Mathematical Conversations

Today the prevailing method in mathematics is 'analytic', in the sense that all mathematical objects are broken down into very small bits. For instance, a space or manifold is regarded as merely a set of points with structure. By contrast, in a 'synthetic' approach, we study objects via basic axiomatic properties, such as in Euclid's geometry. I will describe a synthetic approach to topological and smooth objects, due to Lawvere, and sketch how its 'stacky generalization', due to Schreiber, encodes notions like flat connections on principal bundles.