We consider monads over varying categories, and by defining the morphisms of Kleisli and of Eilenberg-Moore from a monad to another and the appropriate transformations (2-cells) between morphisms of Kleisli and between morphisms of Eilenberg-Moore, we obtain two 2-categories MndKl and MndEM. Then we prove that MndKl and MndEM are, respectively, 2-isomorphic to the conjugate of Kl and to the tra...

We study the module category associated to the quantum double of a finite abelian group G twisted by a 3-cocycle, which is known to be a braided monoidal category, and investigate the question of when two such categories are equivalent. We base our discussion on an exact sequence which interweaves the ordinary and Eilenberg-Mac Lane cohomology of G. Roughly speaking, this reveals that the data ...

We show that the existence of canonical representatives for the elements of the tensor product (coend) of a simplicial and a cosimplicial set depends only on the Eilenberg-Zilber property of the given cosimplicial set. Thus the second condition which is used in [5] for achieving this result is superfluous. Let X: z"P S be a simplicial set and Y: A -S a cosimplicial set. We consider X as a N-gra...

This chapter uses categorical techniques to describe relations between various sets of operators on a Hilbert space, such as self-adjoint, positive, density, effect and projection operators. These relations, including various Hilbert-Schmidt isomorphisms of the form tr(A−), are expressed in terms of dual adjunctions, and maps between them. Of particular interest is the connection with quantum s...

Let G be an infinite group which is finitely presented. Let X be a finite CW−complex of dimension q whose fundamental group is Z × G. We prove that for some i ≤ q the homotopy group πi(X) is not finitely generated. Let M be a manifold of dimension n whose fundamental group is Zn−2×G. Then the same conclusion holds (for some i ≤ maxn2 ] , 3 } ) unless M is an Eilenberg-McLane space. In particula...

We construct in a systematic way the complete Chevalley-Eilenberg cohomology at form degree two, three and four for the Galilei and Poincaré groups. The corresponding non-trivial forms belong to certain representations of the spatial rotation (Lorentz) group. In the case of two forms they give all possible central and non-central extensions of the Galilei group (and all non-central extensions o...

We set up machinery for recognizing k-cellular modules and k-cellular approximations, where k is an R-module and R is either a ring or a ring-spectrum. Using this machinery we can identify the target of the Eilenberg-Moore cohomology spectral sequence for a fi-bration in various cases. In this manner we get new proofs for known results concerning the Eilenberg-Moore spectral sequence and genera...