© 2015 Pivotal Software, Inc. All rights reserved.© 2015 Pivotal Software, Inc. All rights reserved. 1

Category Theory for Mortal Programmers

Stephan February

@beardpappa

sfebruary@pivotal.io

http://hashbangbin.sh

© 2016 Pivotal Software, Inc. All rights reserved. 2

© 2015 Pivotal Software, Inc. All rights reserved. 3

What is Category Theory

• Abstraction • Objects • Arrows (morphisms)

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Relation to Programming

f : A => B

g : B => C

g o f ~ g( f(A)) : A => C

domain(f)

codomain(f)

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Relation to ProgrammingWhat makes something a Category ?

composition :

f: A => B and g : B => Cgiven

g o f : A => C∃

objects :

morphisms : f, g

A, B, C

identity : id1: A => A or id2 : B => B

id1

id2

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Category Laws

id1

id2

Associative Law

Identity Law

(h o g) o f = h o (g o f)

f o id1 = id1 o f = f

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FunctorsCategories : Functional Design Patterns

Apply a function f: x: Int => x2 to the sequence [1,2,3,4,5]

[1,2,3,4,5].map{ x => x * x }

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Functor

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Functor Recap

(A => B) (T[A] => T[B])

General Shape

Our Category

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Monads (homework) Categories : Functional Design Patterns

(A => T[B]) (T[A] => T[B])flatMap

General Shape

ReferencesProgramming Scala 2nd Edition

StackOverFlow: A Monad Is Just a Monoid in the Category of Endofunctors. Whats The Problem ?

YouTube: Category Theory Lulz - Ken Scambler

YouTube: Category Theory, The Beginner's Introduction

Blog: Functors, Monads, Applicatives can be so simple

An Introduction to Category Theory