-------- 10 new papers this time, from BrownR-Kamps-Porter, Chalupnik, ChornyB, Clarke-Crossley-Whitehouse, GrayB, Hikida, Hornbostel, IsaksenD, Lupton-SmithSB, and Porter. Mark Hovey New papers appearing on hopf between 1/5/04 and 2/1/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR-Kamps-Porter/vkt7 TITLE: A van Kampen theorem for the homotopy double groupoid of a Hausdorff space AUTHORS: R. Brown, K.H. Kamps, T.Porter EMAILS: r.brown@bangor.ac.uk, heiner.kamps@fernuni-hagen.de, t.porter@bangor.ac.uk ADDRESSES: Mathematics Department, Dean St., Bangor, Gwynedd LL57 1UT, UK. Fachbereich Mathematik, FernUniversit\"at in Hagen, D-58084 Hagen, Germany Mathematics Department, Dean St., Bangor, Gwynedd LL57 1UT, UK. ABSTRACT: We show that the homotopy double groupoid of a Hausdorff space defined by the authors in a previous paper satisfies a version of the van Kampen theorem, and so is a suitable tool for non abelian, 2-dimensional, local-to-global problems. The methods are analogous to those developed by Brown and Higgins for similar theorems for other higher homotopy groupoids. There is a detailed discussion of commutative cubes in a double category with connections, and a proof of the key result that any composition of commutative cubes is commutative. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Chalupnik/koeks Title of Paper: Koszul duality and extensions of exponential functors Author: Marcin Chalupnik Email: mchal@mimuw.edu.pl Abstract: We study Koszul duality in the category of strict polynomial functors. We compute Koszul duals for various functors and apply these results to the problem of calculating Ext--groups between exponential functors. The main application is a full description of the Ext--groups between twisted exterior and divided powers and between twisted symmetric and divided powers. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/ChornyB/prospaces Title: A generalization of Quillen's small object argument Author(s): Boris Chorny Author's e-mail address: bchorny2@uwo.ca Abstract: We generalize the small object argument in order to allow for its application to proper classes of maps (as opposed to sets of maps in Quillen's small object argument). The necessity of such a generalization arose with the appearance of several important examples of model categories which were proven to be non-cofibrantly generated. Our current approach allows for the construction of functorial factorizations and localizations in the equivariant model category on diagrams of spaces and in two different model structures on the category of pro-spaces. The examples above suggest a natural extension of the framework of cofibrantly generated model categories. We introduce the concept of a class-cofibrantly generated model category, which is a model category generated by classes of cofibrations and trivial cofibrations satisfying some reasonable assumptions. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Clarke-Crossley-Whitehouse/ccwops Title: Algebras of operations in K-theory Authors: Francis Clarke, Martin Crossley, Sarah Whitehouse Authors' e-mail addresses: F.Clarke@Swansea.ac.uk, M.D.Crossley@Swansea.ac.uk, S.Whitehouse@sheffield.ac.uk Abstract: We describe explicitly the algebras of degree zero operations in connective and periodic p-local complex K-theory. Operations are written uniquely in terms of certain infinite linear combinations of Adams operations, and we give formulas for the product and coproduct structure maps. It is shown that these rings of operations are not Noetherian. Versions of the results are provided for the Adams summand and for real K-theory. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/GrayB/decomposition On Decompositions in Homotopy Theory Brayton Gray 55P35, 5P30, 55P45 Dept. of Mathematics, Statistics and Computer Science (M/C 249) University of Illinois at Chicago 851 South Morgan Street Chicago, IL 60607-7045 brayton@uic.edu We first describe Krull-Schmidt theorems decomposing $H$ spaces and simply connected co-$H$ spaces into atomic factors in the category of pointed nilpotent $p$-complete spaces of finite type. We use this to construct a 1-1 correspondence between homotopy types of atomic $H$ spaces and homotopy types of atomic co-$H$ spaces, and construct a split fibration which connects them and illuminates the decomposition. Various properties of these constructions are analyzed. