Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master

SUBJECT

Code
Subject
MML1005 Special Topics of Modern Algebra
Section
Semester
Hours: C+S+L
Category
Type
Mathematics
4
2+1+0
speciality
optional
Teaching Staff in Charge
Prof. CALUGAREANU Grigore, Ph.D.,  calumath.ubbcluj.ro
Aims
This course presents the theory of categories, functors and natural transformations with emphasis on concrete examples of categories: sets, groups, abelian groups, modules, but topology, ordered sets and lattices as well. This way a useful extensive revision is realized of all fundamental notions taught durring the first 5 semesters of mathematics (algebra, topology, geometry, lattices, algebraic topology etc).
Content
1. Definition of category: objects and morphisms. Examples.
2. Equalizers, pullbabcks, pushouts, intersections, unions.
3. Images, inverse images, zero objects (morfisms), kernels.
4. Normal, exact categories, 9-lemma, products.
5. Additive, exact additive and abelian categories.
6. Functors and natural transformations. Equivalences of categories, Functor Categories.
7. Proiective and injective objects, generators.
8. Adjoint functors, embedding theorems.
9. Equivalences of module categories.
References
0. J.AdŽamek, H. Herrlich, G. Strecker, Abstract and Concrete Categories, The Joy of Cats
[http://katmat.math.uni-bremen.de/acc]
1. P. Freyd, Abelian Categories: An Introduction to the Theory of Functors. Harper and Row, 1964
[http://www.tac.mta.ca/tac/reprints]
2. H. Herlich, G. Strecker, Category Theory: An Introduction. Sigma Series in Pure Math. No. 1, Heldermann, Berlin, Second Edition, 1979.
3. R.L. Knighten, Notes on Category Theory, 2007.
[http://math.ubbcluj.ro/~calu/Notes.pdf]
4. S. Mac Lane, Categories for the Working Mathematician. Graduate Texts in Math, No. 5. Second Edition, 1997.
5. B. Mitchell, Theory of Categories. Pure and Applied Math, No. 17. Academic Press, 1965.
6. I. Purdea, Tratat de Algebra Moderna, vol. 2, Editura Academiei, 1982.
7. H. Schubert, Categories. Springer-Verlag, 1972.
8. B. Stenstrom, Rings of Quotients, Springer-Verlag, Berlin, 1975.
Assessment
The students will be graded for the home works (exercises stated durring the lectures, each exercise awarded to a different student).
The final exam consists in definitions and/or results proved durring lectures and exercises from the home works.
Links: Syllabus for all subjects
Romanian version for this subject
Rtf format for this subject