Frobenius and the derived centers of algebraic theories

We show that the derived center of the category of simplicial algebras over every algebraic theory is homotopically discrete, with the abelian monoid of components isomorphic to the center of the category of discrete algebras. For example, in the case of commutative algebras in characteristic $p$, this center is freely generated by Frobenius. Our proof involves the calculation of homotopy coherent centers of categories of simplicial presheaves as well as of Bousfield localizations. Numerous other classes of examples are discussed.Comment: 40 page

Homotopy nilpotent groups

We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define the simplicial theory of homotopy n-nilpotent groups. This notion interpolates between infinite loop spaces and loop spaces. We prove that the set-valued algebraic theory obtained by applying $\pi_0$ is the theory of ordinary n-nilpotent groups and that the Goodwillie tower of a connected space is determined by a certain homotopy left Kan extension. We prove that n-excisive functors of the form $\Omega F$ have values in homotopy n-nilpotent groups.Comment: 16 pages, uses xy-pic, improved exposition, submitte

Bredon Homology of Partition Complexes

We prove that the Bredon homology or cohomology of the partition complex with fairly general coefficients is either trivial or computable in terms of constructions with the Steinberg module. The argument involves developing a theory of Bredon homology and cohomology approximation.Comment: 48 pages. Minor revisions. A typo in the statement of Corollary 1.2 was corrected, along with other typos. Some references have been adde

Normalizers of tori

We determine the groups which can appear as the normalizer of a maximal torus in a connected 2-compact group. The technique depends on using ideas of Tits to give a novel description of the normalizer of the torus in a connected compact Lie group, and then showing that this description can be extended to the 2-compact case.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper31.abs.htm

Homotopy theory of small diagrams over large categories

Let be a large category which is cocomplete. We construct a model structure (in the sense of Quillen) on the category of small functors from to simplicial sets. As an application we construct homotopy localization functors on the category of simplicial sets which satisfy a stronger universal property than the customary homotopy localization functors d

Obstruction Theory in Model Categories

Many examples of obstruction theory can be formulated as the study of when a lift exists in a commutative square. Typically, one of the maps is a cofibration of some sort and the opposite map is a fibration, and there is a functorial obstruction class that determines whether a lift exists. Working in an arbitrary pointed proper model category, we classify the cofibrations that have such an obstruction theory with respect to all fibrations. Up to weak equivalence, retract, and cobase change, they are the cofibrations with weakly contractible target. Equivalently, they are the retracts of principal cofibrations. Without properness, the same classification holds for cofibrations with cofibrant source. Our results dualize to give a classification of fibrations that have an obstruction theory.Comment: 17 pages. v3 includes improved introduction and several other minor improvement

Spaces of null homotopic maps

Abstract. We study the null component of the space of pointed maps from B to X when is a locally nite group, and other components of the mapping space when is elementary abelian. Results about the null component are used to give a general criterion for the existence of torsion in arbitrarily high dimensions in the homotopy of X. In 1983 Haynes Miller [M] proved a conjecture of Sullivan and used it to show that if is a locally nite group and X is a simply connected nite dimensional CW-complex then the space of pointed maps from the classifying space B to X is weakly contractible, ie. Map(B;X) ’ . This result had immediate applications. Alex Zabrodsky [Z] used it to study maps between classifying space