Authors: P.J.Ehlers and T Porter
Date: February 6, 1998
Abstract: We introduce a notion of join for (augmented) simnplicial sets generalising the classical join of geometric simplicial complexes. The definition comesnaturally from the ordinal sum on the base simplicial category $\Delta$.
Keywords: Join, (augmented) simnplicial sets
AMS Subject Classification: 18G30
Type of file: Postscript.
No. of pages: 8
Authors: Z. Arvasi and T. Porter
Title: Freeness Conditions for 2-Crossed Modules of Commutative Algebras
Date: April 1, 1998
Abstract: In this paper we give a construction of free 2-crossed modules. By the use of a `step-by-step' method based on the work of André, we will give a description of crossed algebraic models for the steps in the construction of a free simplicial resolution of an algebra. This involves the introduction of the notion of a free 2-crossed module of algebras.
Keywords: Free 2-crossed modules, free simplicial algebras
Title: Freeness conditions for crossed squares and squared complexes
Date: February 3, 1999
Abstract: Following Ellis, we investigate the notion of totally free crossed squares arndrelated square complexes. It is shown how to interpret the information in a free simplicial group given with a choice of CW-basis, in terms of the data for a totally free crossed square. Results of Ellis then apply to give a description in terms of tensor products of crossed modules. The paper ends with a purely algebraic derivation of a result of Brown and Loday.
Keywords: Crossed square, squared complex, simplicially enriched groupoid.
No. of pages: 23
Title: A homotopy 2-groupoid from a fibration.
Date: October 20, 1998
Abstract: In this paper we give an elementary derivation of a 2-groupoid from a fibration. This extends a previous result for pointed fibrations due to Loday. Discussion is included as to the translation between 2-groupoids and cat^1-groupoids
Keywords: 2-groupoid, cat^1-groupoid, crossed module, fibration.
No. of pages: 16
Author: R. Brown and T.Porter
Title:
The intuitions of higher dimensional algebra for the study of structured
space
Abstract: We discuss some of the origins and applications of the notion of higher dimensional algebra, with the hope that this will encourage new insights into the development of the mathematics of complex hierarchical systems.
Published in:
Title: Crossed squares and 2-crossed modules
Abstract:
Author: T.Porter
Title: Geometric aspects of multiagent systems
Abstract:
Recent advances in Multiagent Systems (MAS) and Epistemic Logic within
Distributed Systems Theory, have used
various combinatorial structures that model both the geometry of the
systems and the Kripke model structure of
models for the logic. Examining one of the simpler versions of these
models, interpreted systems, and the related
Kripke semantics of the logic S5_n (an epistemic logic with n-agents),
the similarities with the geometry/homotopy
theoretic structure of groupoid atlases is striking. These latter objects
arise in problems within algebraic K-theory, an
area of algebra linked to the study of decomposition and normal form
theorems in linear algebra. They have a
natural well structured notion of path and constructions of path objects,
etc., that yield a rich homotopy theory.
In this paper, we examine what a geometric analysis of the model may
tell us of the MAS. Also the analogous notion of
path will be analysed for interpreted systems and S5_n-Kripke models,
and is compared to the notion of `run' as used
with MASs. Further progress may need adaptions to handle S4_n rather
than S5_n and to use directed homotopy
rather than standard `reversible' homotopy.