# ALGEBRA SUPERIORE A

- Overview
- Assessment methods
- Learning objectives
- Contents
- Bibliography
- Teaching methods
- Contacts/Info

Bachelor's degree in Mathematics or equivalent mathematical maturity.

The exam will consist of a seminar on a topic, chosen in agreement with the lecturer, which extends the contents of the course. The evaluation of this seminar will constitute the two thirds of the final grade, while the remaining third will be determined by the resolution of exercises assigned by the lecturer in the course of the lessons and discussed in the exercise sessions.

This course aims to provide an ample introduction to category theory, a sector of great relevance for many fields of contemporary mathematics, notably including homological algebra, algebraic geometry and algebraic topology. By the end of the course it is expected that the student will have acquired a solid theoretical preparation in this field and, through the exercise sessions, the ability to profitably use the categorical intuition in a great variety of mathematical situations.

CATEGORIES, FUNCTORS AND NATURAL TRANSFORMATIONS

Notion of category. Abstract and concrete categories. Monoids, groupoids and preordered sets. The duality principle. Covariant and contravariant functors. Forgetful functors. Natural transformations and isomorphisms. The example of the double dual of a k-vector space. Categorical equivalences and dualities; the case of Stone-type dualities. Full, faithful and essentially surjective functors and their relationship with the notion of categorical equivalence. Constructions on categories: product of categories, functor categories, slice categories. Monomorphisms and epimorphisms.

REPRESENTABILITY

Hom functors. The Yoneda lemma; naturality of the bijection and examples. The Yoneda embedding. Representable functors: definition, properties and examples.

UNIVERSAL PROPERTIES, LIMITS and COLIMITS

Diagrams, cones and cocones. Concept of universal property and examples: free group on a set, tensor product, cartesian product, etc. Abstract definition of limit and colimit. Products, coproducts, terminal and initial objects, equalizers, coequalizers, pullbacks and pushouts, direct and inverse limits. Construction of arbitrary limits starting from products and equalizers and of finite limits starting from pullbacks and the terminal object. Preservation, reflection and creation of (co)limits by a functor.

ADJUNCTIONS

Concept of pair of adjoint functors and examples. Uniqueness of adjoints up to isomorphism. Characterization of adjunctions by means of a universal property and by means of the triangular identities. Every equivalence is an adjunction. Reflections. Preservation of limits (resp. colimits) by right (resp. left) adjoint functors. Adjoint functor theorems.

MONADS

Monads, comonads and examples. Monads induced by an adjunction. Algebras for a monad. Eilenberg-Moore and Kleisli categories associated with a monad and universal properties of the adjunctions that they induce. Monadicity theorems and examples of monadic and non-monadic functors.

ELEMENTS OF HOMOLOGICAL ALGEBRA

Regular and Barr-exact categories. Internal equivalence relations and quotients. Image of a morphism. Regular epimorphisms and their orthogonality property. Pointed, semi-additive and additive categories and their properties. Biproducts. Kernels and cokernels. Regular and normal monomorphisms and epimorphisms. Abelian categories, their properties and their characterization as the categories which are additive and Barr-exact. Complexes and exact sequences. The five lemma and the snake lemma. Homology groups of a complex and Meyer-Vietoris long exact sequence. Homotopies between morphisms of complexes and invariance of homology with respect to them. Projective and injective resolutions and derived functors.

ELEMENTS OF TOPOS THEORY

Sheaves on a topological space. Equivalence with the Ã©tale maps. Sites and sheaves on them. Definition of a Grothendieck topos and examples. Internal structure of a topos (limits, colimits, exponentials, subobject classifier), category of internal abelian groups and its properties. Cohomology.

S. Awodey, "Category theory", Oxford University Press, 2nd edition, 2010

F. Borceux, "Handbook of categorical algebra", Cambridge University Press, 1994

P.T. Johnstone, "Category theory", course of the Part III of the Mathematical Tripos, lecture notes available at the address https://www.math.cornell.edu/~dmehrle/notes/partiii/cattheory_partiii_no...

T. Leinster, "Basic Category Theory", Cambridge University Press, 2014

S. Mac Lane, "Categories for the working mathematician", Springer, 2nd edition, 1997

S. Mac Lane, I. Moerdijk, "Sheaves in geometry and logic. A first introduction to topos theory", corrected reprint of the 1992 edition, Universitext, Springer-Verlag, New York, 1994

E. Riehl, "Category theory in context", Cambridge University Press, 2016

The frontal theoretical lessons, given at the blackboard, will be accompanied by sessions of exercises assigned by the lecturer in the previous lessons, in which the students who desire to do so will be able to expose at the blackboard their solutions and discuss them with the lecturer in front of the other students. The active participation in these exercise sessions will be valorized in the context of the course examination (contributing by one third to the final grade).

The students of the course can reach the lecturer in her office in the hour immediately following the end of each lecture to ask for more explanations, clarifications or further study.

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