Code: UP2122A

Project Title: Categories and Classifying Spaces

Description:
A category consists of a class of objects together with a set of maps (called morphisms or arrows) between any pair of objects and an operation of composition of arrows. The composition has to be associative, and each object has to have an identity arrow with domain and codomain equal to the object itself that behaves like an identity element for the composition. You have already encountered several categories in your mathematical life without even knowing it: The category of vector spaces and linear maps is the central object of study in Linear Algebra, the category of topological spaces and continuous maps is another one. 

Algebraic topology provides a construction that associates to any (small) category a topological space, called the classifying space of the category. These spaces often have very interesting properties and reflect certain topological features of your starting category. As an example: Any discrete group G gives rise to a category that has a classifying space denoted by BG. It turns out that the fundamental group of BG is isomorphic to G and topological invariants of BG often measure interesting features of the group G.

We will start the project by exploring a bit of category theory first. After that we will look into simplicial sets, simplicial spaces and nerves of categories. These constructions form the intermediate steps between categories and classifying spaces. Depending on how ambitious the candidate for this project feels, we may then have a look at a central theorem in K-theory called Bott periodicity, which has a proof in terms of classifying spaces given by Harris. This part would involve reading and understanding a research paper.

Type: Year 4 MMath Project

Supervisor: Ulrich Pennig

Prerequisite modules:
– Algebraic Topology (MA3008)
– Groups (MA0213) or Algebra II: Rings (MA3014)

Max. number of students: 1