Groupoids: a local theory of symmetry


Angelo Vistoli (Scuola Normale Superiore di Pisa)


The theme of symmetry is of great interest to mathematician, physicists, chemists, biologists, psychologists, philosophers, and others. The very word \symmetry” is used with a wide variety of meanings; I will only discuss the way it is used in mathematics. In fact, even this seems to me too ambitious a goal. Symmetry permeates every eld of mathematics, and I do not have the intention, and even less the ability, to give a comprehensive picture of its multifaceted aspects. The mathematical analysis of the concept has been traditionally based on the theory of group actions. As we shall discuss, this notion is global; that is, the symmetries of a structure (geometric or otherwise) always involve the whole structure. It is natural, on the other hand, to talk aboutlocal symmetries, symmetries that appear only among certain parts of the structure itself. Mathematicians have a local theory of symmetry, which is known as the theory of groupoids. However, its existence does not seem to have been really noticed outside of the communities of mathematicians and theoretical physicists; the only place in the philosophical literature where I have seen it discussed is Coreld’s book (Coreld 2003), which, I am afraid, has not been read by many philosophers, because of the vast mathematical background it requires. The very modest purpose of this note is to give a quick introduction to symmetry in mathematics, and point out the existence of a mathematical analysis of the notion of local symmetry to philosophers and others who may be interested in this theme. No originality whatsoever is claimed for any of the ideas presented here.

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