These are notes for the 2021-2022 algebraic topology reading group on \(\infty\)-category theory. These notes provide a brief overview of important material in chapter one of our reference text, Land’s Introduction to Infinity-Categories. Section 3 includes some further remarks. The notation and presentation is (mostly) consistent with Land’s. This is not a comprehensive discussion of these topics, and so I also recommend Emily Riehl’s stellar set of notes A Leisurely Introduction to Simplicial Sets, and the material found in Kerodon. I can further recommend Friedman’s survey on simplicial sets [arXiv:0809.4221] and Groth’s survey on \(\infty\)-categories [arXiv:1007.2925]. Chapter 6 (6.1.2 in particular) of Lurie’s Higher Topos Theory, 2012 reprint, contains more information on classifying spaces.

On 29 November these notes were updated to include more about \(\infty\)-categories themselves.

On 6 December these notes were updated again to include more about homotopy coherence. I take the approach of cosimplicial sets without necessarily saying as much. New things in the last section are described at a partially heuristic level, so a useful resource for further reading is Riehl’s set of notes on cosimplicial objects, Cosimplicial Sets, an Overture.

Notes: Some Vocabulary and Motivation for \(\infty\)-Category Theory