MEMOT Elements of monoidal topology

Faculty of Science
Spring 2022
Extent and Intensity
2/0/0. 2 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
Teacher(s)
Sergejs Solovjovs, Dr., Ph.D. (lecturer)
Guaranteed by
prof. RNDr. Jan Paseka, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Thu 15:00–16:50 M5,01013
Prerequisites
The course requires some basic knowledge on category theory, general topology, and quantales. All the additional required concepts will be introduced during the lecture course. The course language is English.
Course Enrolment Limitations
The course is offered to students of any study field.
Course objectives
The course aims at introducing its listeners into the theory of monoidal topology, which is an approach to general topology based in category theory and quantales. Course attendants will get to know how to represent a number of well-known mathematical structures, e.g., preordered sets, (generalized) metric spaces,topological spaces, approach spaces, and closure spaces as particular categorical structures and how then to describe and study the properties of these categorical structures using the standard tools of category theory.
Learning outcomes
On finishing the course its listeners should be able to apply methods of category theory to different mathematical settings, e.g., general topology. Since the influence of category-theoretic tools in modern mathematics is growing, course attendants will be more competitive in their knowledge of pure mathematics.
Syllabus
  • The course consists of 12 lectures on the following topics. 1. Monads and their algebras. 2. Quantale-valued relations and lax extensions of monads. 3. The category (T, V)-Cat. 4. Fundamental examples of the category (T, V)-Cat. 5. Properties of the category (T, V)-Cat I: Eilenberg-Moore algebras and topological categories. 6. Properties of the category (T, V)-Cat II: induced preorders and algebraic functors. 7. Properties of the category (T, V)-Cat III: change-of-base functors. 8. (T, V)-categories as generalized spaces. 9. Generalized Kuratowski-Mrówka theorem I: proper, closed, and perfect maps. 10. Generalized Kuratowski-Mrówka theorem II: proper (T, V)-functors and compact (T, V)-categories. 11. Symmetric monoidal closed structure on the category V-Cat. 12. The category V-Mod of V-categories and V-modules.
Literature
  • Monoidal topology : a categorical approach to order, metric and topology. Edited by Dirk Hofmann - Gavin J. Seal - W. Tholen. 1st pub. Cambridge: Cambridge University Press, 2014, xvii, 503. ISBN 9781107063945. info
  • ROSENTHAL, Kimmo I. Quantales and their applications. Essex: Longman Scientific & Technical, 1990, 165 pp. Pitman Research Notes in Mathematics Series 234. ISBN 0582064236. info
Teaching methods
The course consists of lectures only. It is planned to have one lecture (two hours) per week. There will be no seminars or any additional activities in the course.
Assessment methods
The course consists of lectures only. It is planned to have one lecture (two hours) per week. There will be no seminars or any additional activities in the course.
Language of instruction
English
Further Comments
Study Materials
The course is taught only once.
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