Elements of monoidal topology: exam questions
Sergejs Solovjovs*
Department of Mathematics, Faculty of Engineering, Czech University of Life Sciences Prague (CZU) Kamýcká 129, 16500 Prague - Suchdol, Czech Republic
Exam questions: set 1
(1) Give the definition and an example of monad.
(2) Give the definition of the category y-Rel. Describe the embedding of the category Set into y-Rel.
(3) Give the definition and an example of the category V-Cat.
Exam questions: set 2
(1) Give the definition and an example of unital commutative quantale.
(2) Give the definition and an example of lax extension of a functor on the category Set to the category V-Rel.
(3) Give the definition and an example of the category (T, V)-C&t. Exam questions: set 3
(1) Give the definition of the powerset monad P on the category Set. Give an example of the category (P, V)-Cat.
(2) Give the definition and an example of lax extension of a monad on the category Set to the category V-Rel.
(3) Give the definition of topological category. Describe initial structures in the category (T, V)-C&t. Exam questions: set 4
(1) Give the definition of the ultrafilter monad § on the category Set. Give an example of the category (J3, V)-Cat.
(2) Describe and give an example of the induced preorder functor (T, V)-Cat Ind> Prost.
(3) Give the definition and an example of morphism of lax extensions of monads.
Exam questions: set 5
(1) Give the definition and an example of the Eilenberg-Moore category XT of a monad T on a category X.
(2) Describe the embedding of the Eilenberg-Moore category SetT into the category (T, V)-Cat.
A A°
(3) Describe the algebraic functor (T, V)-C&t —^ V-C&t and its left adjoint functor V-C&t —> (T, V)-C&t.
•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
July 1, 2022
Exam questions: set 6
(1) Give the definition and an example of lax homomorphism of unital quantales compatible with lax extensions of monads.
(2) Describe and give an example of the change-of-base functor induced by a lax homomorphism of unital quantales compatible with lax extensions of monads.
(3) Given a monad T on the category Set, describe a 2-functor from the 2-quasicategory Quant(T) to the 2-quasicategory CAT.
Exam questions: set 7
(1) Give the definition of closed, proper, and perfect maps. Give the definition of proper (T ,V)-functor.
G
(2) Describe and give an example of the functor (T, y)-Cat —> V-C&t as an extension of the induced preorder functor.
(3) Give the definition of compact topological space. Give the definition of compact (T, V)-category. Exam questions: set 8
(1) Give the definition and an example of ultrafilter on a set.
(2) Give the definition of the category App and represent it as a category (T, V)-C&t.
(3) Give the definition of the category Cls and represent it as a category (T, V)-C&t.
Exam questions: set 9
(1) Give the definition and an example of flat lax extension of a functor on the category Set to the category V-Hel.
(2) Give the definition and an example of fibration.
(3) Describe and give an example of discrete and indiscrete (T, y)-category structures on a set. Exam questions: set 10
(1) Describe the Alexandroff topology functor Prost —>• Top and represent it as a left adjoint functor to an algebraic functor (T, V)-C&t —^ V-C&t.
(2) Given a lax homomorphism of unital quantales V     W, describe the lax functor y-Rel y-RelW7.
(3) Give the definition and an example of taut functor on the category Set.
2
Name, surname:
Uco:
Exam questions: set 1
(1) Give the definition and an example of monad.
(2) Give the definition of the category y-Rel. Describe the embedding of the category Set into y-Rel.
(3) Give the definition and an example of the category V-C&t.
Name, surname:
Uco:
Exam questions: set 2
(1) Give the definition and an example of united commutative quantale.
(2) Give the definition and an example of lax extension of a functor on the category Set to the category V-Rel.
(3) Give the definition and an example of the category (T, V)-Cat.
Name, surname:
Uco:
Exam questions: set 3
(1) Give the definition of the powerset monad P on the category Set. Give an example of the category (P, V)-Cat.
(2) Give the definition and an example of lax extension of a monad on the category Set to the category V-Rel.
(3) Give the definition of topological category. Show that the category (T, V)-C&t is a topological construct.
Name, surname:
Uco:
Exam questions: set 4
(1) Give the definition of the ultrafilter monad § on the category Set. Give an example of the category (J3, V)-Cat.
(2) Describe the induced preorder functor (T, V)-Cat Ind> Prost.
(3) Give the definition and an example of morphism of lax extensions of monads.
Name, surname:
Uco:
Exam questions: set 5
(1) Give the definition and an example of the Eilenberg-Moore category XT of a monad T on a category X.
(2) Describe the embedding of the Eilenberg-Moore category SetT into the category (T, V)-Cat.
A A°
(3) Describe the algebraic functor (T, V)-C&t —^ V-C&t and its left adjoint functor V-C&t —> (T, V)-C&t.
Name, surname:
Uco:
Exam questions: set 6
(1) Give the definition and an example of lax homomorphism of united quantales compatible with lax extensions of monads.
(2) Describe the change-of-base functor induced by a lax homomorphism of unital quantales compatible with lax extensions of monads.
(3) Given a monad T on the category Set, describe a 2-functor from the 2-quasicategory Quant(T) to the 2-quasicategory CAT.
Name, surname:
Uco:
Exam questions: set 7
(1) Give the definition of closed, proper, and perfect maps. Give the definition of proper (T ,V)-functor.
G
(2) Describe the functor (T, V)-C&t —> V-C&t as an extension of the induced preorder functor. Describe
G
the respective functor Top —> Prost.
(3) Give the definition of compact topological space. Give the definition of compact (T, V)-category.
Name, surname:
Uco:
Exam questions: set 8
(1) Give the definition and an example of ultrafilter on a set.
(2) Give the definition of the category App and represent it as a category (T, V)-C&t.
(3) Give the definition of the category Cls and represent it as a category (T, V)-C&t.
Name, surname:
Uco:
Exam questions: set 9
(1) Give the definition and an example of flat lax extension of a functor on the category Set to the category V-Rel.
(2) Give the definition and an example of fibration.
(3) Describe and give an example of discrete and indiscrete (T, y)-category structures on a set.
Name, surname:
Uco:
Exam questions: set 10
(1) Describe the Alexandroff topology functor Prost —>• Top and represent it as a left adjoint functor to an algebraic functor (T, V)-Cat —^ V-Cat.
(2) Given a lax honioniorphism of unital quantales V     W, describe the lax functor y-Rel y-RelW7.
(3) Give the definition and an example of taut functor on the category Set.