It is shown how the theory of commutative monads provides an axiomatic framework for several aspects of distribution theory in a broad sense, including probability distributions, physical extensive quantities, and Schwartz distributions of compact support. Among the particular aspects considered here are the notions of convolution, density, expectation, and… (More)
In terms of synthetic differential geometry, we give a variational characterization of the connection (parallelism) associated to a pseudo-Riemannian metric on a manifold.
We analyze the Bianchi Identity as an instance of a basic fact of com-binatorial groupoid theory, related to the Homotopy Addition Lemma. Here it becomes formulated in terms of 2-forms with values in the gauge group bundle of a groupoid, and leads in particular to the (Chern-Weil) construction of characteristic classes. The method is that of synthetic… (More)
Preface It is a striking fact that differential calculus exists not only in analysis (based on the real numbers R), but also in algebraic geometry, where no limit processes are available. In algebraic geometry, one rather uses the idea of nilpotent elements in the " affine line " R; they act as infinitesimals. (Recall that an element x in a ring R is called… (More)
For any locally cartesian closed category , we prove that a local fibered right adjoint between slices of is given by a polynomial. The slices in question are taken in a well known fibered sense.
Preface It is a striking fact that differential calculus exists not only in analysis (based on the real numbers R and limits therein), but also in algebraic geometry, where no limit processes are available. In algebraic geometry, one rather uses the idea of nilpotent elements in the " affine line " R; they act as infinitesimals. (Recall that an element x in… (More)
The theory of combinatorial differential forms is usually presented in simplicial terms. We present here a cubical version; it depends on the possibility of forming affine combinations of mutual neighbour points in a manifold, in the context of synthetic differential geometry.
We take some first steps in providing a synthetic theory of distributions. In particular, we are interested in the use of distribution theory as foundation, not just as tool, in the study of the wave equation.
We prove that certain categories arising from atoms in a Grothendieck topos are themselves Grothendieck toposes. We also investigate enrichments of these categories over the base topos; there are in fact often two distinct enrichments.