MD135 Higher categories

Faculty of Science
Autumn 2011
Extent and Intensity
2/0. 2 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
Teacher(s)
prof. RNDr. Mihály Makkai (lecturer)
prof. RNDr. Jiří Rosický, DrSc. (assistant)
Guaranteed by
prof. RNDr. Jiří Rosický, DrSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Contact Person: prof. RNDr. Jiří Rosický, DrSc.
Timetable
Thu 13:00–14:50 M5,01013
Prerequisites
Familiarity with basic category theory, essentially, the contents of S. Mac Lane’s book, “Categories for the Working Mathematician”.
Course Enrolment Limitations
The course is offered to students of any study field.
Course objectives
The aim of the course is the exposition of a part of the theory of higher dimensional categories (HDC’s), the part that uses the methods of categorical logic. The lectures will outline various subjects (see below for details). The outlines of the lectures will be complemented by written materials, as well as tutorial sessions. The tutorial sessions will be of a free character, both in terms of timing and in terms of the materials covered. Decisions about the timing and the contents of the tutorials will be based on the needs and demands of the attendees of the course.
Syllabus
  • Basics of categorical logic. Tarskian syntax and semantics reformulated as categorical concepts. Lawvere’s concepts of logical operations as adjoints; doctrines of categorical logic; Grothendieck topologies, hyperdoctrines. First Order Logic with Dependent Sorts (FOLDS). Basic results and examples of applications to the established notions of HDC: bicategories, tricategories, Gray categories, and derived notions. Computads. General results and useful special classes. Multitopic sets and categories. Concepts of algebraic operations versus virtual operations in HDC’s. Equivalence of distinct notions of HDC, using the framework of FOLDS. Open problems, with illustrations of special cases.
Language of instruction
English
Further Comments
Study Materials
The course is taught only once.
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