Let M be a torus bundle, i.e. an (n + 1)-dimensional manifold, which fibers over the circle with the fiber an n-dimensional torus. A covariant functor from M to the Cuntz-Krieger algebra (a C∗-algebra) is constructed. The functor maps homeomorphic torus bundles to the stably isomorphic Cuntz-Krieger algebras. It is shown that in general the K-theory of the Cuntz-Krieger algebra counts the torsi...

For an arbitrary-type functor F, the notion of split coalgebras, that is, coalgebras for which the canonical projections onto the simple factor split, generalizes the well-known notion of simple coalgebras. In case F weakly preserves kernels, the passage from a coalgebra to its simple factor is functorial. This is the simplification functor. It is left adjoint to the inclusion of the subcategor...

The purpose of these notes is to define an equivalence between the natural homology theories associated to operads and the homology of functors over certain categories of operators (PROPs) related to operads.