Julia E. Bergner

A model category structure on the category of simplicial categories

In this paper we put a cofibrantly generated model category struc- ture on the category of small simplicial categories. The weak equivalences are a simplicial analogue of the notion of equivalence of categories.

A Survey of (∞, 1)-Categories

In this paper we give a summary of the comparisons between different definitions of so-called (∞, 1)-categories, which are considered to be models for ∞-categories whose n-morphisms are all invertible for n > 1. They are also, from the viewpoint of homotopy theory, models for the homotopy theory of homotopy theories. The four different structures, all of which are equivalent, are simplicial categories, Segal categories, complete Segal spaces, and quasi-categories.

Homotopy limits of model categories and more general homotopy theories

Generalizing a deflnition of homotopy flber products of model cat- egories, we give a deflnition of the homotopy limit of a diagram of left Quillen functors between model categories. As has been previously shown for homo- topy flber products, we prove that such a homotopy limit does in fact corre- spond 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.

Complete Segal spaces arising from simplicial categories

In this paper, we compare several functors which take simplicial categories or model categories to complete Segal spaces, which are particularly nice simplicial spaces which, like simplicial categories, can be considered to be models for homotopy theories. We then give a characterization, up to weak equivalence, of complete Segal spaces arising from these functors.

Comparison of models for (∞,n)–categories, I

In this paper we complete a chain of explicit Quillen equivalences between the model category for

A characterization of fibrant Segal categories

\Theta_{n+1}

Group actions on Segal operads

-spaces and the model category of small categories enriched in

Reedy categories which encode the notion of category actions

\Theta_n

EQUIVALENCE OF MODELS FOR EQUIVARIANT (∞, 1)-CATEGORIES

-spaces.

Groupoid Cardinality and Egyptian Fractions

In this note we prove that Reedy fibrant Segal categories are fi- brant objects in the model category structure SeCatc. Combining this result with a previous one, we thus have that the fibrant objects are precisely the Reedy fibrant Segal categories. We also show that the analogous result holds for Segal categories which are fibrant in the projective model structure on simplicial spaces, considered as objects in the model structure SeCatf.