Categorical algebra in analysis, geometry, and topology

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Much of mathematics is concerned with the study of space and quantity, and the relation of each to the other. The notion of space here comprises not only the familiar three-dimensional physical space that surrounds us but also the configurations of various shapes and figures therein, as well as higher-dimensional spaces, and various mathematical concepts of space that abstract from these beginnings. The notion of quantity here includes not only the familiar counting numbers, rational, and real numbers, but also numerical quantities that vary from point to point over a space, such as the temperature at the surface of the earth, and also quantities that are distributed through space, such as the quantity of certain gas distributed through the earth's atmosphere. The mathematical notions of function, measure, and distribution, as well as various other related notions, were devised to represent mathematically such variable and distributed quantities. There are various subtly different mathematical notions of space, such as the notions of topological space, manifold, and variety, each studied by its own branch of mathematics, such as topology, differential geometry, and algebraic geometry. Correspondingly, there are numerous subtly different notions of function and distribution associated with different notions of space, and these notions end up being applied in unexpected ways, e.g. to describe the distribution of likelihood across a space of possible outcomes of an experiment, as in the study of probability and statistics. The proposed research employs the methods of category theory, a general study of structure in mathematics, to identify and understand the common structural characteristics shared by all these subtly different notions of space and quantity, and to develop a system of category-theoretic frameworks, axiomatics, and results that foster a new and unified understanding of structures that generate and characterize variable and distributed quantities in general.  This will serve to make the notions and methods of each of various branches of mathematics more accessible to the practitioners of the others, and the insights gained by these structural studies enable new and effective methods in mathematics.  Further, these structural studies allow mathematical insights to be more readily transferred to application domains, e.g. in the design of probabilistic programming languages, and they support the effective variation and adaptation of these ideas to that end.