Categories in Algebra and Topology

Code:

M556

Acronym:

M556

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2015/2016 - 2S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Doctoral Program in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
IUD-M	4	PE do Prog Inter-Univ Dout Mat	1	-	9	60	243

Teaching language

English

Objectives

This unit intends to be an introduction to the main ideas, methods and applications of category theory, crucial tool in several areas of mathematics like algebra, topology, logic or computer scince, among others.

Learning outcomes and competences

The students should improve their skills on: abstraction and generalization; capacity of formulating and solving problems; conception, analysis and use of categorical methods; work as a team.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Linear Algebra, Algebra and Topology at an undergraduate level.

Program

Part I: Introduction to Category Theory:
Categories, functors and natural transformations. Isomorphism and equivalence of categories. Construction of new categories: subcategories, product of categories and dual category. Categorical duality principle. Limits and colimits. Functor categories. Representable functors. Yoneda Lemma and Yoneda embedding.
Adjoints and limits. Existence of adjoints (Freyd's Theorem).

Part II: It includes topics from the list below, chosen according to the interests of the students:
Monads and categories of Eilenberg-Moore algebras.
Cartesian closed categories. Toposes.
Locales.
Exact and regular categories. Additive, abelian, semi-abelian categories and homological categories.

Mandatory literature

Saunders Mac Lane; Categories for the Working Mathematician, Springer

Complementary Bibliography

Francis Borceux; Handbook of Categorical Algebra, Cambridge University Press

Teaching methods and learning activities

Face to face class. The students will be asked to solve individually problems.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	60,00
Participação presencial	15,00
Trabalho escrito	25,00
Total:	100,00

Calculation formula of final grade

The evaluation consists of:
- written works, throughout the semester - 25%
- presencial evaluation - 15%
- final written exam - 60%.