2014 Columbia UMS Summer Seminar

Category Theory

Wednesdays, 7:30pm; Room 507, Mathematics
« MacLane, Categories for the Working Mathematician »
Date Speaker Title Abstract
May 28 Nilay Kumar Motivating abstract nonsense Category theory is a powerful language that formalizes many of the patterns and idioms of mathematics. As the UMS talks this summer will follow MacLane's rather formal book, I will try to motivate, in this introductory lecture, some of the abstract nonsense to come. I will loosely define some basic categorical terms and give a number of examples ranging from Galois theory to algebraic topology to algebraic geometry. These examples will illustrate how natural a language category theory really is for thinking about mathematics.
June 4 Nilay Kumar Categories, functors,
and natural transformations I
In this lecture, we will follow the first half of MacLane's chapter I entitled "Categories, Functors, and Natural Transformations." We will first describe categories rather loosely via the concept of a metacategory before introducing categories as interpreted through set theory. Next we will define functors - morphisms between categories, in some sense - as well as subcategories. Using this language, we can formalize what it means for a morphism to be natural via the definition of a natural transformation. We will end by defining the notion of an equivalence of categories. Numerous examples will be provided throughout.
June 11 Nawaz Sultani Categories, functors,
and natural transformations II
In this lecture, we will be continuing, and concluding, Mac Lane's riveting first chapter. We will start by defining monics, epis, and zeros, all of which should appear familiar as their meanings are revealed. Afterwards, we will revisit the foundations of Category Theory that were addressed in the first lecture. Here, we will tackle the pesky Universe set, defining a set of properties to give it a more rigorous shape, as well as focus on small versus large categories. Our discussion will subsequently focus on generating a proper understanding of large categories, before reaching the last topic to be addressed, which is redefining categories in terms of hom-sets.
June 18 Kendric Schefers Constructions on categories In this week's installment, I will tackle the remaining basic concepts and definitions of the subject. I will first introduce the fundamental notions of duality and contravariance of functors. I will then go over products of categories, functor categories, comma categories, and quotient categories. In an exciting turn of events, we will encounter our first theorem. Because there are many topics to be covered in this week's lecture, there will be fewer examples than in previous weeks' lectures. Rather, emphasis will be placed on the careful and deliberate presentation of the basic concepts within the chapter.
June 25 Kendric Schefers Universals and limits I In this week's lecture, I will wrap up last week's topic of constructions on categories with a discussion of comma categories and quotient categories. We will then dive into some of the meatier topics of category theory with an introduction to universals and limits. The talk will conclude with a presentation of the Yoneda lemma, a major result in category theory. If time permits, I will speak a little bit about coproducts and colimits.
July 2 Nawaz Sultani Universals and limits II This week's talk will focus on getting through the bulk of chapter three's dense material. After a quick review of universals, I will proceed to talk about the Yoneda Lemma. Afterwards, I will spend most of the time on talking about coproducts and colimits, followed by products and limits, important notions for the aspiring category theorist. If time permits, I will proceed to touch on categories with finite products, and groups in categories. The lecture will go quite fast, so I recommend some preparation of the chapter, or a review of the notes afterwards.
July 9 Leonardo Abbrescia Universals and Limits III We will begin by reviewing the definitions of a universal, coproducts, cokernels, and pushouts. I will then introduce the notions of colimits, limits, products, kernels, and pull backs. I will attempt to give examples of these things using categories that you are used to. Unfortunately, this will be another talk without any lemmas or proofs.
July 14
Yifei Zhao Adjoints I I will do a rapid review of some highlights of chapter III, and prove a theorem on colimits of representable functors. Then I will introduce the concept of adjoint functors, and prove some basic facts about them.
July 16 Nilay Kumar Adjoints II In this lecture, we will quickly review adjunctions before covering reflective subcategories, the relation between adjoints and an equivalence of categories, as well as the idea of a Galois connection. We will end by discussing transformations and compositions of adjoints.
July 23 Irit Huq-Kuruvilla Adjoints III Today, we will finish the adjoints chapter. I will talk about how adjoints generalize the notion of sameness, and make that precise by defining the notion of an equivalence of categories. I will also talk about the various categories in which adjoints appear naturally, by defining transformation and composition of adjoints (treating adjoints as objects and adjoints as morphisms, respectively). If there is some time left I will talk about subobject classifiers, a generalization of characteristic functions in Set, as well as elementary topoi.
July 30 Linus Hamann Limits In this lecture we will advance the notions of the limit, defining what it means for a category to be small-complete and for a functor to preserve and create limits. This will give a variety of ways of looking at limits and how they behave under various transformations. In addition, these notions will motivate some concrete examples of limits such as the canonical constructions of the ring of p-adic integers and the topological space of the p-adic solenoid.
July 31
Yifei Zhao Freyd's Adjoint Functor Theorem I will prove Freyd's Adjoint Functor Theorem.
designed by Nilay Kumar