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Mathematics Preprints 1995


Category Theory & Homotopy Theory

95.09 : J.-M. Cordier & T. Porter

Categorical Aspects of Equivariant Homotopy

Abstract:

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.

AMS Subject Classification:

55P91, 18G55, 18D20, 18G30

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95_09.ps.gz

Published in:

Applied Categorical Structures 4 (1996) 195-212.


95.10 : J.-M. Cordier & T. Porter

Homotopy Coherent Category Theory

Abstract:

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.

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95_10.ps.gz

Published in:

Trans. Amer. Math. Soc. 349 (1997) 1-54.


95.11 : Timothy Porter

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.

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95_11.pdf

Published in:

J. Knot Theory and its Ramifications 5 (1996) 687-720.


95.12 : Timothy Porter

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.

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95_12.pdf

Published in:

J. London Math. Soc.(2) 58 (1998) 723-732.


95.15 : EHLERS, P.J. & PORTER, T.

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.


95.16 : ARVASI, Z. & PORTER, T.

Simplicial and Crossed Resolutions of Commutative Algebras

Abstract:

Published in:


95.18 : BROWN, R.

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.

gzipped Postscript file:

basechpr.ps.gz

Published in:

Applied Categorical Structures, 4 (1996) 175-193.


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