Virtual Topology and Functor Geometry
This book has a special character. Its main theme is to describe development of new branches of non-commutative geometry on a different level of realizations, ranging from areas already fully developed to many different suggestions for possible future investigations. In particular, a lot of attention in the book is concentrated on a formulation of a suitable version of non-commutative topology and sheaves in this situation. A standard version of non-commutative geometry consists of an associative algebra, which is a generalization of the commutative algebra of functions on an ordinary space. It is a pointless geometry, which makes the formulation of a topology seemingly hopeless. The author has earlier developed a version of scheme theory over a non-commutative algebra based on module theory and quasi-coherent sheaves. The language used in the book is that of category theory (summarized briefly in Chapter 1). In the book, the author discusses possibilities of extending it to a more general setting, using a non-commutative version of lattices as a tool. This is contained in Chapter 2, ending with the two representative examples of the theory (the lattice of torsion theories and the lattice of closed linear subspaces of Hilbert space). Chapter 3 includes the use of a general notion of a quotient representation for a description of a relation between non-commutative affine spaces and non-commutative projective spaces. A dynamical version of topology and sheaf theory is introduced in the last chapter. A very particular feature of the book is a formulation of numerous suggestions for research projects throughout the book. Some of them are more accessible but often they are quite advanced. On the whole, the book is very inspiring and worth reading.