The first goal of this survey paper is to argue that if orbifolds are groupoids, then the
collection of orbifolds and their maps has to be thought of as a 2-category. Compare this with …

We show how general principles of symmetry in quantum mechanics lead to twisted notions
of a group representation. This framework generalizes both the classical threefold way of …

This book develops rigorous foundations for parametrized homotopy theory, which is the
algebraic topology of spaces and spectra that are continuously parametrized by the points of …

We formulate differential cohomology and Chern-Weil theory--the theory of connections on
fiber bundles and of gauge fields--abstractly in the context of a certain class of higher …

Twisted complex $ K $-theory can be defined for a space $ X $ equipped with a bundle of
complex projective spaces, or, equivalently, with a bundle of C $^* $-algebras. Up to …

We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we
study S^1-bundles and S^1-gerbes over differentiable stacks. In particular, we establish the …

This is the first in a series of papers investigating the relationship between the twisted
equivariant K-theory of a compact Lie group G and the “Verlinde ring” of its loop group. In …

In this book we prove (Thm. 4. 3.24) unified classification results for stable equivariant Γ-
principal bundles when the underlying homotopy type SΓ of the topological structure group Γ …

We introduce quasi-symplectic groupoids and explain their relation with momentum map
theories. This approach enables us to unify into a single framework various momentum map …

We discuss two aspects of the presentation of the theory of principal ∞∞-bundles in an ∞∞-
topos, introduced in Nikolaus et al.(Principal ∞∞-bundles: general theory, 2012), in terms of …