Using the theory of homotopy coherent Kan extensions,
results of Elmendorf and Dwyer and Kan are generalised.
This produces simplicially enriched equivariant versions of the
singular complex / geometric realisation adjunction of the non-
equivariant theory.
This article is an introduction to the categorical theory
of homotopy coherence. It is based on the construction of the
homotopy coherent analogues of end and coend,
extending idear of Meyer and others.
The paper aims to develop homotopy coherent analogues
of many of the results of elementary category theory,
in particular it handles a homotopy coherent form of the
Yoneda lemma and of Kan extensions.
The latter area is linked with the theory of generalised
derived functors.
Interpretations of Yetter's notion of G-coloring :
simplicial fibre bundles and non-abelian cohomology
Abstract:
Yetter showed how a notion of colouring a triangulation of a
triangulation of a manifold with elements of a finite group G
leads to a construction of a topological quantum field theory.
He later adapted his construction to give colourings
with a categorical group as coefficients.
In this paper Yetter's work is reinterpreted in terms of simplicial
groups and is shown to yeild interpretations of the resulting
topological quantum field theory in terms either of
simplicial fibre bundles or of non-abelain cohomology.
Toplogical quantum field theories from homotopy n-types
Abstract:
Using simplicial methods developed in an earlier note (95.11),
the paper constructs topological quantum field theories
using an algebraic model of a homotopy n-type as initial data,
generalising a construction of Yetter in
(J. Knot Theory and its Ramifications, 1 (1992) 1-20) for n=1 and in
(J. Knot Theory and its Ramifications, 2 (1993) 113-123) for n=2.
Varieties of simplicial groupoids I: Crossed complexes
Abstract:
It is usual to use algebraic models for homotopy types.
Simplicial groupoids provide such a model.
Other partial models include the crossed complexes of Brown and Higgins.
In this paper, the simplicial groupoids that correspond to crossed complexes
are shown to form a variety within the category of all simplicial groupoids
and the corresponding verbal subgroupoid is identified.
Published in:
Journal of Pure and Applied Algebra
120 (1997) 221-233.
Erratum:
Journal of Pure and Applied Algebra
134 (1999) 207-209.
Homotopy theory, and change of base for groupoids and multiple groupoids
Abstract:
This survey article is an expanded version of a talk given at the European
Category Theory Meeting, Tours, July 1995.
It shows how the notion of ``change of base'',
used in some applications to homotopy theory of the fundamental groupoid,
has surprising higher dimensional analogues, through the use of
certain higher homotopy groupoids with values in forms of multiple groupoids.
