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Seminar: Higher Category Theory
- Schedule: Thursdays 14-16
- Place: Rheinsprung 21, Mathematisches Institut Basel, Seminarraum
- First seminar: Thursday, 26. February 2009
- For: Mathematicians, physicists, computer scientists, logicians, biologists, neuroscientists, philosophers, etc.
- Prerequisites: (Ordinary) category theory, but not necessarily
- Credit points: None (things could possibly be arranged to get some if you need them)
- Language: Depends on the speaker (German/English)
- Blog: Can be found here
Talks:
- February 26, 2009: Giordano Favi - Organisational stuff and informal introduction
- March 5, 2009: No seminar (Fasnacht)
- March 12, 2009: Giordano Favi - Problems concerning categorification and a theory of n-categories
- March 19, 2009: Kay Werndli - Simplicial Sets, the nerve functor and Ross Street's approach to ω-categories (abstract)
- March 26, 2009: Roland Lötscher - Fibered categories and a first look at 2-functors (abstract)
- April 2, 2009: Roland Lötscher - Fibered categories and a first look at 2-functors (part 2)
- April 9, 2009: Ostern
- April 16, 2009: Giordano Favi - Aspects of Monads (abstract)
- April 23, 2009: Giordano Favi - Aspects of Monads (part 2)
- April 30, 2009: Kay Werndli - Penon's definition of an ω-category (cancelled)
- May 7, 2009: Kay Werndli - Penon's Higher Categories (abstract)
- May 14, 2009: No talk
- May 21, 2009: Auffahrt
- May 28, 2009: No talk
Web"master": For informations, remarks or to send potential content of this site (e.g. an abstract/handout/script to your talk or some other papers/links to n-categorical topics), please send an eMail to kay.werndli-which sign was that again?-stud.unibas.ch
Description:
In 1942 Samuel Eilenberg and Saunders Mac Lane introduced the now
common structure (or rather language) of a category, which has objects (0-arrows)
and morphisms (1-arrows) between such. The categorical approach
turned out to be very successful in the subsequent years and decades in
very different branches of mathematics. However, maybe we're just seeing
the tip of the iceberg. Some natural and easy considerations of topology
(i.e. homotopy theory) or category theory itself lead to the notions of
weak and strict 2-categories respectively, having objects (0-arrows),
morphisms between objects (1-arrows) and 2-arrows between 1-arrows.
Many people now tried to generalise this idea, leading to very different
notions of a n-category - having 0-,1-,...,n-arrows - or even
ω-categories (i.e. ∞-categories), having n-arrows for all
n in N.
Although the many different (technical) definitions of a n-category are very
different from each other, there is still a very intuitive idea behind the
technicalities on which most people agree in some form. Still, making this idea
precise and give an explicit definition of a n-category in terms of sets of
arrows, compositions of such and coherence axioms is a rather difficult task,
even for rather small n. To illustrate this a little better, consider the following
quote made by Todd Trimble as a comment to his explicit (he calls it "primitive")
definition of a "tetracategory" (i.e. a weak 4-category) which is 51 pages long
by the way:
To read the diagrams for each coherence condition, it is recommended that the reader
print them out (large) and spread them out on a wide surface, and arrange them in a circular
pattern.
The aim of this seminar is to explain and analize different definitions of
n- or ω-categories or presenting potential (or existing) applications of these.
The approach to higher categories and the general topic of the talk is very open
and can reach from combinatorics and group theory, to topology, including homological/homotopical
algebra, to even logics.
"This must seem very boring to the people who understand it and very mystifying to those who don't."
Quoted from John Baez's "Tale of n-Categories"
(Well, then let us be grateful that we might just understand it a little bit.)
Useful Links & Literature:
- First a must (re)read: It's Baez-Dolan paper from finite sets to Feynman diagrams (30 pages, 2000):
Everybody should read at least the introduction to get the idea of Higher Dimensional Algebra (=the study of n-categories). It's pretty informal but the ideas can be easily understood. If you already read it, it's worth a rereading.
- Again Baez-Dolan, on Categorification this time (51 pages, 1998):
The paper is a nice read with many informal ideas, it gives many guiding ideas for the future of higher dimensional categories. I think that if we really understand Categorification we make a huge progress! You can go through this paper without reading all the details, it's fairly easy to read (imo)
- Baez on Quantum Quandaries (21 pages, 2004):
This is a very nice paper (for physicists in particular), it tries to explain a link between General Relativity and Quantum Theory (using higher Category Theory)!
- Baez's introduction to n-Categories (34 pages, 1997)
- Baez-Lauda on the prehistory of n-categorical physics (58 pages)
This is a draft paper (contains a lot of mistakes at the end, unfinished sentences) but it explains some nice ideas and makes a nice historical account of the main ideas in physics and their relation to n-category theory.
- Leinster on topology and higher cats (15 pages):
It's a section from his rather large book which can also be found online. It's easily read and informal but gives many important ideas in my opinion.
- Leinster on bicategories (10 pages):
Formal, this basic definitions and a proof of the coherence theorem for bicategories. In order to understand the subject we'll have to go through that somehow.
- Baez's cohomology lectures (66 pages):
Very informal but full of crucial ideas for higher category theory. It's not so easy to read and requires some background knowledge. Another must read at some point (not for everyone at the beginning though).
- Simpson on limits in n-categories (90 pages):
Here is discussed the notion of limit in a n-category. He uses a def of n-cat given by Tamsamani, so this requires a lot of luggage. I wouldn't recommend this as a first reading, but this the only reference for higher limits and it also gives a lot of potential applications of this theory.
- Niles Johnson homepage:
This guy seems to do Brauer groups in 2-categories and triangulated bicategories (a notion which I might find very useful for my own research). His papers look nice, but require some prior knowledge of Algebraic Topology and (non-)commutative Algebra (especially Brauer groups).
- A paper of Kharlamov/Turaev on 2-knots: V. Kharlamov and V. Turaev. On the definition of the 2-category of 2-knots. Amer. Math. Soc. Transl, 174:205-221, 1996. (I haven't read it...)
- Li on combinatorics (26 pages):
These are slides of a talk (so it's informal) but it gives some references (to Joyal especially) on categorical combinatorics. This is the only online link I found to combinatorics and category theory
Some other links:
The wikipedia entry page on higher category theory
John Baez's n-Category Café
The nLab (the so called 'Backroom of the n-Category Café')
The Website of last year's lecture/seminar on (ordniary) category theory
"In mathematics, the n-category number of a mathematician is a humorous construct invented by Dan Freed, intended to measure the capacity of that mathematician to stomach the use of higher categories. It is defined as the largest number n such that he or she can think about n-categories for a half hour without getting a splitting headache."
Quoted from Wikipedia
Back to Giordano's homepage
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