# teaching higher algebra

Has anyone ever (successfully or unsuccessfully) taught a course in higher algebra (in the $\infty$-categorical sense)? I'm asking out of curiosity (and also hoping for more resources).

The kind of course I had in mind would be a class for graduate students, with background in standard topology and algebra, with goal to develop homological algebra from the "higher" point of view (but I'd like to know of any attempt at teaching anything related, for example a "derived commutative algebra" or "derived affine schemes" course).

One could start with explaining $\infty$-categories in an axiomatic way (ie one just assumes things like limits, adjunctions etc. exist and behave the way you think they do; intuition for example coming from categories enriched in Top). Then one moves on to stable categories (and hence triangulated categories) and to the examples arising in nature. Finally, functors between these categories include the theory of derived functors and one goes through many examples there too.

One could argue this is the next logical step of a progression. Older books in homological algebra refused to use spectral sequences. Then Weibel's highly praised book does the opposite and introduces them early on, but relegates derived categories to a final chapter. Then Gelfand-Manin take it one step further and start with derived categories. They discuss dg-algebras and model categories at the very end and stop short of discussing non-abelian derived functors. Lurie's higher algebra is the next step but it's also quite big and not meant to be used for lectures (I would argue that the best place for a quick introduction to derived stuff is Lurie's thesis or Toen's notes in the simplicial operads thing).

• In 2013 we had a seminar (under the supervision of Gabriele Vezzosi) on derived algebraic geometry, for which notes were produced, see dma.unifi.it/~vezzosi/seminar. Some parts of it might be relevant, although it wasn't strictly speaking a course being conducted by someone. The same year there was a winter school on derived algebraic geometry, www2.math.ethz.ch/mathphysics/news/DAG_School. – pbelmans Dec 10 '15 at 12:21

## 2 Answers

Some years ago I taught some homotopy coherence theory mentioning quasi-categories, etc in a Masters course on cohomology in Ottawa, and then used the material from that in several graduate level mini-courses at conferences. The course used crossed modules etc. and was not as detailed on Higher Algebra as perhaps you are meaning. I wrote up the notes and then extended them adding in material for the mini-courses as I went and filling in background. Anyone is welcome to have a look and to use the notes . The notes have got very long in their current form but I have another copy available on my n-lab page at: http://ncatlab.org/timporter/files/menagerie11.pdf

Other forms of the notes are available although none fits exactly the original questioner's requirements. They have been used on several occasions with moderate success. Have a look at the other notes that are linked to from my n-Lab page: http://ncatlab.org/timporter/show/HomePage as they may have some material that is usable.

That deals with available material. Prior to that(<2006) we taught numerous masters level courses at Bangor (when there was still a Maths Dept there) which incorporated material on crossed modules, crossed complexes, their infinity groupoid interpretations, etc. and their use in non-Abelian cohomology and homological algebra. The links with various variants of infinity category theory were explored in those courses.

• Only a very small part of it was used in the Ottawa course, of course. – Tim Porter Dec 11 '15 at 6:26
• Like a lot of people, I find writing notes on a subject helps me get to grips with the details. If then they are useful to others that is great. Another point is that fairly detailed notes can be `mined' for explanatory parts of articles which one is writing. A quick copy and paste gives you a starting draft which you can then augment to fill in details or extension that will be needed for the new results later on. Again I find this helps with improving my knowledge of an area and sometimes gives, without much effort, some new insights. – Tim Porter Aug 19 '16 at 8:03

Re "Has anyone ever [...] taught a course in higher algebra (in the ∞-categorical sense)?":

the following very new "resource" may be seen to spiritually qualify for this thread:

François Métayer: Homotopy theory of strict ω-categories and its connections with homology of monoids. Three-lecture course at the conference 'Categories in Homotopy Theory and Rewriting September 25 - 29, 2017' CIRM Marseille FRANCE

Needless to say, only spiritually (and arguably, not even that) since the central concept of the monograph 'Higher Algebra' are $(\infty,1)$-categories, which are what the OP is asking about, and which are quite different from the 'strict $\omega$-categories' in the sense of [R. Street: The algebra of oriented simplexes. J. Pure Appl. Algebr. 49 (1987) 283-335], which are what the CIRM course is about.

Abstract for the course on the CIRM website.

Abstract : In the first part, we describe the canonical model structure on the category of strict ω-categories and how it transfers to related subcategories. We then characterize the cofibrant objects as ω-categories freely generated by polygraphs and introduce the key notion of polygraphic resolution. Finally, by considering a monoid as a particular ω-category, this polygraphic point of view will lead us to an alternative definition of monoid homology, which happens to coincide with the usual one.

MSC Codes : 18D05 - Double categories, 22-categories, bicategories, hypercategories 18G10 - Resolutions; derived functors 18G50 - Nonabelian homological algebra 18G55 - Homotopical algebra

References mentioned at the beginning of the course.

In Lecture 1, the following are mentioned by the lecturer as references for the course (maybe giving them here helps readers to decide whether to watch):

• D. G. Quillen: Homotopical Algebra (Lecture Notes in Mathematics)

• M. Hovey: Model categories Mathematical Surveys and Monographs 63, AMS, Providence, RI, 1999 (x + 209 pages).

• T. Beke: Sheafifiable homotopy model categories Math. Proc. Camb. Phil. Soc. vol.129 (2000), no.3, pp.447-475

• T. Beke: Sheafifiable homotopy model categories, Part II Journal of Pure and Applied Algebra vol.164 (2001), no.3, pp.307-324

Remarks.

• It seems preferable to use the course via the CIRM website instead of a large video hosting site, since CIRM site has additional structure, e.g.(so far only for Lecture 1) clickable buttons named by keywords, which take one to relevant points in the lectures. These are, in chronological order

Lecture 1

lifting properties : at 05:27

model structures : at 17:40

Smith's theorem : at 23:42

$\omega$-categories : at 32:24

weak equivalences : at 55:59

cylinder category : at 1:21:07

Lecture 2,3: sadly, the clickable buttons on the CIRM version cease after Lecture 1

• The course isn't primarily addressed at students, rather is a "mini-course" (this is the description on the conference's page integrated into a conference. It is therefore only an approximation to a graduate course. In particular, there aren't any exercises or tutorials.

Also, the above short course does not have the "goal to develop homological algebra from the "higher" point of view" (OP, 1st par.), yet it perhaps meets the rather permissive

"I'd like to know of any attempt at teaching anything related, "

requirement in the OP.