`Stable homotopy of algebraic theories'
to appear in Topology
Stefan Schwede
Fakultaet fuer Mathematik
Universitaet Bielefeld
33615 Bielefeld, Germany
schwede@mathematik.uni-bielefeld.de
ABSTRACT: The simplicial objects in an algebraic category admit an abstract homotopy theory via a Quillen model category structure.
We show that the associated stable homotopy theory is completely determined by a ring spectrum functorially associated with the algebraic theory.
For several examples of algebraic theories the parameterizing ring spectrum can be identified with something familiar:
for the theory of sets we obtain the standard model of the sphere spectrum; the theories of monoids and groups give different, but stably equivalent models for the sphere spectrum; for sets with an action of a fixed groups one gets the spherical group ring; the theory of modules over a fixed ring leads to the Eilenberg-MacLane ring spectrum. For many other algebraic theories we obtain new examples of ring spectra.
For the theory of commutative algebras we obtain a ring spectrum which is related to Andre-Quillen homology via certain spectral sequences.
We show that the (co-)homology of an algebraic theory is isomorphic to the topological Hochschild (co-)homology of the parameterizing ring spectrum.