Elements of monoidal topology
Sergejs Solovjovs∗
Department of Mathematics, Faculty of Engineering, Czech University of Life Sciences Prague (CZU)
Kam´yck´a 129, 16500 Prague - Suchdol, Czech Republic
1. Pre-requisites in the language of instruction
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.
2. 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.
3. 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.
4. Course contents
The course consists of 14 lectures on the following topics.
1. Monads and their algebras.
2. Quantale-valued relations and lax extensions of monads.
3. The categories (T, V )-Cat and V -Cat. Examples of the category 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.
13. Topological spaces via neighborhood filters.
14. Power-enriched monads and Kleisli monoids.
∗Tel.: (+420) 224 383 239
Email address: solovjovs@tf.czu.cz (Sergejs Solovjovs)
URL: http://home.czu.cz/solovjovs (Sergejs Solovjovs)
Preprint submitted to the Masaryk University in Brno June 15, 2022
5. Recommended or required reading
The main source of information for the course is ”D. Hofmann, G. J. Seal, and W. Tholen (eds.), Monoidal
Topology: A Categorical Approach to Order, Metric and Topology, Cambridge University Press, 2014” (freely
available at http://sweet.ua.pt/dirk/artigos/HST14_Monoidal_topology_A_categorical_approach_
to_order_metric_and_topology.pdf). Additional recommended references include the following:
• category theory: “J. Ad´amek, H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories:
the Joy of Cats, Repr. Theory Appl. Categ. 17 (2006), 1–507” (this reference is freely available at
http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf);
• quantales: “K. I. Rosenthal, Quantales and Their Applications, Pitman Research Notes in Mathematics,
vol. 234, Addison Wesley Longman, 1990”;
• general topology: “R. Engelking, General Topology, Sigma Series in Pure Mathematics, vol. 6, Heldermann
Verlag, 1989”(https://www.heldermann.de/SSPM/SSPM06/sspm06.htm).
The transcripts of the course lectures can be found at https://home.czu.cz/solovjovs/ke-stazeni
under “Elements of monoidal topology”.
6. Planned learning activities and 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.
7. Assessment methods and criteria
The course will end with an oral examination. Course attendants will be asked three questions on the
topics of the course and will be given 40 minutes to prepare the answers. Every question will require an
answer of moderate length. The examination result will depend on the number of correct answers to the
questions as follows:
• zero correct answers: “failed” (exam mark “F”);
• one correct answer: “good” (exam mark “C”);
• two correct answers: “very good” (exam mark “B”);
• three correct answers: “excellent” (exam mark “A”).
Course attendants will have three attempts to pass the exam.
8. Examination dates
The following are the planned course examination dates for the spring semester of 2022.
Date Time Place
June 2, 2022 (Thursday) 15:00 – 17:00 M5, 01013
June 16, 2022 (Thursday) 15:00 – 17:00 M5, 01013
June 30, 2022 (Thursday) 15:00 – 17:00 M5, 01013
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