August 2004 - Categories - categories.org.au

Question about wfs
by Philippe Gaucher 02 Sep '04

02 Sep '04

Dear All Let (L1,R1) and (L2,R2) be two weak factorization systems (by weak, it is usually meant that the diagonal map is not unique) such that L1 \subset L2. Example of such a thing: in a locally presentable category, take two sets of maps I1 and I2 with I1\subset I2, and let (L1,R1)=(cof(I1),inj(I1)) and (L2,R2)=(cof(I2),inj(I2)). Two such wfs are called cofibrantly generated. Now come back to (L1,R1) and (L2,R2) with L1 \subset L2. Question: if (L2,R2) is cofibrantly generated, is (L1,R1) cofibrantly generated ? if we work in a locally presentable category, can one say something under Vopenka's principle ? The only example I have in mind is the wfs (Serre cofibration, trivial Serre fibration) which is cofibrantly generated by {S^{n-1}->D^n} and (Hurewicz cofibration, Hurewicz fibration which are homotopy equivalences) which is probably not cofibrantly generated. Unfortunately, the inclusion is in the wrong direction : {Serre cofibration}\subset {Hurewicz cofibration} since {Hurewicz fibration which are homotopy equivalences} \subset {trivial Serre fibration} so that does not give a counterexample. Thanks in advance. pg.

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Preprints available
by Walter Tholen 01 Sep '04

01 Sep '04

The following fairly recent articles are available from my home page at http://www.math.yorku.ca/Who/Faculty/Tholen/research.html Walter Tholen. G. Janelidze and W. Tholen: Facets of descent III: monadic descent for rings and algebras Abstract: We develop an elementary approach to the classical descent problems for modules and algebras, and their generalizations, based on the theory of monads. D. Hofmann and W. Tholen: Kleisli operations for topological spaces Abstract: The axioms for a topology in terms of open sets follow necessarily from the intuitive relation of this concept with ultrafilter convergence. By contrast, the intuitive relations between neighbourhood systems or closure operations on the one hand and ultrafilter convergence on the other lead only to pretopologies. Kleisli operations greatly facilitate categorical descriptions of topological spaces, both in terms of neighbourhood systems and (ultra)filter convergence relations.

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Octoberfest meeting in Montreal, Oct 16-17.
by Robert Seely 31 Aug '04

31 Aug '04

-------------------------------------------------------------------------- Category Theory OctoberFest McGill University, Montreal Saturday - Sunday, October 16 - 17, 2004 -------------------------------------------------------------------------- We invite you to join us in Montreal next October for a weekend meeting in Category Theory, the "not-quite-annual" OctoberFest. As has been the tradition with these meetings, we invite talks from all participants. If you wish to give a talk, send your request along with a short abstract (before the end of September please) to Robert Seely at the address below. The final schedule of talks will be handed out at the meeting, but a provisional schedule will be made available, if possible, before the meeting. Details are still being worked out, but we hope that the talks will be held in Burnside Hall, 805 Sherbrooke West, starting on Saturday morning, October 16th. Coffee will be available from 8:30 am. The first talk will be at 9:00. Registration will take place during the morning, before the first talk and during the first coffee break. There will be a registration fee: we are currently estimating $CAN60 ($US50), discounted to $CAN30 ($US25) for students. (No promises! but if we can reduce this we shall, and if we need to increase it, we'll do that instead!) We are planning a dinner/party on Saturday evening, also in Burnside Hall. (Details will be announced at the meeting.) Please let us know if you intend to join us by sending a short email (before the end of September if possible) to Robert Seely, also indicating whether or not you will bring a guest to the party. (We would like to have a good estimate of how many people will attend the party about a week before the meeting.) A list of hotels and B&B style accomodation will be posted soon, as well as an updated notice of the details of our plans. In the meantime, if you have any further questions, please contact one of us. Michael Makkai [makkai(a)math.mcgill.ca] Robert Seely [rags(a)math.mcgill.ca] Dept of Mathematics and Statistics McGill University 805 Sherbrooke West Montreal, Quebec CANADA H3A 2K6 ---------------------------------------------------------------------- Updated information will be available (when ready!) on the "triples" website: http://www.math.mcgill.ca/triples/octoberfest04.html ---------------------------------------------------------------------- -- <rags(a)math.mcgill.ca> <www.math.mcgill.ca/rags>

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open-access edition of Adamek-Herrlich-Strecker
by Jiri Adamek 30 Aug '04

30 Aug '04

The book "Abstract and Concrete Categories (The Joy of Cats)" by Jiri Adamek, Horst Herrlich and George E. Strecker, though out print for some time, is available again via the internet address http://katmat.math.uni-bremen.de/acc This is an open-access publication and can be freely used and distributed. It contains corrections and small improvements and some new sketches by Marcel Erne. We are grateful to Christof Schubert whose skill and untiring effort made this possible. Take a look, enjoy, and let others know!

