The Kan Extension Seminar
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.
This (northern hemisphere) spring, from early January to late June 2014, I will run an online (“extension”) Kan seminar in category theory with the aim of reading the twelve papers listed below. I am seeking between 6 and 12 participants who, in addition to engaging with all of the papers, will compose one or two blog posts for the n-Category Café over the course of the six months, which will be published every other week. The other participants will be expected to comment. On the week preceding each blog entry, the class will have a private discussion (likely via Google hangout) on the paper in question, tentatively to take place at 9pm GMT on alternate Mondays, with some time adjustment later in the term to account for daylight savings time. The course will conclude with a series of short public expository lectures given, by those able to attend, on June 29th in conjunction with the 2014 International Category Theory Conference at Cambridge, UK.
Please feel free to contact me with any questions regarding the course.
- F.W. Lawvere, An elementary theory of the category of sets, 1964, Repr. Theory Appl. Categ. 11 (2005) 1-35.
- R. Street, The formal theory of monads, J. Pure Appl. Algebra 2(2) (1972) 149-168.
- P.J. Freyd, G.M. Kelly, Categories of continuous functors, I, J. Pure Appl. Algebra 2(3) (1972) 169-191.
- F.W. Lawvere, Metric spaces, generalized logic and closed categories, 1973, Repr. Theory Appl. Categ. 1 (2002) 1-37.
- G.M. Kelly and R. Street, Review of the elements of 2-categories, Lecture Notes in Math. 420 (1974) 75-103.
- R. Street, R. Walters, Yoneda structures on 2-categories, J. Algebra 50(2) (1978) 350-379.
- P.T. Johnstone, On a topological topos, Proc. London Math. Soc. 3(38) (1979) 237–271.
- G.M. Kelly, Elementary observations on 2-categorical limits, Bull. Austral. Math. Soc. 39 (1989) 301-317.
- R. Blackwell, G.M. Kelly, A.J. Power, Two-dimensional monad theory, J. Pure Appl. Algebra 59 (1989) 1-41.
- J. Adámek, F. Borceux, S. Lack, J. Rosický, A classification of accessible categories, J. Pure Appl. Algebra 175 (2002) 7-30.
- S. Lack, Codescent objects and coherence, J. Pure Appl. Algebra 175 (2002) 223-241.
- M. Shulman, Enriched indexed categories, Theory Appl. Categ. 28(21) (2013) 616-695.
As a prerequisite, students should be comfortable with the material found in Mac Lane's Categories for the Working Mathematician or its equivalent. Anyone is welcome to apply but preference will be given to current graduate students (at either the masters or PhD level).
To apply, please email me a single PDF file containing the following:
You may append your CV if you wish. Finally, please indicate whether or not you expect to attend the 2014 International Category Theory conference in Cambridge (June 29-July 5th).
- 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 list of the four papers (selected from the list above) that you would most like to present together with an explanation of your preferences.
Application deadline: November 30th, 2013.
I am delighted to announce the following participants in the Kan extension seminar. You will be hearing from them shortly on the n-Category Café.
- Tom Avery, Edinburgh, Scotland — Metric Spaces, Generalized Logic, and Closed Categories
- Eduard Balzin, Nice, France — Formal Theory of Monads (Following Street)
- Alexander Campbell, Sydney, Australia — An Exegesis of Yoneda Structures
- Tim Campion, Westwood, MA, USA — Classifying by Generalizing: The Theory of Accessibility Relative to a Limit Doctrine
- Alexander Corner, Sheffield, England — Codescent Objects and Coherence
- Joe Hannon, Boston, MA, USA — Enriched Indexed Categories, Again
- Fosco Loregian, Padua, Italy — Categories of Continuous Functors
- Sean Moss, Cambridge, England — On a Topological Topos
- Clive Newstead, Pittsburg, PA, USA — An Elementary Theory of the Category of Sets
- Sam van Gool, Paris, France — On Two-Dimensional Monad Theory
- Christina Vasilakopoulou, Cambridge, England — Elementary Observations on 2-Categorical Limits
- Dimitri Zaganidis, Lausanne, Switzerland — Review of the Elements of 2-Categories
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.
