Although we do not yet know how long the social distancing related to the Covid-19 pandemic will last, and regardless of the changes that had to be made in the evaluation of the June 2020 session in relation to what is provided for in this learning unit description, new learnig unit evaluation methods may still be adopted by the teachers; details of these methods have been - or will be - communicated to the students by the teachers, as soon as possible.
LMAT1131 - linear algebra (first year of the bachelor program in mathematics) or any equivalent cours.
LMAT 1231 - multilinear algebra and group theory (second year of the bachelor program in mathematics) or any equivalent cours.
Categories, functors, natural transformations. Adjoint functors and equivalences of categories. Limits and colimits. Regular, exact and abelian categories. Exact sequences and homological lemmas.
At the end of this learning unit, the student is able to :
Contribution of the course to learning outcomes in the Master in Mathematics programme. By the end of this activity, students will have made progress in:
- Recognise and understand a basic foundation of mathematics.He will have made progress in:
-- Recognise the fundamental concepts of some important current mathematical theories.
-- Establish the main connections between these theories.
- Show evidence of abstract thinking and of a critical spirit. He will have made progress in:
-- Identify the unifying aspects of different situations and experiences.
-- Argue within the context of the axiomatic method.
-- Construct and draw up a proof independently, clearly and rigorously.
Learning outcomes specific to the course. By the end of this activity, students will be able to:
- Identify, in his mathematical knowledge, several meaningful examples of categories, functors and natural transformations.
- Establish the adjointness of some pairs of functors and the equivalence of some categories.
- Construct limits and colimits, eventually using adjoint functors and equivalences of categories.
- Recognise and prove some important exactness properties of regular, exact and abelian categories.
- Concretely explain different notions and results in the categories of sets, groups, abelian groups and topological groups.
The contribution of this Teaching Unit to the development and command of the skills and learning outcomes of the programme(s) can be accessed at the end of this sheet, in the section entitled “Programmes/courses offering this Teaching Unit”.
The following subjects are studied:
- Definition and examples of categories, functors, natural transformations.
- Isomorphisms, monomorphisms and epimorphisms in a category.
- Adjoint functors (unit, counit, triangular identities) and their fundamental properties.
- Reflective subcategories and equivalences of categories.
- Examples of limits and colimits.
- Limits and adjoint functors.
- Definition of regular and of exact category, main properties and examples. Barr-Kock theorem. Mal'tsev categories.
- Abelian categories, exact sequences, five lemma, nine lemma, snake lemma.
Students may choose the examination language (English or French).
F. Borceux : Handbook of categorical algebra, Vol. 1-2 (Cambridge University Press).
P. Freyd : Abelian categories (disponible sur MoodleUCLouvain).
S. Mac Lane : Categories for the Working Mathematician (Springer).
T. Leinster : Basic Category Theory (Cambridge studies in advanced mathematics)