Keith Pinson
I am a software developer who is getting into speaking.
A Little Bit of
Category Theory
Goes A Long Way
Mathematics
Abstractions without Leaks
Composition: Categories
The essence of a category is composition. Or, if you prefer, the essence of composition is a category.
—Bartosz Milewski
What is a Category?
Objects
Morphisms
Composition
Two Laws of a Category
1. Composition of morphisms is associative
2. An identity morphism exists for every object
Refresher: Associativity
("a" + "b") + "c"= "a" + "b" + "c"= "a" + ("b" + "c")
Theory Applied
Function Composition
λfλgλx.(f (g x))var compose = function(f, g) {
return function(x) {
return f(g(x));
};
};Is this composible?
Code that composes
Compiler composition
https://github.com/CraigStuntz/TinyLanguage
module Compiler
let compile =
Lexer.lex
>> Parser.parse
>> Binder.bind
>> OptimizeBinding.optimize
>> IlGenerator.codegen
>> Railway.map OptimizeIl.optimize
>> Railway.map Il.toAssemblyBuilderFunctional Programming is Math... so it must only be in its prime for stuff like financial software, right?
WRONG
Functional Programming is Math... so it must only be in its prime for stuff like financial software, right?
Algebraic Data Types
type ParseResult =
| Success of Token * string
| Failure of ErrorProducts
and
Coproducts
first : (A, B) -> A
first (a, b) = a
second : (A, B) -> B
second (a, b) = bdata AorB =
OnlyA A
| OnlyB B
My Journey Into FP
?
Map and endofunctors
map : Functor f => (a -> b) -> f a -> f b
safeLength : Maybe String -> Maybe Int
safeLength maybeStr = map length maybeStr
render : Signal UIState -> Signal HTML
render signal = Signal.map stateToHtml signalWadler and Java Generics
List v = new ArrayList();
v.add("test");
Integer i = (Integer)v.get(0);List<String> v = new ArrayList<>();
v.add("test");
Integer i = v.get(0);https://en.wikipedia.org/wiki/Generics_in_Java
Runtime: Boom!
Compiler: Uh, no.
Problem Solving
1. Compose atomic solutions
2. Lift the result a level of abstraction
3. Treat it as an atomic solution & repeat
0. Decompose big problems
Software Design
State -> Event -> (State, Event)State -> Request -> (State, Response)State -> ProjectionTesting Software
Algebra
All
the
Way
Down
For Further Reference
https://bartoszmilewski.com/2014/10/28 /category-theory-for-programmers-the-preface/
Just a Little
A Little Bit of Category Theory Goes a Long Way
By Keith Pinson
A Little Bit of Category Theory Goes a Long Way
Composition is the sweet spot in programming. The longtime wisdom in the object-oriented community (for those willing to listen) has been "Favor composition over inheritance." Functional programming is fundamentally compositional, and so is the internet. This talk will use category theory to unpack the how and why of composition and argue we should wholeheartedly embrace it at all levels of abstraction. Don't worry if you are intimidated by abstract math; I am a category theory n00b myself and this is a practical examination rather than a deep dive. (For Pittsburgh TechFest 2016)