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Hikida/Acycle Title: Some acyclic relations in the lambda algebra Author: Mizuho Hikida Author's e-mail address: hikida@bus.hiroshima-pu.ac.jp Author's mailing address: Hiroshima Prefectural University, Shobara-shi, 727-0017, Japan AMS Classification numbers: 55Q40 Abstract: We consider the relations omega gamma = 0 in Lambda, and show that if omega alpha = 0 then alpha = gamma beta for some beta. These relations give the acyclic chain complex Lambda -gamma-> Lambda -omega-> Lambda . We consider various cases, e.g. omega = lambda_n and gamma = lambda_{2n+1}. Especially, we consider the case omega = w_n = d lambda_n for n=2^{e+r} + 2^{e}-1, where gamma = (h_{e+r})^r. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Hornbostel/Motchrom2 Author : Jens Hornbostel Author's e-mail address : jens.hornbostel@mathematik.uni-regensburg.de Author's mailing address: Universitaet Regensburg, NWF I - Mathematik, D- 93047 Regensburg, Germany Abstract: We construct a motivic version of the chromatic filtration and the chromatic spectral sequence. This should be used to study the stable ${\bf A}^1$-homotopy groups of the motivic sphere spectrum. We also study different localization techniques both for classical and motivic spectra. This is an improved version of a preprint posted in June 2003. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/IsaksenD/flasque Title: Flasque model structures for simplicial presheaves Author: Daniel C. Isaksen Author's e-mail address: isaksen@math.wayne.edu Abstract: By now it is well known that there are two useful (objectwise or local) families of model structures on presheaves: the injective and projective. In fact, there is at least one more: the flasque. For some purposes, both the projective and the injective structure run into technical and annoying (but surmountable) difficulties for different reasons. The flasque model structure, which possesses a combination of the convenient properties of both structures, sometimes avoids these difficulties. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Lupton-SmithSB/GseqII Title: Rationalized Evaluation Subgroups of a Map II: Quillen Models and Adjoint Maps Authors: Gregory Lupton and Samuel Bruce Smith Authors' e-mail addresses: G.Lupton@csuohio.edu and smith@sju.edu Authors' mailing addresses: Department of Mathematics, Cleveland State University, 2121 Euclid Ave., Cleveland OH 44115 and Department of Mathematics, Saint Joseph's University, Philadelphia, PA 19131 AMS classification number: 55P62, 55Q52 Other useful information: 33 pages; http://arXiv.org/abs/math.AT/0401178 Abstract: Let w: Map(X,Y;f) -> Y denote a general evaluation fibration. Working in the setting of rational homotopy theory via differential graded Lie algebras, we identify the long exact sequence induced on rational homotopy groups by w in terms of (generalized) derivation spaces and adjoint maps. As a consequence, we obtain a unified description of the rational homotopy theory of function spaces, at the level of rational homotopy groups, in terms of derivations of Quillen models and adjoints. In particular, as a natural extension of a result of Tanre, we identify the rationalization of the evaluation subgroups of a map f: X -> Y in this setting. As applications, we consider a generalization of a question of Gottlieb, within the context of rational homotopy theory. We also identify the rationalization of the G-sequence of f and make explicit computations of the homology of this sequence. In a separate result of independent interest, we give an explicit Quillen minimal model of a product AxX, in the case in which A is a rational co-H-space. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Porter/s-catsv2 Title: S-categories, S-groupoids, Segal categories and quasicategories Author: Timothy Porter Author's e-mail address: t.porter@bangor.ac.uk Author's mailing address: Mathematics Department, School of Informatics, University of Wales Bangor, Bangor, Gwynedd, LL57 1UT, United Kingdom. Included ps or eps files: 5 epsi files AMS classification number: 55U35 Other useful information: arXive submission number: math.AT/0401274 Abstract: These notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Laguna, the Canary Islands, in September, 2003. They aim (i) to revisit some oldish material on abstract homotopy and simplicially enriched categories, that seems to be being used in today's resurgence of interest in the area and to try to view it in a new light, or perhaps from new directions;(ii) to introduce Segal categories and various other tools used by the Nice-Toulouse group of abstract homotopy theorists and link them into some of the older ideas;(iii) to introduce Joyal's quasicategories, and show how that theory links in with some old ideas of Boardman and Vogt, Dwyer and Kan, and Cordier and Porter; and finally to ask lots of questions of myself and of the reader. ----------------