Introduction to the language of stacks and gerbes

This is an introduction to gerbes for topologists, with emphasis on non-abelian cohomology.Comment: 30 page

On the derived category of an algebra over an operad

We present a general construction of the derived category of an algebra over an operad and establish its invariance properties. A central role is played by the enveloping operad of an algebra over an operad.Comment: References and remark 2.5 adde

Topological Representation of Geometric Theories

Using Butz and Moerdijk's topological groupoid representation of a topos with enough points, a `syntax-semantics' duality for geometric theories is constructed. The emphasis is on a logical presentation, starting with a description of the semantical topological groupoid of models and isomorphisms of a theory and a direct proof that this groupoid represents its classifying topos. Using this representation, a contravariant adjunction is constructed between theories and topological groupoids. The restriction of this adjunction yields a contravariant equivalence between theories with enough models and semantical groupoids. Technically a variant of the syntax-semantics duality constructed in [Awodey and Forssell, arXiv:1008.3145v1] for first-order logic, the construction here works for arbitrary geometric theories and uses a slice construction on the side of groupoids---reflecting the use of `indexed' models in the representation theorem---which in several respects simplifies the construction and allows for an intrinsic characterization of the semantic side.Comment: 32 pages. This is the first pre-print version, the final revised version can be found at http://onlinelibrary.wiley.com/doi/10.1002/malq.201100080/abstract (posting of which is not allowed by Wiley). Changes in v2: updated comment

Axiomatic homotopy theory for operads

We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.Comment: 29 pages, revised for publicatio

Orbifolds as Groupoids: an Introduction

This is a survey paper based on my talk at the Workshop on Orbifolds and String Theory, the goal of which was to explain the role of groupoids and their classifying spaces as a foundation for the theory of orbifolds