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addendum to enriched fibrations
by Thomas Streicher 25 Aug '04

25 Aug '04

As the question was originally about internal enriched categories I want to add the following easy observation. If the fibration U : UU -> BB is *locally small* and C is a U-enriched fibration over BB (as defined in Gouzou & Grunig) then the associated (non-enriched) fibration over BB is small (i.e. the externalisation of an internal category). The family of morphisms of this internal category is given by hom(I_{C_0 x C_0},H) : C_1 ---> C_0 x C_0 where hom comes from U being locally small. Thomas

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precise reference on enriched fibrations
by Thomas Streicher 25 Aug '04

25 Aug '04

Dear Bill, I have now looked up the paper I mentioned in my previous mail. It is a report form the Seminaire B'enabou authored by Gouzou and Grunig entiteled Fibrations Relatives dated from Novemeber 1976. In the first chapter they introduce fibred monoidal categories U : UU -> BB. In the second chapter they define UU-enriched BB-categories as follows: an object C_0 in BB, a family C_1 in U(C_0 x C_0), a map \eta : 1_{C_0} --> \Delta^*C_1 (here \Delta is the diagonal on C_0) providing identities and a map \mu : p_3^*H \otimes p_1^*C_1 --> p_2^*H in U(C_0xC_0xC_0) providing multiplication. These data have to satisfy the obvious identities. The problem I ran into in my previous mail is avoided here by assuming that C_0 is a small discrete fibration in Psh(BB) and thus has definable equality. It is not clear to me how to extend their approach to enriched fibred categories with a non-small set of objects. Best, Thomas PS The report of Gouzou and Grunig contains much more material (about 90 pages) I haven't looked at yet.

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fibred enriched categories
by Thomas Streicher 25 Aug '04

25 Aug '04

Dear Bill, you have asked whether somebody has clarified the notion of enriched internal category. I don't know of any written account but it most certainly has been studied by Jean B\'enabou in his work on fibred categories. I think one should start as follows. Consider some base category BB with finite limits and consider some fixed fibred monoidal category C : CC -> BB. (The latter notion (due to B\'enabou) I think can be found in some seminar reports by Gouzou & Grunig from the 70ies. The idea is to consider a cartesian functor \otimes : C x_BB C --> C together with cartesian natural transformations \alpha, \lambda, \rho satisfying literally the same coherence conditions usually required for ordinary monoidal cats. Similarly, one gets a notion of fibred symmetric monoidal category etc.) Now given a fibration P : XX --> BB one may ask what it means that P is enriched over C ? An essential ingredient of such an enrichment should consist of a cartesian functor H : P^op x P ---> C whose I-th component enriches P(I) in C(I). The tricky thing is to conceptualize composition in the fibred enriched category. I think it must be something like a cartesian natural transformation c_{X,Y,Z} : H(X,Y) \otimes H(Y,Z) ----> H(X,Z) satisfying associativity requirements etc. But the problem is that Y has two occourrences, one co- and one contravariant. Thus, H(X,Y) \otimes H(Y,Z) should rather be replaced by a coend \int_Y H(X,Y) \otimes H(Y,Z) (\int stands for the integral symbol) Thus, one has to assume that P is locally small and that C is a cocomplete fibration. I am afraid one has to get into such a mess as one cannot simply consider a H : |P|^op x |P| ---> C with |P| the "presheaf of objects" because of the "equality of objects" problem. Hm, I see when starting to write things down I run into problems. At home I will have a look at this rapport by Gouzou & Grunig and hope that I can tell you more. Best, Thomas

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cubical omega-categories
by Ronnie Brown 24 Aug '04

24 Aug '04

The following paper has been put on the arXiv on behalf of Philip Higgins Ronnie Brown Title: Thin Elements and Commutative Shells in Cubical omega-categories Author: Philip J. Higgins Report-no: University of Wales, Bangor, Maths Preprint 04.16 Subj-class: Category Theory; Algebraic Topology MSC-class: 18D05,18D35,55U99 ABSTRACT: The relationships between thin elements, commutative shells and connections in cubical omega-categories are explored by a method which does not involve the use of pasting theory or nerves of omega-categories (both of which were previously needed for this purpose; see previous work by Al-Agl/Brown/Steiner). It is shown that composites of commutative shells are commutative and that thin structures are equivalent to appropriate sets of connections; this work extends to all dimensions the results proved in dimensions 2 and 3 by Brown/Mosa, and Brown/Kamps/Porter. Available from the arXiv math.CT/0408306 Preprint 04.16 on http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/0… -- Professor Emeritus R. Brown, Department of Mathematics, University of Wales, Bangor Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom Tel. direct:+44 1248 382474|office: 382681 fax: +44 1248 361429 World Wide Web: home page: http://www.bangor.ac.uk/~mas010/ (Links to survey articles: Higher dimensional group theory Groupoids and crossed objects in algebraic topology) Centre for the Popularisation of Mathematics: http://www.popmath.org.uk/

