On first glance, homotopy theory and category theory are not directly connected. One deals with abstract "objects" and "maps between them," while the other deals with classifying spaces up to certain deformations. (In fact, some students do not believe that adjoint functors were first invented by a homotopy theorist!) The link between them first appears historical: algebraic topologists discovered category theory while working with homology theories, and thus it makes sense that the two fields would be closely related. However, the link goes far deeper, and it is this link that we will explore in this talk. Analogously to how algebra and geometry are linked by the fact that polynomials describe geometric objects (and geometric phenomena can therefore explain certain algebraic phenomena), categories can be used to describe topological objects. The natural "equality" relationship between categories, that of equivalence, is closely related to homotopy equivalences between spaces. In this talk we will describe this connection explicitly, linking the topological world and the categorical world through the use of objects called "simplicial sets." Time permitting, we will discuss Quillen's Theorem A, which explicitly describes an important connection between categories and homotopy.

Zoom Link: https://cornell.zoom.us/j/91057396562?pwd=aTRPMVUyVVZyL3pxNXJYVVpsYTBBQT09

Meeting ID: 910 5739 6562
Passcode: 0m654o