Unifying theory
Unification programme
Motivated by the results obtained so far,
I intend
to continue the research along the lines
described above
in order to further develop
this unification programme.
Central themes in this
project
will be:
-
Deriving specific Morita-equivalences from the common mathematical practice (much as in the spirit of the papers
"A topos-theoretic approach to Stone-type dualities"
and "Gelfand spectra and
Wallman compactifications")
-
Introducing new methods for generating Morita-equivalences
-
Introducing new topos-theoretic invariants admitting natural site characterizations
- Compiling an 'encyclopedia of invariants and their characterizations' so that the 'working mathematician' can identify properties of sites and toposes which directly relate to his questions of interest
- Investigating the possible limitations of the classical concept of site and/or of geometric logic for expressing properties arising in the mathematical practice and possibly introducing appropriate generalizations of the concept of Grothendieck topos
- Concretely applying these methods in specific situations of interest in classical mathematics
- Automatizing the methodology 'toposes as bridges' on a computer to generate new and non-trivial mathematical results in a mechanical way
Some of these tasks will be of purely topos-theoretic nature, while
others will require, or at least greatly benefit from, interaction with
specialists in the various mathematical fields. Indeed, these methods
are meta-mathematical in nature and hence are not meant to substitute
the
more specialized
techniques
but to complement them; a joint effort between the specialist and
the
'meta-mathematical topos-theorist'
(who might
possibly
be the same
individual
in some cases!) is certainly needed
to obtain new and striking results in the specialist’s field of
interest.