Combinatorial Group Theory and Applications to Geometry by D.J. Collins, R.I. Grigorchuk, P.F. Kurchanov, H. Zieschang,

By D.J. Collins, R.I. Grigorchuk, P.F. Kurchanov, H. Zieschang, A.I. Kostrikin, I.R. Shafarevich, P.M. Cohn

From the stories: "... The booklet lower than evaluate includes monographs on geometric features of crew idea ... jointly, those articles shape a wide-ranging survey of combinatorial team idea, with emphasis a great deal at the geometric roots of the topic. this may be an invaluable reference paintings for the specialist, in addition to delivering an outline of the topic for the outsider or beginner. many various themes are defined and explored, with the most effects offered yet no longer proved. this permits the reader to get the flavor of those subject matters with out turning into slowed down intimately. either articles provide entire bibliographies, in order that it truly is attainable to take advantage of this publication because the place to begin for a extra distinctive learn of a selected subject of curiosity. ..." Bulletin of the London Mathematical Society, 1996

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This is equivalent to other approaches (in particular, to those of Mac Lane and Segal), but looks much more natural. , finitely presented or finite limit theories). The idea of parametrized and functional geometry (see Chap. 3), was already present in the literature, for example in the work of Grothendieck and Souriau’s schools, and in the synthetic geometry community, grounded by Lawvere’s categorical approach to dynamics. However, its systematic use in the formalization of the known physical approaches to quantum field theory is new.

Note that this formulation cannot be obtained without the use of homotopical methods. 11 Doctrines and Theories We now briefly describe the general setting that allowed us to treat all the constructions described in this introduction in a unified setting, basing our approach on (higher) categorical logic. 1 The Categorical Approach to Geometry We first motivate our systematic use of categories by explaining how it lies at the basis of parametrized (see Sect. 1) and functional (see Sect. 4) geometry.

First, we have omitted a lot of the many applications of quantum field theoretic tools in mathematics. Moreover, this books essentially treats the formalization of quantitative aspects of quantum field theories, the main aim being to give mathematicians direct access to the standard physics textbooks in this subject. A very interesting mathematical research direction that we have not treated at all in this book is the study of qualitative aspects of quantum field theories, that one may putatively call the theory of quantum dynamical systems.

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