The category of von Neumann correspondences from B to C (or von Neumann B–C–modules) is dual to the category of von Neumann correspondences from C to B via a functor that generalizes naturally the functor that sends a von Neumann algebra to its commutant and back. We show that under this duality, called commutant, Rieffel’s Eilenberg-Watts theorem (on functors between the categories of represen...

We devise a generic framework where a weakest precondition semantics, in the form of indexed posets, is derived from a monad whose Kleisli category is enriched by posets. It is inspired by Jacobs’ recent identification of a categorical structure that is common in various predicate transformers, but adds generality in the following aspects: 1) different notions of modality (such as “may” vs. “mu...

The motivation for this paper was to construct approximations to a conformal version of homotopy quantum field theory using 2-categories. A homotopy quantum field theory, as defined by Turaev in [9], is a variant of a topological quantum field theory in which manifolds come equipped with a map to some auxiliary space X . From a geometrical point of view a 1+1 dimensional homotopy quantum field ...

The purpose of this paper is to present a new approach to the homology of the S orbits of the free loop space ΛX. Let X be a space, and Cn the cyclic group considered as acting on X n by permuting coordinates. The (Z/2-)homology of ECn×CnX can be computed from R = H∗(X;Z/2) by a Serre spectral sequence. It is a fundamental fact, derived from the equivariance of the Eilenberg-Zilber map, that th...

We consider partial liftings k : A → E of maps f : X → B, where (X,A) is a relative CW-complex and E → B is a fibration. In this situation, we have a primary obstruction to extend the partial lifting to a lifting of f on all of X, and there is an obstruction to extend k as an ordinary map into the space E. A relation between these two cohomology classes is proved when the fibre of E → B is an E...

For a real central arrangement H, Salvetti introduces a construction of a finite complex which is homotopy equivalent to the complement of the complexified arrangement in [Sal87]. We compare the skeletal filtration on the Salvetti complex for the braid arrangement and the filtration on the space of n little squares C2(n) introduced in [Tam94] under the Σn-equivariant homotopy equivalence betwee...

We discuss actions of Lie n-groups and the corresponding action Lie n-groupoids; discuss actions of Lie n-algebras (L∞-algebras) and the corresponding action Lie n-algebroids; and discuss the relation between the two by integration and differentiation. As an example of interest, we discuss the BRST complex that appears in quantum field theory. We describe it as the Chevalley-Eilenberg algebra o...