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Module CS4002: Category theory

Credit weighting (ECTS)

5 credits

Semester/term taught

Hilary term 2013-14

Contact Hours

11 weeks, 3 lectures including tutorials per week

Lecturer

Prof. Arthur Hughes (Computer Science)

Learning Outcomes

On successful completion of this module, students will be able to explain why:

Many objects of interest in mathematics congregate in concrete categories.
Many objects of interest to mathematicians are themselves small categories.
Many objects of interest to mathematicians may be viewed as functors from small categories to the category of Sets.
Many important concepts in mathematics arise as adjoints, right or left, to previously known functors.
Many equivalence and duality theorems in mathematics arise as an equivalence of fixed subcategories induced by a pair of adjoint functors.
Many categories of interest are the Eilenberg-Moore or Kleisli categories of monads on familiar categories2
Many data types of interest to computing science are algebras for endofunctors.
Many process of interest to computing science are coalgebras for endofunctors.

Module Content

What is category theory? As a first approximation, one could say that category theory is the mathematical study of (abstract) algebras of functions. Just as group theory is the abstraction of the idea of a system of permutations of a set or symmetries of a geometric object, category theory arises from the idea of a system of functions among some objects.

We think of the composition g â°¦ f (f ; g often used in CS) as a sort of â``product'' of the functions f and g, and consider abstract â``algebras'' of the sort arising from collections of functions. A category is just such an â``algebra'', consisting of objects A, B, C, ¼ and arrows f \colon A â® B, g \colon B â® C, ..., that are closed under composition and satisfy certain conditions typical of the composition of functions1.

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 objects, 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 - subobjects, pullbacks, properties of pullbacks, limits, preservation of limits, colimits.
Exponentials - exponential in a category, cartesian closed categories, Heyting algebras, equational definition, l-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.

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

Module Prerequisite

Assessment Detail