2012/13, second term, 7 credits.

Theacher: Marco Grandis.

INTRODUCTION

    Category Theory yields a general framework for the study of mathematical structures and 'universal constructions'; it originated in the 1940's within Algebraic Topology and Homological Algebra. Several mathematical constructions can be presented and characterised as adjoint functors of other, obvious procedures.
    It is understood that all notions introduced will be applied to various structures of algebra, topology and basic functional analysis.
    The course consists of five hours per week. One of them is devoted to exercises, which are done by students on assigned subjects. Evaluation will be based on the exposition of a seminar and an examination. Lectures can be given in English, according to students attending.

PROGRAM

1. Categories. Introduction. Definition of category and groupoid. Small categories. Isomorphisms, monomorphisms, epimorphisms, retracts. Initial and terminal object. Opposite category and duality. Subcategory, quotient category, cartesian product of categories.

2. Functors and natural transformations. Functors and their composition; isomorphisms of categories. Faithful and full functors, concrete categories. Natural transformations and vertical composition. Equivalence of categories and adjoint equivalence; the characterisation theorem of equivalences. Skeletal categories. Horizontal composition of natural transformations; interchange law. Hints at 2-categories. Representable functors and Yoneda lemma. Contravariant functors.

3. Limits and universal problems. Products, equalisers, pullbacks; limit of a functor. Complete categories; construction and preservation theorem for limits. Colimits. Limits and colimits as functors. Universal arrow; limits as a particular case; comma categories. Subobjects, regular and normal subobjects; quotients.

4. Adjoint functors. Definition and characterisation theorem. Preservation of limits. Composition of adjunctions. Examples of interest: free algebraic structures; abelianised group; ring of fractions; limits and colimits; completion of a metric space; sheafification of a presheaf; Stone-Cech compactification; tensor product of modules; geometric realisation of a simplicial or cubical set. Galois connections. Reflective and coreflective subcategories, examples. Faithful and full adjoints. Adjoint Functor Theorem.

5. Complements. Algebraic structures in a cartesian or monoidal category. Monads, Eilenberg-Moore algebras for a monad, monadic categories; examples. Some notions about additive, exact and abelian categories. Hints at 2-categories and bicategories. Relations and spans.

TEXTS

Online Notes: in pdf (24 p.)

S. Mac Lane, Categories for the working mathematician, Springer 1971.

J. Adámek - H. Herrlich - G. Strecker, Abstract and concrete categories, Wiley Interscience Publ., 1990.
Online version: in pdf

F. Borceux, Handbook of categorical algebra. 1-2-3, Cambridge University Press, Cambridge, 1994.

A. Grothendieck, Sur quelques points d'algèbre homologique, Tôhoku Math. J. 9 (1957), 119-221.

Office Hours:   Monday 15-18. Other afternoons, when possible.