16 research outputs found
Rings, modules, and algebras in infinite loop space theory
We give a new construction of the algebraic -theory of small permutative
categories that preserves multiplicative structure, and therefore allows us to
give a unified treatment of rings, modules, and algebras in both the input and
output. This requires us to define multiplicative structure on the category of
small permutative categories. The framework we use is the concept of
multicategory, a generalization of symmetric monoidal category that precisely
captures the multiplicative structure we have present at all stages of the
construction. Our method ends up in Smith's category of symmetric spectra, with
an intermediate stop at a new category that may be of interest in its own
right, whose objects we call symmetric functors.Comment: 59 pages, 1 figur
Homotopy Theoretic Models of Type Theory
We introduce the notion of a logical model category which is a Quillen model
category satisfying some additional conditions. Those conditions provide enough
expressive power that one can soundly interpret dependent products and sums in
it. On the other hand, those conditions are easy to check and provide a wide
class of models some of which are listed in the paper.Comment: Corrected version of the published articl
A model structure for coloured operads in symmetric spectra
We describe a model structure for coloured operads with values in the
category of symmetric spectra (with the positive model structure), in which
fibrations and weak equivalences are defined at the level of the underlying
collections. This allows us to treat R-module spectra (where R is a cofibrant
ring spectrum) as algebras over a cofibrant spectrum-valued operad with R as
its first term. Using this model structure, we give suficient conditions for
homotopical localizations in the category of symmetric spectra to preserve
module structures.Comment: 16 page
Topological Andr\'e-Quillen homology for cellular commutative -algebras
Topological Andr\'e-Quillen homology for commutative -algebras was
introduced by Basterra following work of Kriz, and has been intensively studied
by several authors. In this paper we discuss it as a homology theory on CW
-algebras and apply it to obtain results on minimal atomic -local
-algebras which generalise those of Baker and May for -local spectra and
simply connected spaces. We exhibit some new examples of minimal atomic
-algebras.Comment: Final revision, a version will appear in Abhandlungen aus dem
Mathematischen Seminar der Universitaet Hambur
K-theory of truncated polynomial algebras
Suppose that A is a regular noetherian ring that is also an Fp-algebra. Then it was proved by the author and Madsen [5, 7] that there is a long-exact sequence · · · � � ⊕ q−2i i