Calculations with simplicial and cubical groups in AXIOM
Abstract:
Work on calculations with simplicial and cubical groups in AXIOM
was carried out using loan equipment and software from IBM UJ
and guidance from L. Lambe.
We report on the results of this work,
and present the AXIOM code written by the second author during
this period.
This includes an implementation of the monoids which model cubes
and simplices, together with a new AXIOM category of near-rings
with which to carry out non-abelian calculations.
Examples of the use of this code in interactive AXIOM sessions
are also given.
Covering groups of non-connected topological groups revisited
Abstract:
The purpose of this paper is to relate Taylor's result on the
obstruction class of a topological group
to modern work on coverings of groupoids
and the equivalence between group-groupoids and crossed modules.
This leads to a clear relation of the theory with the classical theory
of abstract kernels and with the theory of extensions
of the type of a crossed module.
The results of this paper form Part 1 of Mucuk's Ph.D. thesis.
We show that under general circumstances,
the disjoint union of the universal covers of the stars of a Lie groupoid
admits the structure of a Lie groupoid, such that the projection has
a monodromy property on the extension of local smooth morphisms.
This completes a detailed account of results announced by J Pradines.
Published in:
Cahiers de Topologie Geometrie Differentielle categoriques
36 (1995) 345-369.
We show that a paracompact foliated manifold determines a locally Lie groupoid
(or piece of a differentiable groupoid, in the sense of Pradines).
This allows for the construction of holonomy and monodromy groupoids
of a foliation to be seen as particular cases of constructions for
locally Lie groupoids.
Published in:
Cahiers de Topologie Geometrie Differentielle categoriques,
37 (1996) 61-71.
We prove a `slightly non-abelian' version of the classical
Eilenberg-Zilber theorem:
if K,L are simplicial sets, then there is a strong
deformation retraction of the fundamental crossed complex
of the cartesian product K x L onto the tensor product
of the fundamental crossed complexes of K and L.
This satisfies various side conditions and associativity/interchange laws,
as for the chain complex version.
Given simplicial sets K_0,...,K_r, we discuss the r-cube
of homotopies induced on \pi(K_0 x ... x K_r)
and show these form a coherent system.
We introduce a definition of a double crossed complex,
and of the associated total (or codiagonal)
crossed complex.
We introduce a definition of homotopy colimits of
diagrams of crossed complexes.
We show that the homotopy colimit of crossed complexes
can be expressed as the total complex of a certain `twisted'
simplicial crossed complex, analagous to Bousfield and Kan's
definition of simplicial homotopy colimits as the diagonal
of a certain bisimplicial set.
Using the Eilenberg-Zilber theorem we show that the fundamental
crossed complex functor preserves these homotopy colimits
up to a strong deformation retraction.
This is applied to give a small crossed resolution
of a semidirect product of groups.
We consider a simplicial enrichment of the category of crossed complexes,
and investigate the coherent homotopy structure up to which a simplicial
enrichment may be given to the fundamental crossed complex functor.
We end with a definition of homotopy coherent functors
from a small category to the category of crossed complexes,
and suggest a definition of homotopy colimits of such functors
and of a small crossed resolution of an arbitrary group extension.
Representation and computation for crossed modules
Abstract:
This paper discusses the notion of ``structural computation'',
and illustrates it with the problem of translating a notion of
tensor product between several equivalent categories,
in this case crossed modules, cat-1-groups, double groupoids with connection,
2-groupoids. In this case, the easy definition
of tensor product is for double groupoids with connection.
This paper advertises the joining of two themes: groups and symmetry; and
categorical methods and analogues of set theory. The basic idea is that the
`symmetry object' of a directed graph should be both a group and a directed
graph. From this is obtained the notion of `inner automorphism' of a directed
graph. The work of J. Shrimpton completely describes these.
