Intro
Recently I started looking into the origin of computer science monad, and found out that at least some of them originate in pretty simple things. Like segments of simplicial category Δ (for a better explanation/definition, see MacLane, p.175). Here I'll give a couple of examples, Option and Reader monads, how they derive from a very simple subcategory of Δ.
0, 1, 1+1
I'm going to introduce a very simple category, consisting of three, well, sets: empty, singleton and a union of two singletons (1+1), together with two arrows,
0 → 1 and 1+1 → 1:
There's not much to talk about this category; you can think of it as living in the category of all sets, but actually any category with finite limits and colimits contains it.
Now I'll show how it generates two popular monads.
Option Monad
Given an object
X, if we add X to each part of the diagram (a), we get this:
which is exactly Option Monad.
Exception Monad
First, we multiply the diagram
(a) by E:
Then we can do the same trick as in Option Monad, add an
X:
This is Exception Monad.
Reader Monad
Given an
X, we can build the following diagram by applying X- functor to the diagram (a):
Which is, simply speaking, just
Now this can be used to build the following diagram, for any
A:
Do you see Reader diagram? It is the functor
, with all the natural properties that follow from the naturality of the morphisms above.This post is work in progress; comments are very welcome.
