MSGSS
Fibrations, or how to make a bundle out of anything
A “fibre bundle” in topology is a space E that resembles a product space of spaces B x F, but only locally on open neighborhoods of B. These come in various flavours, characterized by the type of fibre, and by what happens when you move from one local patch to another. In the context of topology, a “fibration” is an abstraction of this concept defined in terms of being able to lift certain classes of maps X to B to maps X to E. Using the more abstract notion, we can think about fibrations in non-geometric settings. Indeed, the famous “Grothendieck construction” is a way to construct a fibration p : E to B that is a functor between two categories. In this informal talk, I’ll give some examples of fibrations of various kinds, explain how Grothendieck’s construction works, and sketch what it could possibly mean to “make a bundle out of anything”. This is a topic I am still learning for myself, so I hope to remain accessible to non-experts, and am also looking forward to any comments from the knowledgeable.