Universal Simplicial Monoid Constructions on Simplicial Categories and their Associated Spectral Sequences

Abstract

We explore a simplicial group construction of Carlsson in this Thesis. On one hand, we provide a deep conceptual explanation of Carlsson's construction and its geometric realization. We use the machinery of category theory and adjoint functors to describe Carlsson's construction as the universal monoid on the action category.

On the other hand, we compute the mod 2 homology of Carlsson's construction in the case of actions by the discrete group with two elements. We utilize the algebraic nature of Carlsson's group construction to create filtrations. These filtrations in turn define spectral sequences which we analyze to obtain information about the homology of Carlsson's construction.

Throughout this thesis, we emphasize the applications of our work to equivariant homology. We hope to indicate how category theory, simplicial homotopy theory and spectral sequences can yield fruitful applications to the field of equivariant homology.