TALKS AT CT2014
Please join us at the Winstanley Lecture Theatre in Trinity College from 2pm-6pm on Sunday 29 June for a series of short expository talks. At the conclusion, we will walk together to the welcome reception for the 2014 International Category Theory Conference.
2:00pm-2:30pm — tea & conversation
3:50pm-4:10pm — tea & conversation
- Fosco Loregian — For the sake of well-completeness
- A pair of mutually orthogonal classes of arrows in a category C often "extends" to a factorization system (i.e. every arrow in C can be factored according to these two classes). This is rather useful in many occasions.
A sufficient condition for this extension to be true is a rather peculiar completeness request on C: every class (even large ones) of subobjects of an object X must admit an intersection.
- Tom Avery — The Cauchy completion is the Cauchy completion
- The Cauchy completion of a category is the universal extension of that category in which all idempotents split. When we move from ordinary categories to enriched categories, it turns out that the appropriate notion of Cauchy completion is given by replacing "splittings of idempotents" with "absolute colimits". A student of category theory meeting these concepts for the first time might wonder why they are named after Cauchy, an early pioneer of analysis who worked long before category theory was developed. The reason is that when a metric space is regarded as an enriched category, the Cauchy completion is the metric space obtained by adjoining limits of Cauchy sequences. I will prove this result and along the way talk about bimodules and absolute colimits. The talk is based on Lawvere's "Metric spaces, generalized logic, and closed categories", in which these ideas were first introduced.
- Alexander Campbell — An Exegesis of Yoneda Structures
- We motivate the notion of Yoneda structure by expressing the Yoneda lemma and the notion of universal arrow in a natural 2-categorical language of liftings and extensions.
- Sean Moss — On "On a topological topos"
- I will describe the 'topological topos' of Johnstone in which we model spaces using convergent sequences, rather than open sets, as the primitive structure on points. I will give some examples of 'local truth' and calculating with a site to show some of the nice properties of this topos, including the good colimit-preservation properties of the inclusion of CW-complexes.
- Christina Vasilakopoulou — Comma-objects in 2-categories
- We will explicitly construct the comma-object of two arrows in a 2-category, and see how the comma-category is expressed via inserters and binary products in Cat.
- Tim Campion — D-Accessible Categories and Free Colimit Completions
- Adámek, Borceux, Lack, and Rosický extend the theory of locally presentable and accessible categories to work relative to a ``doctrine" D of limits. In this talk, I will consider one part of the theory which they generalize -- namely, the fact that a D-accessible category is precisely a free cocompletion of a small category under D-filtered colimits.
If Φ is a class of colimit weights, then the free Φ-cocompletion of a category C is a category containing C which has all colimits weighted by elements of Φ, with the obvious universal property. I will review the ``fundamental theorem of free cocompletions", which identifies the free Φ-cocompletion of C as the ``transfinitely iterated Φ-colimits of representables" in the presheaf category [Cop, Set]. I will also discuss two cases when this transfinite completion process terminates after one step: one case is when Φ is saturated and the other comes from sound doctrines.
- Alex Corner — Coherence for categorical structures
- The classic coherence theorem of Mac Lane tells us that each monoidal category is monoidally equivalent to a strict monoidal category. Strict monoidal categories are the strict algebras for the free monoid 2-monad T on Cat, whilst (unbiased) monoidal categories are the pseudoalgebras for this 2-monad. Another way of phrasing the coherence theorem for monoidal categories is then to say that the inclusion 2-functor of strict algebras into pseudoalgebras, U : T-Alg → Ps-T-Alg, has a left adjoint for which the components of the unit are equivalences in Ps-T-Alg.
This framework for coherence can then be extended to arbitrary 2-monads. For monoidal categories this follows from a result of Power, whereas the more general case (and that the equivalences are the unit of the given adjunction) follows from the work of Lack in the paper `Codescent objects and coherence'. In this talk I will state one version of Lack's coherence theorem (which generalises that of Power) and give examples of many categorical structures that it applies to.
- Clive Newstead — Overview of Lawvere's ETCS
- Whereas a model of ZFC is a universe for set theory, based on set membership, a model of ETCS is a category of sets, based on isomorphism-invariant structure. In this talk I will give a high-level overview of ETCS, discuss to what extent it achieves its goal, and compare it with ZFC.
My contact infomation can be found on my personal website.