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cattheory.tex
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cattheory.tex
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\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\title{Category Theory Notes}
\author{Krystal Maughan }
\date{May 30th 2017}
\begin{document}
\maketitle
\mathversion{bold}{Catgory Theory}
\mathversion{normal}
\section{Eilenberg-MacLane Spaces}
was regarded as abstract nonsense, dogmatic
working in Topology
\section{Grothendieck}
60s, algebraic geometry
\section{Lawvere, Tierney}
logic, alternative foundations for mathematics
\\
\section{Category Theory}
Homological Algebra
\\
new kinds of algebraic objects
\\
combinatorial
\\
simplicial
\\
graph
\\
computer science
\\
John Baez
\\
Braid diagrams
\\
higher and enriched categories
\\
\section{What is Category Theory?}
Category Theory vs Set Theory
\\
How do objects behave and relate to each other?
\\
relations between things
\\
Algebraic co-product
\\
common vocabulary among areas in mathematics
\\
categorical vocabulary
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Duality
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Comma Categories
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Yoneda Lemma
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Colimits and Limits
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Adjunctions
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Monads
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Beck's Theorem
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Abelian Categories
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Kan extensions
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Homotopical Algebra : Theory of Model Categories
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Topoi
\\
\section{Practical Category Theory : David Koontz}
Monoids, SemiGroups
\\
\section{SemiGroup}
class Semigroup a where
\\
append :: $a$ $\rightarrow a \rightarrow a$
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Append is Associative
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$(a <> b) <> c == a <> (b <> c)$
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(A appended with B) appended with C
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is the same as
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A appended with (B appended with C)
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Monoid
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class (Semigroup a) <= Monoid a where
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empty :: a
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SemiGroup is append + associativity law
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Monoid is semigroup and it has an empty value
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\\
Monoid Laws
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The empty value doesn't change the meaning of a monoidal value
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a <> empty == a
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empty <> a == a
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(left and right identity)
\\
\\
\section{Higher Kinded Types : David Koontz}
Constraints
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Generic in structure
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\section{Functor}
f a
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Apply a function to the value(s) in the box
\\
map : Functor $f$ => $(a \rightarrow b) \rightarrow f$ $ a \rightarrow f$ $b$
\\
\end{document}