." It seems to me that a great deal more is known about mathematical rigor than about mathematical intuition. Economics Nobelist Daniel Kahneman recently published at www.edge.org a survey of several men-decades of research on flaws of human statistical intuition. A paper, "The Intuitive Experience" in "The View from Within" (Journal of Consciousness Studies, V.6 1999) discusses in considerable detail schema and methods of invocation of intuition in psychotherapy, art, and biology research. My overall question is whether there really are different kinds of intuition depending on the research discipline. In particular, is there some kind of kinetic intuition specific to category theory that crucially involves visualization of time-varying diagrams? Do conjectured adjoint functors arise from distinct algebraic, or geometric, or logical intuitions? Do categorists deploy special methods to access their intuition, or do intuitions just happen to those with a knack for category theory? Does categorical intuition just develop with experience, or is there a specialized training to enhance it? Is categorical intuition any different from mathematical intuition in general? Ellis D. Cooper [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

7 8

What else do simplicial sets classify?
by Andrej Bauer 05 Oct '11

05 Oct '11

The presheaf category of simplicial sets is the classifying topos for the theory L of a bounded linear order. In general, there could be other theories which are "Morita equivalent" to L in the sense that their classifying toposes are equivalent to simplicial sets. Are any such known, preferably occurring in nature? With kind regards, Andrej [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

8 7

Re: Reference requested
by Peter May 03 Oct '11

03 Oct '11

Thanks everybody for comments, although I guess the use goes so far back into antiquity that the request for an original reference is unanswerable. For context, with two young collaborators (Bertrand Guillou and Mona Merling), I have a draft in progress tentatively entitled ``Chaotic categories and equivariant classifying spaces''. I prefer `chaotic' to `indiscrete' not just because of the `coarse' implications of the latter, but because indiscrete spaces are boring, `null or banal', whereas chaotic categories have genuinely significant applications. They are quite surprisingly central to the theory of universal bundles, equivariant or not. Via the (product-preserving) classifying space construction from categories (especially categories internal to spaces) to spaces, they provide a rich source of contractible spaces that can very easily be given interesting additional structure. That is just what one wants when constructing universal bundles. More fun, it is just what one wants to construct an E infinity operad of G-categories that defines `genuine' symmetric monoidal G-categories (which are not merely symmetric monoidal categories on which a group G acts in the obvious `naive' way). These which give rise to `genuine' G-spectra. Genuine G-spectra that define equivariant algebraic K-theory arise in precisely this way. All starting from chaotic trivialities. Cheers, Peter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

5 7

simplicial vs (cubical with connections) Ronnie Brown
by F William Lawvere 02 Oct '11

02 Oct '11

categories: Re: Simplicial versus (cubical with connections)‏ Ronnie Brown ronnie.profbrown(a)btinternet.com     One basic intuition is that cubes are products. Yet almost none of the combinatorial toposes commonly called “cubical sets” have that feature. Is there some profound disadvantage in allowing projection maps to have their universal property (yielding diagonal maps, etc)? If so, I have never seen it spelled out. An advantage to having finite products in a site is that the (iterated) pathspace functor has a right adjoint, leading to a very natural construction of Eilenberg-Mac Lane spaces. Dan Kan told me that the reason for his switch was that cubical groups do not have the extension property (i.e., the Kan property), as simplicial groups do. But later I realized that there is ambiguity about what “cubical” means. Since these combinatorial categories are usually toposes, some light is shed on their particularity by determining what kind of structure they classify (in the established categorical sense, e.g.,the simplicial topos classifies total orders with distinct endpoints, and a simple cubical example classifies strictly bipointed objects). Concretely, there are many different theories of algebraic structure for which the unit interval is a model, and having chosen one, this structure should be preserved by geometric realization. Bill Lawvere [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

2 1

partial categories
by Emily Riehl 30 Sep '11

30 Sep '11

A colleague of mine is wondering if anyone has studied "partial categories," by which she means directed graphs with identities but with only some compositions (including all identity compositions) defined. A partial category can be thought of as a category enriched in pointed sets (with smash product as tensor and S^0 as unit). The slogan is that the basepoint in each hom-set stands in for "does not exist". But enriched functors don't give the right notion of maps; these should preserve identities and all specified compositions. Enriched functors behave appropriately with regards to the identites but may "forget" extant arrows and in particular need not preserve composites. So perhaps this perspective is not useful. I'll happily pass along any suggestions. Thanks, Emily Riehl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

9 8

Graphical language and compact closed categories
by Aleks Kissinger 30 Sep '11

30 Sep '11

Categorists, Everyone (as far as I know) believes that appropriately defined string diagrams can be used to construct the free compact closed category on a monoidal signature. Kelly-Laplaza proved this only for the case of boxes that have a single wire in and out (i.e. free CCC's on a category, not on an arbitrary moniodal signature). However, at least at the time of publishing his survey (2009), Selinger writes that a general proof doesn't exist in the literature. So, my question is: Is this still the case, and why? Is it simple a question of someone putting in the hours to write this up, or are there serious technical obstacles to the general result? Thanks! Aleks Kissinger [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

2 1

Re: partial categories
by Emily Riehl 30 Sep '11

30 Sep '11

Thank you. This seems very helpful. I've already passed your message along. Emily On Wed, 28 Sep 2011, Peter Selinger wrote: > Dear Emily, > > one notion of "partial category" that has been studied is Freyd's > notion of "paracategory". It differs from what you wrote below as > follows: instead of taking compositions of two arrows as the primitive > operation, one takes compositions of n arrows as primitive operations, > for all n. So if f1, ..., fn are n arrows (so that the codomain of > each is the domain of the next), then [f1, ..., fn] is their > composition, which may be defined or undefined. > > In the case of (total) categories, adding n-ary compositions makes no > difference, since they are already definable in terms of identities > and binary composition. But in the partial case, it does make a > difference, as it is possible, for example, that [f,g,h] is defined, > but [f,g] and [g,h] are undefined. > > The axioms are: > > (a) [] : A -> A is defined (the composition of the empty path from A to A) > (b) [f] is defined and equal to f, for all arrows f, > (c) if ff, gg, hh are are (composable) paths, and if [gg] is defined, > then [ff,[gg],hh] is defined if and only if [ff,gg,hh] is defined, > and in this case, they are both equal. > > The main representation theorem is: > > Every reflexive subgraph of a category is a paracategory; conversely, > every paracategory can be faithfully completed to a category. > > It is not the only possible notion of partial category, and may not > always be the notion that one wants. > > -- Peter > > Emily Riehl wrote: >> >> A colleague of mine is wondering if anyone has studied "partial >> categories," by which she means directed graphs with identities but with >> only some compositions (including all identity compositions) defined. >> >> A partial category can be thought of as a category enriched in pointed >> sets (with smash product as tensor and S^0 as unit). The slogan is that >> the basepoint in each hom-set stands in for "does not exist". But enriched >> functors don't give the right notion of maps; these should preserve >> identities and all specified compositions. Enriched functors behave >> appropriately with regards to the identites but may "forget" extant >> arrows and in particular need not preserve composites. So perhaps this >> perspective is not useful. >> >> I'll happily pass along any suggestions. >> >> Thanks, >> Emily Riehl >> >> >> [For admin and other information see: http://www.mta.ca/~cat-dist/ ] >> > >

2 1