Prof. Dr. Moritz Weber

Daniel Gromada

Alexander Mang

Block seminar on category theory

(Summer term 2020      ---      7-11 September 2020)

News

The deadline for the written assignments (Seminarausarbeitung) is 15 October.

In the assignment you should work out your talk on 5-10 pages, possibly involving some proofs or remarks that had to
be left out in your talk. You may send a preliminary version of your assignment to Moritz Weber in order to receive
some feedback before submitting the final version.

Contents

In this seminar we will cover basic notions and more advanced topics in category theory. The main focus will be on applications
of category theory and concrete examples. See below for the concrete topics.

The seminar will be in English unless all participants speak German. If you feel uncomfortable giving your talk in English,
please indicate to us.

There are no prerequisites for the seminar. The talk may or may not include a "Ausarbeitung", depending on your conditions
of study and the amount of credit points you aim for.

Schedule

1 June - 31 August
Individual reading of the basic reading part; preparation of the talks

Monday, 31 August, 2:00 pm
Discussion on questions regarding the basic reading part (participation voluntary)

Monday, 7 September
9:00 - 10:00 Introduction and summary/discussion of the basic reading part Slides
10:15 - 11:15 Talk 1 (Leon) Slides
11:30 - 12:30 Talk 2 (Roberto) Slides

Tuesday, 8 September
9:00 - 10:00 Talk 3 (Steven) Definitions, Slides
10:15 - 11:15 Talk 4 (Lennard) Definitions, Slides, Notes
11:30 - 12:30 Talk 5 (Adrian) Definitions, Slides

Wednesday, 9 September
9:00 - 10:00 Talk 6 (Daniel) Definitions, Slides
10:00 - 11:00 Talk 7 (Alexander R.) Definitions, Slides

Thursday, 10 September
9:00 - 10:00 Talk 8 (Simon) Definitions, Slides
10:15 - 11:15 Talk 9 (Alexander W.) Definitions, Slides

Friday, 11 September Definitions (Talks 10-13)
9:00 - 10:00 Talk 10 (Eileen) Slides
10:10 - 11:10 Talk 11 (Friedrich) Slides
11:20 - 12:20 Talk 12 (Nick) Slides
12:30 - 13:30 Talk 13 (Friedrich and Nick) Slides (Part 1), Slides (Part 2)

Talks and readings

In this seminar we will cover basic notions and more advanced topics in category theory. The main focus will be on applications
of category theory and concrete examples.

Basic reading (for everyone): Talks (basic topics):
  1. (Leon; MW) Objects and morphisms [AHS, Chpt. II]
  2. (Roberto; DG) Universal property [Br, Chpt. 5]
  3. (Steven; AM)) Limits, colimits, sources, sinks, pullbacks, pushouts [AHS, Chpt. III; Ba16, Chpt. 2; Br, Chpt. 6]
  4. (Lennard; MW) From posets to categories - a concrete approach to category theory [A, Chpt. 1&2]
  5. (Adrian; MW) Adjoints [Br, Chpt. 7; AHS, Chpt. V]
  6. (Daniel; DG) Monoidal categories and the example of the Temperley-Lieb category [NT 2.1, EGNO Ch. 2, Ba16, Chpt. 8]
  7. (Alexander R.; AM) Enriched categories [R14, Chpt. 3]
  8. (Simon; AM) Abelian categories [ML, Chpt. VIII]
Talks (special topics):
  1. (Alexander W.; DG) (C*-)Tensor categories, reconstruction and Tannaka-Krein [EGNO, Chpt. 4 and 5][NT, Chpt. 3][D; DM; JS;]
  2. (Eileen; MW) About chains and snakes - chain complexes and homology [Bl]
  3. (Friedrich; MW) Projective and injective resolutions [Bl]
  4. (Nick; MW) Derived Functors [Bl]
  5. (Friedrich and Nick; MW) Examples are manifold - de Rham Cohomology, Ext and others [Bl]
See also some additional information on the topics of the talks.

Hints for the preparation of the talks

Each participant should deliver a talk of 50 minutes plus 10 minutes for discussions and questions. The talk should cover
some parts of the given references (not necessarily all of it).

