Daniel Kan's influence at MIT persists through something called the Kan seminar, a graduate reading course in algebraic topology. Over the course of a semester, each student is asked to give a few one-hour lectures summarizing classic papers in the field and to engage with each other paper by writing a reading response. The lectures are preceded by a practice talk of unbounded length that is conducted in private, i.e., in the absence of the lead instructor, before the reading responses are due. This format aims to teach students how to read papers quickly and at various levels of depth, as well as to work on presentation skills. At the semester's conclusion, Kan traditionally hosted a party that took advantage of Boston's high concentration of mathematicians, giving his students an opportunity to meet senior people in the field.

The Kan Extension Seminar, piloted in early 2014, was conceived as an online (“extension”) Kan seminar for peridoctoral category theorists. A dozen category theorists plus one facilitator met biweekly for videochat presentations on classic papers in the field. After each seminar meeting, the presenter wrote a blog post summary that was published on the n-Category Café. Some reflections on the first iteration of the Kan Extension Seminar can be found in the December 2014 issue of the Notices of the AMS.

**SEMINAR**** STRUCTURE**

For the second Kan Extension Seminar, I am delighted to announce that I'll be joined by two co-facilitators:

- Alexander Campbell is an Australian category theorist who recently completed his PhD under the supervision of Ross Street at the Centre of Australian Category Theory at Macquarie University. In the first year of his PhD studies he participated in the first Kan Extension Seminar. His research interests lie in higher and enriched category theory, with an eye towards their applications in homotopy theory.
- Brendan Fong is a postdoctoral scholar at the University of Pennsylvania, working with Rob Ghrist on applications of topology and category theory. The central theme of his research is reasoning about composite systems -- whether they comprise electrical components, chemical reactions, dynamical systems, programs, data, and so on -- with an emphasis on the tools of category theory. Earlier this year he completed a DPhil in Computer Science at the University of Oxford, under the guidance of Bob Coecke and John Baez (UC Riverside).

Please feel free to contact any of the organizers with any questions regarding the course.

**READING**** LIST**

- M. Hyland and J. Power, The category theoretic understanding of Universal Algebra: Lawvere Theories and Monads, Electronic Notes in Theoretical Computer Science Volume 172, 1 April 2007, Pages 437-458
- J. Beck, “Distributive laws,”
*Seminar on triples and categorical homology theory*, ETH 1966/67, edited by B. Eckmann, LNM 80, Springer 1969, pages 119–140. - P. Freyd, Algebra valued functors in general and tensor products in particular, Colloquium Mathematicum, XIV, Warsaw, 1966.
- G.M. Kelly, On the operads of J.P. May, Reprints in Theory and Applications of Categories, No. 13, 2005, pp. 1–13 (original 1972).
- G.M. Kelly, Structures defined by finite limits in the enriched context, I, Cahiers de topologie et géométrie différentielle catégoriques 23(1), 1982, 3-42.
- G.M. Kelly, “On Clubs and Data-type Constructors,” Applications of Categories in Computer Science: Proceedings of the London Mathematical Society Symposium, Durham 1991, 163-190.
- S. Lack and J. Rosicky, Notions of Lawvere Theory, Applied Categorical Structures, February 2011, Volume 19, Issue 1, pp 363–391
- C. Berger, P-A. Mellies, and M. Weber, Monads with arities and their associated theories, Volume 216, Issues 8–9, August 2012, Pages 2029-2048.

The seminar will meet nine times, the first week for introductions with each of the remaining eight devoted to one of the papers in the order listed above. We will meet according to the following schedule (altered midway through to minimize the pain caused by daylight savings time switches):

- Monday 9pm UTC: January 16, January 30, February 13, February 27, March 13, March 27, April 28, May 8
- Monday 1pm UTC: April 10

As a prerequisite, participants should feel comfortable with the material found in *Categories for the Working Mathematician* or its equivalent and demonstrate enthusiasm for engaging with more sophisticated categorical topics. Anyone is welcome to apply, but some preference will be given to current graduate students.

To apply, please send a single PDF file to “alexanderpcampbell at gmail dot com” or “fo at seas dot upenn dot edu” containing the following information:

- Your contact information and educational history.
- The name of a reference as well as his or her contact information.
- A brief paragraph explaining your interest in this course.
- A paragraph or two describing one of your favorite topics in category theory.
- A ranked list of the three papers (selected from the list above) that you would most like to present together with an explanation of your preferences.