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Internal enriched ??
by wlawvere＠buffalo.edu 24 Aug '04

24 Aug '04

Concerning the basic question raised by Hendryk Pfeiffer : If C is a symmetric monoidal category, it seems that there does not exist any category with finite limits whose representations in C are the C-enriched categories. An object in C is not suitable to serve as the Space of Objects of a C-category because the latter would require a well-behaved Diagonal Map. Unless of course C is cartesian. "Internal category" in a cartesian category is a coherent concept, as is "enriched category " in a general monoidal category because the latter is always presumed to be itself enriched in the cartesian category of sets where the spaces of objects are taken. I know of no way, for example, to define the enriched hom of a strong functor category without using the diagonal map on the space of objects of the domain because the very definition of natural transformation requires that the same thing ocurs in two different places. (Of course the problem is compounded for hypothetical enriched n-categories). The combination of internal and enriched features is implicit in mathematics. The most immediate way of making it explicit would seem to be simply to take as basic C->E , an enrichment of a monoidal category in a cartesian category. For any category based on that, any object I from E can parameterize its objects and any object V from C can parameterize its homs. (Giving a meaning to "smooth" parameterizations like these is after all the basic point of both enrichment and internalizations, because they do occur). Relative to such contexts, a suitable notion of theory can probably be found. There are plenty of examples: C= chain complexes in a topos E ... Are there publications which treat these issues ? A similar problem arises with the "comprehension scheme" when we try to set up an adjunction between modules (actions, presheaves) and "fibrations" . This does not seem routine when not only the fibers but also the base and the hypothetical total are to be, for example , Ab-enriched. Is there a coherent approach to this ? Note that for example the "trivial extension" idea of adjointing infinitesimals serves well as a way of making modules into rings (additive categories) but does not appear as any kind of cousin to the notion of fibered category.

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literature on internal categories
by Hendryk Pfeiffer 23 Aug '04

23 Aug '04

Hi, I would be most grateful if somebody could point me to the appropriate literature on internal categories. I am interested in the following. Let C be a category with finite limits. Cat(C) denotes the 2-category whose objects are internal categories in C, whose morphisms are internal functors in C, and whose 2-morphisms are internal natural transformations. If C,D have finite limits and F:C->D is a functor that preserves finite limits, then there is a 2-functor Cat(F):Cat(C)->Cat(D). Similarly, for a natural transformation e:F=3D>G between two such functors F,G:C->D, one obtains a 2-natural transformation Cat(e):Cat(F)=3D>Cat(G). In total, there is a 2-functor Cat(-):LECat -> 2Cat where LECat is the 2-category whose objects are categories with finite limits, whose morphisms are finite-limit preserving functors and whose 2-morphisms are natural transformations. I am interested in the following. If the category with finite limits C has additional properties or additiona= l structure, what does this imply for the 2-category Cat(C)? In particlar if, * C is enriched * C is symmetric|braided monoidal * C is symmetric monoidal and closed Maybe the following point of view (which views Cat(C) as a 1-category) is more familiar. If one studies essentially algebraic theories, one writes, (1) Cat(C)=3DLEFunc(Th(Cat),C). Here LEFunc(D,C) is the category of finite-limit preserving functors D->C and natural transformations. C,D are categories with finite limits. Th(Cat) is a suitable category with finite limits (`theory of categories') such tha= t (1) holds. Cat(C) is then called a model of the theory of categories in the category C. Then my above questions come down to the following: If C is enriched/monoidal/..., what does this imply for LEFunc(D,C)? I am also interested in the 2-categorical structure of Cat(C) which is not visible in this picture. I guess all this is known and has been written somewhere. But where? I woul= d appreciate any sort of comments. Hendryk Pfeiffer --=20 NEU: WLAN-Router f=FCr 0,- EUR* - auch f=FCr DSL-Wechsler! GMX DSL =3D superg=FCnstig & kabellos http://www.gmx.net/de/go/dsl

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