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42 pages, yr 2003

Abstract:

Higher dimensional algebra (H.D.A.) came to light as such in the 1970s in several different forms and in several different contexts. From algebraic topology, developments of descriptions of homotopy coherent structures developed and then used tools such as operads, (symmetric) monoidal categories and monads. Other work used higher dimensional analogues of groups and groupoids to model the homotopy types of spaces. In algebra, representation theory was beginning to use monoidal category structures, whilst category theory had explored 2-categories, bi-categories and the ubiquitous monoidal categories. Workers in mathematical physics were beginning to consider models of physical processes using the same types of structures, often phrased in different language. The subject developed slowly but was spurred on by Grothendieck's Pursuing Stacks (manuscript, 1984), which began to suggest why the H.D.A. machinery was natural to these diverse areas. In the 1990s, progress was suddenly made on understanding long standing problems related to higher dimensional analogues of monoidal categories, bicategories, and 2-categories. At the same time, workers in other fields, notably logic and computer science, started to rediscover and then to apply some of the higher dimensional algebraic framework in their areas and now new applications are being found `thick and fast'. The reason for the ubiquity of the H.D.A. framework is perhaps due to its combination of both combinatorial (graph theoretic) and monoidal/group theoretic tools within the same mathematical package. An illustration of this can be simply constructed by considering the concept of an equivalence relation. In practice, an equivalence relation expresses the fact that various pairs of points are `reachable', each from the other. The relation records just the data of these $(x_1, x_2)$, such that $x_1$ is reachable from $x_2$ and vice versa. The term `reachable' encodes some process of `going' from $x_1$ to $x_2$. This may be a path in a geometric / topological context, a logical derivation in logic, a computer program, a sequence of `rewrites', or transitions in a computer science context, or a reversible process in the physical setting. To increase the amount of information stored on this situation, one can replace the pair, $(x_1,x_2)$ by the set of processes/paths etc. from $x_1$ to $x_2$, so that not only does one know they are `equivalent' but also have the reason or reasons why they are. It is often the case that this set of `reasons' has a natural equivalence relation defined on it, and the bootstrapping to higher dimensions gets going. In Computer Science, higher dimensional automata have recently been introduced to handle concurrency and distributed systems. This is also a young area of research, but it has already produced good insights into hard problems. The models being used are H.D.A. ones or more exactly they bear a similar relationship to H.D.A. that classical automata bear to classical monoid and semigroup theory. This work on modelling distributed systems and use of higher dimensional automata reveals a new exciting possibility for an application of H.D.A. techniques. There is a strong link between distributed systems (in particular their control and optimisation) and modal logics (S4n and S5n in particular). Techniques from algebraic topology (simplicial sets, homotopy, etc.) are being adapted to analyse the problems of computation in such systems and such logics. Given the geometric intuitions behind H.D.A. together with the combinatorial and monoid-like structures in that context, it is highly probable that the use of H.D.A. based concepts and methods would speed up analysis of the applicability of these models of distributed systems to actual systems. The problems of control and optimisation of complex systems again seems an area where H.D.A. insights may help but the development of H.D. algorithms is a pressing need if progress is to be made here.

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Open: http://www.bangor.ac.uk/~mas013/HDA/HDA.html Copyright@2003 by Ronald Brown http://www.informatics.bangor.ac.uk/public/math/research/ftp/cathom/03_05.pdf http://www.springerlink.com/content/a280w10813411982/

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