Higher Categories and Homotopical algebra

News

Completed Chatper 11 of the script. The final digression on the proof of Theorem A will appear soon, but digression 2 (on Smith's theorem) will not see the light of day anymore...

Zoom

I will try to publish the meeting-ID and password at noon each Tuesday and Thursday, at the website linked under "Location" above. The stream will be up by then, and the lectures should start around 12:15pm.
In order to access the website, you will need its credentials. So please write me an eMail if you wish to participate.

Exams

The first batch of exams will take palce in the week of August 3rd to August 7th. IF you want to be examined just send me an eMail with a proposed date. All exams will be oral and via Zoom.
The second batch of exams will take place in early October, preferrably the week Oct 12th - 16th. For the stragglers from last semester's lecture course, the second batch of exams will take place July 20th to July 24th, with the same procedure. Those of you who want/have to take advantage of the free Corona attempt, should contact me sometime in early September for an exam date in late September, and will first have to clear this with the BaMa-office.

Overview

In the lecture we will go through more of the basics of higher category theory. We will assume the contents of the course I taught last semester (its webpage is here), though anyone with a firm grasp of the very basics of quasi-category theory (coherent nerves, inner/left/right fibrations, Joyal's lifting theorem) should be able to follow.
The focus this term will be on techniques for constructing functors between quasi-categories, in particular, to and from the category of anima. Roughly, the idea is to cover:

Localisations

Straightening and Unstraightening

Limits, colimits and Kan extensions

Adjoint functors

Time permitting we will also talk about the basics of some of

symmetric monoidal quasi-categories and higher operads

stable quasi-categories and, in particular, spectra

complete Segal spaces

but this may well turn out to be a fantasy (if there is enough interest there may be a third course covering the remainder, and then moving on to algebraic K-theory).

Notes

The course will roughly follow the notes of Markus Land below, though we will probably deviate further from them than we did last semester. See here for my own notes.

Recommended Literature

Cisinski: Higher categories and homotopical algebra (available here)

Haugseng: Introduction to ∞-categories (lecture notes, available here)

Joyal: Notes on Quasicategories (available here)

Joyal: The theory of quasi-categories and its applications (available here)

Land: Introduction to infinity-categories (lecture notes, available here)

Lurie: Higher Topos theory (available here)

Lurie: Kerodon (Wiki, available here)

Riehl: Categorical Homotopy Theory (available here)

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