Please try to design your talk as concrete as possible. For instance, you could first give a simplified version of your main
definition, then treat a number of examples, and finally give the general definition. Giving concrete examples and concrete
applications is much appreciated.
Here is a list of possible running examples for categories: sets; groups; vector spaces; topological spaces; posets;
modules; monoid as a category. While preparing your talk, please check whether you may illustrate your definition or
construction at some of these running examples (preferably finite sets and groups).

In your notations, please try to stay as close as possible to [AHS].

For the presentation of your talk, you may either produce slides, scan or take a picture of some handwritten notes, or use
some ``live'' devices such as writing on a tablet.
If you have technical problems, please let me know; we also have a document camera projecting live your handwriting
from a paper to zoom (has to be used in a seminar room at the university).

It would be good to prepare a short PDF (1-2 pages) of the main definitions and notions of your talk, which you can
share with all participants. This will help to avoid getting lost in technical details.

The seminar will be in English unless all participants speak German. If you feel uncomfortable giving your talk in English,
please indicate to us.

References

Link to the Semesterapparat of the librabry (including an e-book version of Brandenburg's book [Br])

General references

[AHS]Adámek, Herrlich, Strecker: Abstract and concrete categories. The joy of cats, 1990.
[Br]Brandenburg: Einführung in die Kategorientheorie, 2017. (German only.)
[Ba16]Baez: Category Theory, Lecture Notes , 2016.
[ML]Mac Lane: Category theory for the working mathematician, 1978.
[R14]Riehl: Categorial homotopy theory, 2014 (advanced).

Specific references

[A]Armstrong: Fancy Algebra, Lecture Notes, 2016. (On the foundations of noncommutative algebra.)
[Ba07]Baez: The Homotopy Hypothesis, 2007.
[Bl]Bland: Rings and their modules, De Gruyter Textbook, 2011. (SULB via VPN.)
[BC]Brannan, Collins: Dual bases in Temperley–Lieb algebras, quantum groups, and a question of Jones, Quantum topology 9 (2018), pp. 715-748.
[CO]Comes, Ostrik: On Deligne's category, Algebra & Number Theory 8 (2014), pp. 473-496.
[D]Day: Enriched Tannaka reconstruction, Journal of Pure and Applied Algebra 108 (1996), pp. 17-22.
[DM]Deligne, Milne: Tannakian Categories, 2018.
[E14a]Etingof: Representation theory in complex rank I, Transformation Groups 19 (2014), pp. 359–381.
[E14b]Etingof: Representation theory in complex rank II, Advances in Mathematics 300 (2016), pp. 473–504.
[EGNO]Etingof, Gelaki, Nikshych, Ostrik: Tensor categories, Mathematical Surveys and Monographs, Vol. 205, AMS, 2015.
[I]Isbell : Structure of categories, Bulletin of the American Mathematical Society 72 (1966), pp. 619-655.
[JS]Joyal, Street: An introduction to Tannaka duality and quantum groups in: Carboni, Pedicchio, Rosolini: Category Theory, Proceedings, Como 1990, Lecture Notes in Mathematics 1488 (1991).
[K]Knop: Tensor envelopes of regular categories, Advances in Mathematics 214 (2007), pp. 571-617.
[NT]Neshveyv, Tuset: Compact quantum groups and their representation categories, Cours Spécialisés (20) 2013.
[Q]Quillen: Homotopical Algebra, Lecture Notes in Mathematics 43 (1967).
[R16]Riehl: Category theory in context, 2016.
[Sh]Shulman: Exact completions and small sheaves, Theory and Applications of Categories 27 (2012), pp. 97-173.
[Sto]Stone: The representation of Boolean algebras, Transactions of the American Mathematical Society 40 (1936), pp. 37-111.
[Str]Street: The family approach to total cocompleteness and toposes, Transactions of the American Mathematical Society 284 (1984), pp. 355-369. (SULB via VPN.)
[Wei]Weibel: An introduction to homological algebra, Cambridge studies in advanced mathematics 38 (1994). (SULB via VPN.)
[Wen]Wengenroth: Derived functors in functional analysis, Springer Lecture Notes 1810 (2003). (SULB via VPN.)




Last update: 20 September 2020   Moritz Weber Impressum