Application deadline: **November 30th, 2016.**

** PARTICIPANTS**

We are delighted to announce the following participants in the Kan extension seminar. You'll be hearing from them shortly on the n-Category Café.

- Evangelia Aleiferi, Halifax, Canada — The Category Theoretic Understanding of Universal Algebra
- Pierre Cagne, Paris, France — On Clubs and Data-Type Constructors
- Simon Cho, Philadelphia, PA, USA — On the Operads of J.P. May
- Daniel Cicala, Riverside, CA, USA — A Discussion on Notions of Lawvere Theories
- David Myers, Palm Beach, FL, USA — Enrichment and its Limits & Gluing Together Finite Shapes with Kelly
- Maru Sarazola, Ithaca, NY, USA — Algebra Valued Functors in General and Tensor Products in Particular
- José Siqueira, Brasilia, Brazil — Unboxing Algebraic Theories of Generalised Arities
- Liang Ze Wong, Seattle, WA, USA — Distributive Laws

The eight participants of the Kan Extension Seminar gave a series of short expository talks at the 2017 International Category Theory Conference on Monday, July 17 from 15:30-17:25.

- Evangelia Aleiferi —
*Lawvere theories vs monads*- Lawvere theories were introduced in Lawvere's doctoral thesis, in order to formulate universal algebra from a categorical perspective. A different approach to the understanding of universal algebra was given a few years later, using the theory monads. We will talk about the connection between these two and why monads seemed to be favored over the years.

- Liang Ze Wong —
*Distributive laws, strings attached*- Distributive laws between monads S and T are equivalent to lifts of T to the category of S-algebras. I will illustrate this result using string diagrams.

- Maru Sarazola —
*Detecting algebraic categories with an internal Hom*- Linton defines an autonomous category as one possessing a forgetful functor U: C → Set and an internal hom, Hom: C
^{op}× C → C, such that U ο Hom(X,Y) ~ C(X,Y). In this talk we will define algebraic theories from an equational point of view, and after introducing some basic concepts, show a criterion for determining whether an algebraic category (i.e. the category of models for some algebraic theory) is autonomous.

- Linton defines an autonomous category as one possessing a forgetful functor U: C → Set and an internal hom, Hom: C
- Simon Cho —
*On (On the operads of J.P. May) of G.M. Kelly*- This talk is an exposition on operads, constructed as monoids for a particular monoidal structure on the functor category [P, V], where P is the permutation category and V is a cosmos. That is, the cosmos structure of V endows [P, V] with a closed monoidal structure by Day convolution; and we use this to further define the substitution product ◦, which is a left closed, non-symmetric monoidal structure ◦ on [P, V]. Operads are defined to be monoids for the substitution product. Every operad yields a monad, and we can use this connection to define operad algebras.

- David Jaz Myers —
*Weighted (Co)Limits*- Weighted (co)limits are the right notion of (co)limit in the enriched setting where points no longer reign supreme. This is because they allow for cones of a more general shape. In this brief talk, I will define weighted (co)limits and give a few examples.

- Pierre Cagne —
*When computational monads go clubbing*- As already spotted by Kelly himself in "On clubs and data-type constructors", his abstract notion of club encompasses most of monads encountered in computer science. After a gentle introduction to computational monads, we will recall the definition of a club and see how those monads can be understood from that point of view.

- Daniel Cicala —
*Generalizing Lawvere theories*- In this talk, we outline the paper Notions of Lawvere Theories by Lack and Rosicky. Their central conceit is to generalize Lawvere's functorial sematics along three trajectories: replacing the base category Set, working within an enriched setting, and including a class of limits other than finite products. We discuss what assumptions are made to ensure that, within the new settings, algebraic functors have left adjoints, models are reflective, and theories correspond to monads.

- José Siqueira —
*Dense generators of algebraic categories*- Dense generators of categories were introduced by J. R. Isbell in 1960, under the name of left adequate subcategories. Objects in a category with a dense generator are canonically given as a colimit of ones in the subcategory, a main example being the skeleton א of the category of finite sets within the category of sets. This suggests that the notion is a good replacement for the traditional concept of arity for algebraic operations, an idea that was further explored by C. Berger, P. Melliès and M. Weber. We will introduce the notion of monad with arity and associated (generalised Lawvere) theory and discuss how their categories of algebras inherit dense generators, with examples.

My contact infomation can be found on my personal website.