----- There are 7 new papers this time, from Biedermann, Blanc, Oliver-Ventura, Shipley, Stacey-Whitehouse, Wuethrich, and YauD. Mark Hovey New papers appearing on hopf between 6/5/06 and 7/8/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Biedermann/presh-n-types Title: On the homotopy theory of n-types Author: Georg Biedermann Mail address: Dep. of Mathematics, Middlesex College, UWO, London, Ontario, N5X 2W8, Canada Abstract: We achieve a classification of n-types of simplicial presheaves in terms of (n-1)-types of presheaves of groupoids enriched in simplicial sets. This can be viewed as a different description of the homotopy theory of higher hyperstacks. As a special case we obtain a good substitute for the homotopy theory of (weak) higher groupoids. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc/comp Title: Comparing homotopy categories Author: David Blanc Address: Department of Mathematics University of Haifa 31905 Haifa Israel Abstract: Given a suitable functor T:C -> D between model categories, we define a long exact sequence relating the homotopy groups of any X in C with those of TX, and use this to describe an obstruction theory for lifting an object G in D to C. Examples include finding spaces with given homology or homotopy groups. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Oliver-Ventura/ov1 Extensions of linking systems with $p$-group kernel Bob Oliver and Joana Ventura LAGA Departamento de Matem\'atica Institut Galil\'ee Instituto Superior T\'ecnico Av. J-B Cl\'ement Av. Rovisco Pais 93430 Villetaneuse, France 1049--001 Lisboa, Portugal bobol@math.univ-paris13.fr jventura@math.ist.utl.pt Subject class: Primary 55R35. Secondary 55R40, 20D20 Keywords: Classifying space, $p$-completion, finite groups, fusion. Abstract: We study extensions of $p$-local finite groups where the kernel is a $p$-group. In particular, we construct examples of saturated fusion systems $\calf$ which do not come from finite groups, but which have normal $p$-subgroups $A\nsg\calf$ such that $\calf/A$ is the fusion system of a finite group. One of the tools used to do this is the concept of a ``transporter system'', which is modelled on the transporter category of a finite group, and is more general than a linking system. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Shipley/zdga17 Title: HZ-algebra spectra are differential graded algebras Author: Brooke Shipley Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZ-algebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZ-algebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that basically any rational stable model category is Quillen equivalent to modules over a differential graded Q-algebra (with many objects). 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Stacey-Whitehouse/deloopv2 Title: Stable and Unstable Operations in mod p Cohomology Theories Authors: Andrew Stacey and Sarah Whitehouse Abstract: We consider operations between two multiplicative, complex orientable cohomology theories. Under suitable hypotheses, we construct a map from unstable to stable operations, left-inverse to the usual map from stable to unstable operations. The main example is where the target theory is one of the Morava K-theories in which case our map is closely related to the Bousfield-Kuhn functor. Resubmitted to correct font generation problem with the conversion to postscript and PDF. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Wuethrich/thickenings Title: Infinitesimal thickenings of Morava K-theories Author: Samuel Wuethrich Abstract: A. Baker has constructed certain sequences of cohomology theories which interpolate between the Johnson-Wilson and the Morava K-theories. We realize the representing sequences of spectra as sequences of MU-algebras. Starting with the fact that the spectra representing the Johnson-Wilson and the Morava K-theories admit such structures, we construct the sequences by inductively forming singular extensions. Our methods apply to other pairs of MU-algebras as well. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/YauD/GD2 Title: Gerstenhaber structure and Deligne's conjecture for Loday algebras Author: Donald Yau Abstract: A method for establishing a Gerstenhaber algebra structure on the cohomology of Loday-type algebras is presented. This method is then applied to dendriform dialgebras and three types of trialgebras introduced by Loday and Ronco. Along the way, our results are combined with a result of McClure-Smith to prove an analogue of Deligne's conjecture for Loday algebras. -----------------