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Course, academic year 2022/2023
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Fundamentals of Category Theory for Computer Scientists - NMAI065
Title: Základy teorie kategorií pro informatiky
Guaranteed by: Department of Applied Mathematics (32-KAM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2017
Semester: winter
E-Credits: 3
Hours per week, examination: winter s.:2/0, Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Teaching methods: full-time
Guarantor: prof. RNDr. Aleš Pultr, DrSc.
Class: Informatika Mgr. - Diskrétní modely a algoritmy
Classification: Informatics > Discrete Mathematics
Pre-requisite : NMAI064
Annotation -
Last update: T_KAM (24.03.2004)
Basic notions of category theory: category, functor, transformation. Categorial constructions, in particular limits and colimits. Adjunction and preserving (co)limits. Monads, description of algebras, Kleisli categories.
Course completion requirements -
Last update: prof. RNDr. Aleš Pultr, DrSc. (11.06.2019)

Oral exam.

Literature -
Last update: prof. RNDr. Aleš Pultr, DrSc. (11.10.2017)

S. MacLane, Categories for Working Mathematician, Springer 1989.

Appendix on categories in Picado-Pultr:Frames and Locales

Requirements to the exam -
Last update: prof. RNDr. Aleš Pultr, DrSc. (11.06.2019)

Categories and functors, examples. Natural transformations and natural equivalences. Special morphisms.

Bounds, limits and colimits. Special (co)limits. Complete categories and theorems on completeness.

Adjoint functors. Adjunction units. Adjunction and preserving limits or colimits. Theorem on the existence of adjoints.

Yoneda lemma.

Monads. Monads and adjunction. Description of algebraic structures (Eilenberg - Moore algebras). Kleisli categories; notes on their role in computer science.

Syllabus -
Last update: prof. RNDr. Aleš Pultr, DrSc. (11.10.2017)

Basic concepts of category theory. Categories and functors, examples. Natural transformations and natural equivalences, examples. Special morphisms.

Basic categorial constructions. Factorisation. Image of a morphism. Bounds, limits and colimits. Special (co)limits. Complete categories and theorems on completeness.

Adjoint functors, examples. Descriptions of adjunctions by means of adjunction units, Reflective and coreflective subcategories. Adjunction and preserving limits or colimits. Theorem on the existence of adjoints.

Cartesian closed categories. Categories of functors.

Yoneda lemma. Categorical models of some theories.

Monads. Monads and adjunction. Description of algebraic structures (Eilenberg - Moore algebras). Kleisli categories; notes on their role in computer science.

 
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