Category theory, composition & functional programming презентация

Introduction
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THIS IS NOT A LECTURE
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Lets inquire and discover together!
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What is programming?
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OOP point of view
Class diagram
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OOP point of view
Sequence diagram
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FP point of view
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FP point of view
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What is similar between those POVs?
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Composition
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Programming is Composition
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Is there any science which is focused on composition?
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Yes, it is Category Theory
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Why to study CT?
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Source: http://rs.io/why-category-theory-matters/
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Why to study CT?
Extremely general and wide-applicable
Gives the new way (not set-theoretic) to look at the foundations of mathematics and programming
Connect different fields and shows their similarities
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Why to study CT?
programming
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Why to study CT for software engineer?
Growing complexity of software systems and multicore\distributed systems revolution
CT is a source of extremely useful programming ideas
Gives you a great theoretic foundation for programming
CT is the kind of math that is particularly well suited for the minds of programmers
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What is a Category?
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Category is:
Objects
Arrows between objects
Composition rule:
( . ) :: (a → b) → (b → c) → (a → c)
g . f = \x → g (f x)
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Associativity of composition
What?
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Associativity of composition
What?
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Order of the operations doesn’t matter
Associativity of composition
Why?
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Identity Arrow
What?
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Identity Arrow
What?
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Identity Arrow
What?
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It is neutral element (‘zero’) for the composition (id x = х).
It gives us a possibility to define the notion of sameness - isomorphism:
Identity Arrow
Why?
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Examples
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Category 3
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Category 1
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Category 0
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Integer Category
Objects: singleton (set with only one element in it)
Arrows: all possible integers
Composition: + - integer addition
Identity: 0 - zero.
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String Category
Objects: singleton
Arrows: all possible strings
Composition: ++ - string concatenation
Identity: «» - empty string
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Monoids
Set-theoretic approach
Set M
Binary operation •: M × M → M
For all a, b and c in M:
(a • b) • c = a • (b • c)
Exists e in M that for all a in M:
e • a = a • e = a.
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Examples:
Monoids
Category-theoretic approach
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Monoids
Category-theoretic approach
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Set Category
Objects: all sets.
Arrows: functions between sets.
Identity: id function between sets.
Composition:
( . ) :: (A → B) → (B → C) → (A → C)
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Hask Category
Objects: types in Haskell
Arrows: functions between them (everything is pure!)
Identity: id function: id x = x.
Composition:
( . ) :: (a → b) → (b → c) → (a → c)
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Functions
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Functions in categories: Functors
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Functors: preserving composition
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Functors: preserving identities
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Functors: in brief
Maps objects to objects
Maps arrows to arrows
Preserve composition and identities
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Example
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Why is that useful?
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Wait, but what if you’ll need
one more container type ?
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How to rescue the situation?
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Conclusions and some further thoughts
CT and programming are tightly connected through composition
CT – source of useful patterns for programming: Functors, Monoids, Monads, Lenses, F-algebras, Zippers and more.
CT helps to understand FP better
CT – is in some sense – Science about thinking: the proof is the extremely wide applicability of CT – from logic to biology, from physics and CS to ecology.
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Acknowledgements
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Acknowledgements
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Acknowledgements
Richard Feynman
Jiddu Krishnamurti
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Q&A
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