Algebraic homotopy, Galois theory and Descent
1 Objectives
Classical Galois theory studied extension fields of a given field. Techniques
of descent provided ways of glueing local information together to obtain
global information. Algebraic homotopy concerns the use of both algebraic
models for homotopy types and homotopy theory applied to algebraic objects.
It is closely related to homotopical algebra.
The link between these is clearly seen in Grothendieck's SGA1 in which
the theory of covering spaces is generalised using categorical methods,
to the objects of algebraic geometry. The viewpoint is wide enough to encompass
the fundamental group or groupoid approach to covering spaces and Galois
theory, within the same collection of results. The subgroups of the fundamental
group of a space, X, yield information on the covering spaces of
X
and vice versa via a Galois correspondence. Higher order homotopy invariants
such the higher homotopy groups do not yield any analogue of this result.
The glueing together of covering spaces over open sets from an open cover
of X to form a global covering space gives the covering space version
of van Kampen's theorem.
In the 1970s Grothendieck tried to formulate a higher dimensional analogue
of a covering space. These 'n-stacks' were to be exemplified by
various constructions from non-abelian cohomology and algebraic geometry.
An 'n-stack' was to be a sheaf-like object obtained by gluing algebraic
models of homotopy n-types together. His ideas were influenced by
correspondence with R Brown (Bangor) and the latter's work on generalised
forms of van Kampen's theorem. This resulted in 1984 in a manuscript entitled
'Pursuing Stacks' of over 650 type written pages. Amongst other things,
this put forward a rough conjecture that each n-type should be represented
by an algebraic or categorical model so that if X was a space, the
category of 'n-stacks' using that type of algebraic model would
be equivalent to that of the lax-actions of the corresponding algebraic
(n+1)-type of X. This was one facet of a wide ranging generalisation
of Galois theory. Given the central importance of the Galois / Poincaré
approach to covering spaces within classical pure mathematics the prospect
of a hierarchy of Galois-type theories in higher dimensions was very exciting.
Although much progress towards completion of Grothendieck's programme
has been made since 1984, there is still much to investigate before the
full richness of his ideas can be exploited. This project is part of that
work.
Janelidze in 1989 published a new purely categorical approach to Galois
theory which again included both covering space theory, and classical Galois
theory, and the SGA1 approach of Grothendieck, but also was applicable
to classification problems in other parts of algebra and topology. In particular
it could be directly applied to central extensions of groups, thus making
a known resemblance between that theory and the Galois / Poincaré
theory into just two instances of the same categorical theory.
Descent theory has continued to advance clarifying the problems that
arise when handling topological descent (Sobral) as well as being adapted
to more algebraic contexts. Lax versions of descent theory are being investigated
to enable effective handling of situations where the 'local' information
only coincides up to an equivalence (topological or categorical). Advances
in topological quantum field theory, logic and category theory have produced
more workable n-categorical models for homotopy types and work has
started on the internalisation of these models to within algebraic categories
(monoids, rings, etc.) thus providing new tools for the homotopical algebra
of these contexts. As one would expect, mixes of Galois theory and descent
are being applied back in algebraic geometric settings with results on
ringed topoi, whilst work by Brown continues to look at essential obstructions
to extending local to global information measurable by monodromy groupoids
generalising the fundamental group.
Particular cases of the Grothendieck problem have been investigated.
His own preference was for a setting involving topoi as substitutes for
spaces. These (Grothendieck) topoi have been shown by Joyal and Tierney
to correspond to categories of sheaves on an étale localic groupoid.
Special cases include G-sets and sheaves on orbifolds and orbihedra.
The corresponding Galois theory and resulting algebraic homotopy is being
studied at Bangor, not only for the potential application within equivariant
homotopy and the theory of orbifolds, but also as a test bed for methods
relating to Grothendieck's programme. Other related work has been investigating
structures (including inverse semigroups) that occur naturally within many
branches of mathematics, and are closely related to the localic groupoids
mentioned earlier. Given Janelidze's extension of Grothendieck's SGA1 theory,
these semi-group structures would seem of great potential for widening
the applicability of the Galois / Poincaré theory and related questions
of descent to exciting new settings.
Many of these settings lead to internal categories and groupoids within
that context, yet our knowledge of how to analyse such objects is far from
complete. As these objects often have occurred in other contexts (e.g.
semidirect products of categories), some mainly homological methods have
been used with some success, these homological methods need to be analysed
further and extended to improve the understanding of the objects obtained
as invariants in the Galois theoretic framework.
Symbolic computation within these various Galois theoretic settings
is still in its infancy. New methods have been developed for computations
with internal categories of groups and related areas of polynomial ring
theory. These symbolic computational methods allow, to some extent, calculations
to be made that mirror van Kampen theoretic constructions and problems
of a descent theoretic nature (induced crossed modules).
This overall area of study is ripe for considerable progress. It is 15 years since Grothendieck's Pursuing Stacks made researchers realise there were higher dimensional Galois theoretic results possibly 'out there'. More and more evidence for their existence and importance has accumulated and the various necessary tools would now seem to be there. This project hopes to make a contribution to this overall progression by combining workers in various aspects of the theory into a small international team for active collaboration.
Bangor team:
T. Porter, R. Brown, M. V. Lawson, C. D. Wensley:
School of Mathematics, University of Wales Bangor.
Coimbra team:
M.Sobral, M. M Clemintino, J. Picardo, G. Gutierres:
Departamento de Matematica, Universidade de Coimbra.
Georgian Academy team:
G. Janelidze, T. Datuashvili, B. Mesablishvili, Z. Omiadze, A. Patchkoria,
Technical University of Georgia team:
D. Zanguashvili.