The intuitions of higher dimensional algebra for the study of structured space |
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Authors: | Email author" target="_blank">Ronald?BrownEmail author Timothy?Porter |
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Institution: | (1) FSHN and NPRE Departments, AFC–NMR and NIR Microspectroscopy Facility, University of Illinois at Urbana–Champaign, Urbana, IL 61801, USA;(2) School of Informatics, University of Wales, Dean St., Bangor, Gwynedd, LL57 1UT, UK;(3) Department of Mathematics and Computer Science, Eastern Illinois University, 600 Lincoln Ave., Charleston, IL 61920-3099, USA |
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Abstract: | Higher dimensional algebra frees mathematics from the restriction to a purely linear notation, in order to improve the modelling of geometry and so obtain more understanding and more modes of computation. It gives new tools for noncommutative, higher dimensional, local to global problems, through the notion of «algebraic inverse to subdivision». We explain the way these ideas arose for the writters, in extending first the classical notion of abstract group to abstract groupoid, in which composition is only partially defined, as in composing journeys, and which brings a spatial component to the usual group theory: An example from knot theory is used to explain how such algebra can be used to describe some structure of a space. The extension to dimension 2 uses compositions of squares in two directions, and the richness of the resulting algebra is shown by some 2-dimensional calculations. The difficulty of the jump from dimension 1 to dimension 2 is also illustrated by the comparison of the commutative square with the commutative cube-discussion of the latter requires new ideas. The importance of category theory is explained, and a range of current and potential applications of higher dimensional algebra indicated. |
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