MEMOT Elements of monoidal topology

Přírodovědecká fakulta
jaro 2022
Rozsah
2/0/0. 2 kr. (příf plus uk k 1 zk 2 plus 1 > 4). Ukončení: zk.
Vyučující
Sergejs Solovjovs, Dr., Ph.D. (přednášející)
Garance
prof. RNDr. Jan Paseka, CSc.
Ústav matematiky a statistiky – Ústavy – Přírodovědecká fakulta
Dodavatelské pracoviště: Ústav matematiky a statistiky – Ústavy – Přírodovědecká fakulta
Rozvrh
Čt 15:00–16:50 M5,01013
Předpoklady
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.
Omezení zápisu do předmětu
Předmět je otevřen studentům libovolného oboru.
Cíle předmětu
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.
Výstupy z učení
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.
Osnova
  • 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.
Literatura
  • 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 s. Pitman Research Notes in Mathematics Series 234. ISBN 0582064236. info
Výukové metody
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.
Metody hodnocení
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.
Vyučovací jazyk
Angličtina
Další komentáře
Studijní materiály
Předmět je vyučován jednorázově.
  • Statistika zápisu (nejnovější)
  • Permalink: https://is.muni.cz/predmet/sci/jaro2022/MEMOT