School of Mathematics
School of Mathematics
Module CS4002  Category theory
200910 (JS & SS Mathematics, JS & SS Twosubject Moderatorship
)
Lecturer: Dr. Arthur Hughes (Computer Science)
Requirements/prerequisites:
Duration: Hilary term, 11 weeks
Number of lectures per week: 3 lectures including tutorials per week
Assessment:
Endofyear 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, Homsets.
 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  Setvalued 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, Homset 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
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On 16 Sep 2009, 17:13.