A First Course in Category Theory
by Jaap van Ooosten
Jaap van Oosten has written a first course in category theory which
is intende to contain what's presumed knowledge in not too specialized
papers and theses (in computer science). It's 75 pages long.
The synopsis is:
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Categories and functors. Definitions and examples. Duality.
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Natural transformations. Exponents in Cat. Yoneda lemma. Equivalent
categories; Set^op equivalent to Complete Atomic Boolean Algebras.
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Limits and Colimits. Functors preserving (reflecting) them.
(Finitely) complete categories. Limits by products and equalizers.
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A little categorical logic. Regular categories, regular epi-mono
factorization, subobjects. Interpretation of coherent logic in
regular categories. Expressing categorical facts in the logic.
Example of \Omega -valued sets for a frame \Omega.
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Adjunctions. Examples. (Co)limits as adjoints. Adjoints preserve
(co)limits. Adjoint functor theorem.
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Monads and Algebras. Examples. Eilenberg Moore and Kleisli as
terminal and initial adjunctions inducing a monad. Groups monadic
over Set. Lift and Powerset monads and their algebras. Forgetful
functor from T-Alg creates limits.
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Cartesian closed categories and the \lambda-calculus. Examples of
ccc's. Parameter theorem. Typed \lambda calculus and its
interpretation in ccc's. Ccc's with natural numbers object: all
primitive recursive functions are representable.
The notes are available by
anonymous ftp
.