doceamus . . . let us teach The Seminar: An Experimental Online Graduate Reading Course

Emily Riehl

Daniel Kan (1927–2013) has left an indelible mark interested in are geographically on and category theory, with dispersed without critical mass in any one research highlights including his discovery of adjoint func- center. I therefore decided to move the seminar tors and his work developing simplicial homotopy online, and the “Kan Extension Seminar” was born.1 theory. A lesser-known aspect of his lasting in- In October 2013 I announced the seminar on The fluence, however, is pedagogical: in the 1960s n-Category Café, a blog devoted to higher category he inaugurated what is now known as the Kan theory and related topics of which I am a cohost. seminar, which continues at MIT and has expanded From a surplus of applications, I selected twelve elsewhere. Kan’s seminar is a graduate reading students (three based in the US, eight from Western course in algebraic topology designed to acquaint Europe, and one from Australia).2 Together we read students rapidly with the literature. The format one paper every two weeks from January to June is as follows: over the course of a semester, each 2014. The papers I selected range over fifty years: I student is asked to give a few one-hour lectures aimed to balance commonly acknowledged classics summarizing classic papers in the field and to in the field with more recent papers that exposed engage with the other assigned papers by writing students to a broad range of active research areas. reading responses. Each lecture is preceded by a Each student presented one paper via a private practice talk of unbounded length conducted in class video chat. Over the course of the following private (i.e., in the absence of the lead instructor) week, that student drafted a blog post for The and—importantly—before the reading responses n-Category Café. The other students wrote reading are due. This format aims to teach students how to responses in a shared LATEX file and were expected to read papers quickly and at various levels of depth comment on the blog. At the seminar’s conclusion, while giving them a chance to refine presentation eight of the students convened in person to give skills. The seminar has an additional social element: a series of expository talks as a satellite to the Kan traditionally hosted a Halloween party where 2014 International Conference in Category Theory his students could mingle with members of the in Cambridge, England. Boston algebraic topology community. A video chat is not the most obvious way to Regretting having never had the opportunity to discuss mathematical ideas. Our software was participate myself, I decided to adapt Kan’s model simple—each participant appeared in a little box for graduate students in category theory. As is the on the computer screen. More elaborate video- case for many specialized subdisciplines, students conference technology exists, but I did not want to assume all of my students had access to the Emily Riehl is a Benjamin Peirce Fellow in the Department necessary equipment. A few students prepared of Mathematics at Harvard University. Her email address is [email protected]. 1The title of one section in ’s Categories Members of the Editorial Board for Doceamus are: David for the Working Mathematician asserts that “All Concepts Bressoud, Roger Howe, Karen King, William McCallum, and Are Kan Extensions.” Mark Saul. 2The daylight savings time change in April significantly DOI: http://dx.doi.org/10.1090/noti1187 complicated the scheduling of class meetings.

December 2014 Notices of the AMS 1357 LATEX files that were distributed before their pre- Nonetheless, the most problematic aspects of sentations, but most often we simply talked to the course were the discussions on The n-Category one another. A side benefit of this format is that Café (I say this not to detract at all from the content students can practice mathematical conversation of those conversations). Most of the students were without the aid of a chalkboard. Think of this as a legitimately hesitant about posting comments technologically enabled complement to a skill I’ve early in their careers that will remain publicly romanticized after reading accounts of mathemat- viewable in perpetuity and leave a digital trail. Yet ical collaborators thinking together while taking the larger categorical community clamored for long walks outdoors. greater access to course content: over the semester, The Kan extension seminar functioned almost multiple colleagues asked to observe our private entirely online, yet despite any superficial similari- discussions, as well as have video chats and the ties, it decidedly was not akin to a MOOC (Massive satellite conference uploaded to the Internet. I Open Online Course). Namely, part of the value of resisted these overtures in order to shelter the the experience was its intimacy, with each student class from a surfeit of observers. Still, one could taking a turn to present a paper and the entire class imagine adopting a different strategy that would working in close collaboration. I see my students as benefit the wider mathematical community. colleagues with whom I look forward to interacting For the pilot version of the Kan Extension in the future. Seminar, I selected the best-prepared students As teaching tends to be, the experience was from the applicant pool, most of whom were both more rewarding and more consuming than I attached to a department active in category theory. had predicted. The rewards came from getting to The students’ advanced backgrounds prepared know the students, watching them excel in their them for the challenging reading assignments, and presentations, and seeing them improve in their when confusion struck, many also had local experts exposition over the course of the editing process. to consult. This choice did, however, privilege There was also an unexpected social pleasure: those at established centers of category theory I was able to meet all twelve in person over (principally abroad), leading to an unfortunate the six-month period. Our class discussions were paradox: an online seminar which could reach lively and provocative. I had chosen papers that students anywhere limited itself to participants personally interested me, several of which I had from specific locales. never read carefully. (I had been advised that the An online seminar has the potential to catalyze Kan seminar was traditionally hands-off, more peer specialized conversations and collaboration among teaching than instructor driven.) Yet as students geographically dispersed mathematicians. A more enthusiastically wrote lengthy, thoughtful reading inclusive approach than the one I took might responses, I felt impelled to give each response encourage participants whose research interests the consideration it deserved. diverge from the areas of expertise of their gradu- Based on an exit survey and informal conver- ate departments. This would provide a welcome sations with class participants, the seminar was antidote to the social forces that accrue benefits wildly successful. The reading responses sponta- to those on the “inside track.” neously transformed into an active dialogue as A few other axes for variation: a future seminar students responded to one another’s work. Two could focus on a wider or narrower range of papers, participants even commuted regularly to meet in read fewer papers (or include fewer students), person to discuss the papers. or could be targeted differently to a more or The greater category theory community also less advanced level. Some students were critical responded positively. One colleague reported that of the low-tech software (dreams of peripatetic the seminar “brought new life” to The n-Category mathematical conversations aside). As tablets and Café that hosted us. A large audience, including digital meeting tools become more ubiquitous, it senior colleagues, attended the satellite talks may be easier to host online “chalkboard” lectures. in June. Students and faculty have expressed Dan Kan once told me that if you shake some- enthusiasm for future iterations of the seminar; thing and it rattles, then there is something inside indeed, I’ve already been contacted by several worthy of further contemplation—a particularly prospective students. categorical viewpoint. I don’t submit that the first An online seminar introduces certain issues iteration of the Kan Extension Seminar proceeded beyond those of a typical graduate seminar. For in frictionless silence. Indeed, I do think, following example, I had to negotiate which aspects of Kan, that that fact suggests it is worthy of further the course to make public or keep private. The experimentation. students’ expository blog entries constitute a new resource that will benefit anyone who searches online for one of the published papers.

1358 Notices of the AMS Volume 61, Number 11