Selected Topics in Topology (V5D2)

Homotopy Coherent Algebraic Structures

In algebraic topology we are often not really interested whether two maps between a fixed pair of objects are equal, but only whether they are homotopic. However, it was an important observation (and one of the founding moments of modern homotopy theory) that it is many cases not enough for such a homotopy to simply exist, but that one should consider it as an object in itself. This naturally leads one to consider homotopies between homotopies and so on, leading to an infinite hierarchy of ‘compatibility data.’

In this lecture course we want to give an introduction to this family of ideas by studying one classical example of this philosopy in action, namely homotopy theoretic versions of abelian groups. Following the above pattern, it turns out that topological abelian groups are too rigid for many purposes, while the naive notion of abelian groups up to homotopy does not carry enough structure to support an interesting theory. Instead, we will consider homotopy coherent abelian groups in the form of so-called very special Γ-spaces, and in particular see as the main result of the first part of this course how they account for all cohomology theories concentrated in non-negative degrees. Along the way we will learn about several interesting concepts from homotopy theory, and in particular homotopy colimits and homotopy limits.

In the second part of the course we will consider some related questions, the exact choice of topics depending on the interests of the participants. Some suggestions:

• Segal spaces as a model of ‘categories’ with a composition that is associative and unital only up to coherent homotopies
• equivariant versions of homotopy coherent abelian groups
• other models of homotopy coherent abelian groups, in particular E_∞-algebras and models based on the notion of ultra-commutativity
• homotopy coherent versions of commutative rings

Prerequisites

The contents of last semester's course Algebraic Topology I and the topology lectures before it, in particular: simplicial sets, fibrations and cofibrations, loop spaces, representability of singular cohomology. In addition some familiarity with categories will be assumed (in particular limits, colimits, and adjoint functors).

Literature

You can find lecture notes for this course on eCampus. They also contain additional references to the original research articles as well as other sources.