Adding inverses to diagrams encoding algebraic structures

We modify a previous result, which showed that certain diagrams of spaces are essentially simplicial monoids, to construct diagrams of spaces which model simplicial groups. Furthermore, we show that these diagrams can be generalized to models for Segal groupoids. We then modify Segal's model for simplicial abelian monoids in such a way that it becomes a model for simplicial abelian groups.Comment: 24 pages, final version; erratum included at the end. arXiv admin note: text overlap with arXiv:math/050841

Homotopy limits of model categories and more general homotopy theories

Generalizing a definition of homotopy fiber products of model categories, we give a definition of the homotopy limit of a diagram of left Quillen functors between model categories. As has been previously shown for homotopy fiber products, we prove that such a homotopy limit does in fact correspond to the usual homotopy limit, when we work in a more general model for homotopy theories in which they can be regarded as objects of a model category.Comment: 10 pages; a few minor changes made. arXiv admin note: text overlap with arXiv:0811.317

A model category structure on the category of simplicial categories

In this paper we put a cofibrantly generated model category structure on the category of small simplicial categories. The weak equivalences are a simplicial analogue of the notion of equivalence of categories.Comment: 16 pages, revised version has proof of right properness and a few minor changes; to appear in Transactions of the AM

Homotopy colimits of model categories

Building on a previous definition of homotopy limit of model categories, we give a definition of homotopy colimit of model categories. Using the complete Segal space model for homotopy theories, we verify that this definition corresponds to the model-category-theoretic definition in that setting.Comment: 7 pages; substantial changes to construction. To appear in Arolla proceeding

Rigidification of algebras over multi-sorted theories

We define the notion of a multi-sorted algebraic theory, which is a generalization of an algebraic theory in which the objects are of different "sorts." We prove a rigidification result for simplicial algebras over these theories, showing that there is a Quillen equivalence between a model category structure on the category of strict algebras over a multi-sorted theory and an appropriate model category structure on the category of functors from a multi-sorted theory to the category of simplicial sets. In the latter model structure, the fibrant objects are homotopy algebras over that theory. Our two main examples of strict algebras are operads in the category of simplicial sets and simplicial categories with a given set of objects.Comment: This is the version published by Algebraic & Geometric Topology on 14 November 200