By Frédéric Paugam (auth.)

This formidable and unique booklet units out to introduce to mathematicians (even together with graduate scholars ) the mathematical tools of theoretical and experimental quantum box concept, with an emphasis on coordinate-free shows of the mathematical gadgets in use. This in flip promotes the interplay among mathematicians and physicists by means of delivering a typical and versatile language for the nice of either groups, even though mathematicians are the first objective. This reference paintings presents a coherent and entire mathematical toolbox for classical and quantum box idea, in keeping with specific and homotopical tools, representing an unique contribution to the literature.

The first a part of the booklet introduces the mathematical tools had to paintings with the physicists' areas of fields, together with parameterized and sensible differential geometry, functorial research, and the homotopical geometric conception of non-linear partial differential equations, with purposes to common gauge theories. the second one half provides a wide kin of examples of classical box theories, either from experimental and theoretical physics, whereas the 3rd half presents an advent to quantum box thought, provides a number of renormalization equipment, and discusses the quantization of factorization algebras.

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2 Higher Categories and Categorical Logic Suppose given a notion of n-category for n ≥ 0. All theories, types of theories and their models (also called semantics) that will be considered in this book can be classified, in a coordinate-free fashion, by the following definition of a doctrine. 1 Let n ≥ 1 be an integer. A doctrine is an (n + 1)-category D. A theory of type D is an object C of D. A model for a theory of type D in another theory of the same type is an object M : C1 → C2 of the n-category Mor(C1 , C2 ).

The doctrine of (L, C)-sketches is the (n + 1)-category: 1. whose objects are triples (T , L, C), called sketches, composed of n-categories T , equipped with a class L of I -indexed cones for I ∈ L and a class C of J -indexed cocones for J ∈ C; 2. whose morphisms are functors that respect the given classes of cones and cocones. A sketch (T , C, L) is called a full sketch if C and L are composed of colimit cocones and limit cones indexed by categories I ∈ L and J ∈ C. 2 Note that some doctrines may be described by sketch theories, with additional properties.

10 1. The doctrine FPC AT of finite product theories coincides with the doctrine of (L, C)-sketches, with L the class of all finite discrete categories (whose only morphisms are identities) and C = ∅. 2. The doctrine FPA LG of finitely presented algebraic theories is defined as the doctrine of (L, C)-sketches with L the class of all finite discrete categories and C the class of finite categories. An example of a full sketch for this doctrine is given by the category of finitely presented algebras (coequalizers of two morphisms between free algebras) for a given algebraic theory T .