The Theory of Topos-Theoretic ‘Bridges’—A Conceptual Introduction
Mathematics is divided into several distinct areas: geometry, number theory, algebra, analysis, mathematical logic, etc. Each of these areas has evolved throughout the years by developing its own ideas and techniques, and by now has reached a remarkable degree of specialization. Now, even more than in the past, we feel the need for unifying theories that could intra-disciplinarily connect different areas of mathematics with their different sets of concepts, objects, and methods, in new and powerful ways, hence providing effective tools for solving long-standing problems. It has happened several times that solutions to profound problems in one field have first, or only, been obtained by using methods from other fields, and this indicates that Mathematics should be seen as a coherent whole rather than a collection of separate fields. Think for example of analytic geometry, which allows the study of geometrical shapes using algebraic manipulation, or the Grothendieckian notion of spectra, which allows the study of discrete objects using a geometric continuous intuition.
The importance of ‘bridges’ between different areas lies in the fact that they make it possible to transfer knowledge and methods between the areas, so that problems formulated in the language of one field can be tackled (and possibly solved) using techniques from a different field, and results in one area can be appropriately transferred to results in another.
