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MA938 - Topics in Algebraic Topology

Lecturer: Emanuele Dotto

Term(s): Term 2

Commitment: 30 lectures

Assessment: Oral exam

Prerequisites: Basic knowledge of homology and homotopy groups, some familiarity with the language of category theory.

Content :

The aim of this module is to illustrate some aspects of the interaction between category theory and topology, typically through simplicial techniques.

Many invariants in topology are naturally constructed using simplicial sets, for example singular homology, group homology, classifying spaces of groups and families of groups, Eilenberg-MacLane spaces, algebraic K-theory, higher fundamental groupoids.

Simplicial sets also provide a convenient combinatorial model for topological spaces (up to weak homotopy equivalences), and for higher category theory.

The course we will start with a basic treatment of category theory and simplicial sets. We will then use simplicial tools to construct some of the invariants listed above. We will explain how to construct homotopy limits and homotopy colimits, and eventually introduce some of the basic notions of the theory of quasi-categories.

References:

▪ Goerss, Jardine, Simplicial homotopy theory

▪ Joyal, Terney, Notes on simplicial homotopy theory

▪ Lurie, Higher topos theory

▪ Haugseng, introduction to infinity-categories, lecture note