- On Parity Quasicomplexes and Non-abelian
group cohomology -
1998 (49 pages PS file) (math.CT/9808068).
To characterize the categorical constraints - associativity,
commutativity
and monoidality - in the context of quasimonoidal categories, from a
multiplicative
cohomological point of view, we define the notion of a parity
(quasi)complex.
Applied to groups gives non-abelian cohomology.
The categorification - functor from groups to monoidal categories
-
provides the correspondence between the respective parity
(quasi)complexes
and alows to interprete 1-cochains as functors, 2-cocycles - monoidal
structures,
3-cocycles - associators.
The cohomology spaces $H^3, H^2, H^1, H^0$ correspond as usual to
quasi-extensions,
extensions, split extensions and invariants, as in the abelian case.
A larger class of commutativity constraints for monoidal
categories
is identified. It is naturally associated with representations of
coboundary
Hopf algebras.
- Torsion algebras and differential geometry
-
1998 (DVI file, 15 pages)
Classical differential geometry is built on the notion of space, e.g.
differential manifolds. A rough hierarchy is: space, functions, vector
fields and differential forms, connections etc. Algebraic geometry
starts
at the ``second level'' - functions - by considering an arbitrary
commutative
algebra and then constructing the ``first level'', the (local)
substitute
for a space: its spectrum. Then gluing, etc. A natural question arises:
What can be derived starting from the ``third level'' - vector fields -
and in what extent is it profitable?
Interpreting the elements of a non-associative algebra as "vector
fields"
and the multiplication as a connection, we investigate a natural
candidate
for the algebra of functions, with derivations the former algebra. As a
model: pre-Lie algebras. A generi c example: Hochschild cochains under
Gerstenhaber composition.
- Overview: non-abelian cohomology and parity
quasi-complexes
- 01/1999 (DVI file, 14 pages)
Previous attempts to define non-abelian cohomology either in a direct
way (P.Dedecker, L. Breen) or through objects representing cocycles
(Giraud,
Grothendieck) are mentioned. We define parity quasicomplexes as an
analog
of the Bar construction, generalizing the existing constructions in low
dimensions. It is a specific construction defining non-abelian
cohomology
of groups and cohomology of monoidal categories. It suggests some
future
developments in the context of non-abelian categories. It is a point of
view unifying the cohomology conditions defined by S. Mac Lane in the
study
of extensions of groups (1946), the cohomological conditions for a
monoidal
category defined by R. Saavedra (1972) and the idea of separating the
boundary
of a chain, present in the work of R. Street (1991), and A.A. Davydov
(1997).
- On Categorification - 04/1999 (10
pages,
math.CT/9906038)
We review several known categorification procedures, and introduce
a functorial categorification of group extensions with applications to
non-abelian group cohomology. Categorification of acyclic models and of
topological spaces are briefly mentioned.
- On ideals and homology in additive
categories
- v.2 01/2000 (10 pages, math.CT/9906039; v.1 05/1999 8p.).
Ideals are used to define homological functors for additive categories.
In abelian categories the ideals corresponding to the usual universal
objects
are principal, and the construction reduces, in a choice dependent way,
to homology groups.
An analogue of the axioms for an abelian category is considered.
Derived categories are important examples of categories which are
not
exact: cokernels may be missing. Still the "usual" homological approach
using ideals can be used. In derived categories the cone of a map is a
canonical generator for the corresponding cokernel ideal. Is the
present
approach an alternative approach to triangulated categories? ...
- Remarks on quantum theory and
noncommutative geometry
- 06/1999 (ps.gz file, 17 pages)
I have always believed that there must be a deep reason for the
incompatibility
between general relativity, a classical-phenomenological-macroscopical
theory, and quantum physics, a fundamental-highly experimental theory,
incompatibility accounted for by 50-60 years of unsuccessful attempts
to
unify them.
I consider that, as a first step towards reconciliation, the
classical
mathematical-physics point of view: