Requirements/prerequisites:

Duration: Hilary term, 11 weeks

Number of lectures per week: 3 lectures including tutorials per week

Assessment:

End-of-year Examination: 2 hour examination in Trinity term.

Description: (Preliminary.)

• Categories - functions of sets, definition of a category, examples of categories, isomorphisms, constructions on categories, free categories, foundations: large, small, and locally small.
• Abstract structures - epis and monos, initial and terminal ob jects, generalized elements, sections and retractions, products, examples of products, categories with products, Hom-sets.
• Duality - the duality principle, coproducts, equalizers, coequalizers.

• Groups and categories - groups in a category, the category of groups, groups as categories, finitely presented categories.

• Limits and colimits - subob jects, pullbacks, properties of pullbacks, limits, preservation of limits, colimits.
• Exponentials - exponential in a category, cartesian closed categories, Heyting algebras, equational definition, -calculus.
• Functors and naturality - category of categories, representable structure, stone duality, naturality, examples of natural transformations, exponentials of categories, functor categories, equivalence of categories, examples of equivalence.
• Categories of diagrams - Set-valued functor categories, the Yoneda embedding, the Yoneda Lemma, applications of the Yoneda Lemma, Limits in categories of diagrams, colimits in categories of diagrams, exponentials in categories of diagrams, Topoi.
• Adjoints - preliminary definition, Hom-set definition, examples of adjoints, order adjoints, quantifiers as adjoints, RAPL, locally cartesian closed categories, adjoint functor theorem.

• Monads and algebras - the triangle identities, monads and adjoints, algebras for a monad, comonads and coalgebras, algebras for endofunctors.

Bibliography: Awodey, S. (2006). Category Theory. Oxford Logic Guides 49, Oxford University Press.

Sep 16, 2009