Basic Category Theory

Jaap van Oosten

January 1995

Abstract:

This course was given to advanced undergraduate and beginning Ph.D. students in the fall of 1994 in Aarhus, as part of Glynn Winskel's semantics course. It is, in the author's view, the very minimum of category theory one needs to know if one is going to use it sensibly. Nevertheless, two topics are breathed on, which may be skipped: there is a glimpse of categorical logic, and there is a treatment of the -calculus in cartesian closed categories. These are there to give the reader at least a very rough idea of how the theory ``works''. The text contains a bit over hundred exercises, varying in difficulty, which supplement the treatment and are warmly recommended. There is an elaborate index.

Contents

1

Categories and Functors

1.1

Definitions and examples

1.2

Some special objects and arrows

2

Natural transformations

2.1

The Yoneda lemma

2.2

Examples of natural transformations

2.3

Equivalence of categories; an example

3

(Co)cones and (co)limits

3.1

Limits

3.2

Limits by products and equalizers

3.3

Colimits

4

A little piece of categorical logic

4.1

Regular categories and subobjects

4.2

Coherent logic in regular categories

4.3

The language and theory associated to a regular category

4.4

Example of a regular category

5

Adjunctions

5.1

Adjoint functors

5.2

Expressing (co)completeness by existence of adjoints; preservation of (co)limits by adjoint functors

6

Monads and Algebras

6.1

Algebras for a monad

6.2

-Algebras at least as complete as

6.3

The Kleisli category of a monad

7

Cartesian closed categories and the -calculus

7.1

Cartesian closed categories (ccc's); examples and basic facts

7.2

Typed -calculus and cartesian closed categories

7.3

Representation of primitive recursive functions in ccc's with natural numbers object

Available as PostScript, DVI.

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