Download for better quality. A single slide in which Adelbert Chang explains how Monoids map onto Category Theory

1. Let’s talk about how monoids map onto category theory.
I could say “any category with a single object is a monoid”
and then just walk off stage, which is what some people did
to me when I was first learning category theory, but I think
the actual explanation is more interesting and enlightening.
If you think about integers, the integers with addition and
zero as identity, and then we think about what my definition
of category was, there is nothing in that definiton that said I
can’t have multiple arrows starting and ending at a given
object.
So we’ll have one object and we’ll
just have a bunch of arrows starting
and ending at that object. And think
of these arrows not at functions but
as each element of the monoid set.
So in the case of integers, think of x
and y as 1 or as 2 or as 3 etc, as
actual integers. And categories
require that given any two arrows
where the domain and codomain
match, we should be able to form
their composition.
So the composition between two
arrows here is going to be this
monoid composition that we
talked about before.
And then similarly, the category is
going to require for every object,
in this case a single object, to have
an identity arrow, such that it is
going to be a left and right identity
with respect to arrow
composition.
That’s just going to be the monoid identity that we talked
about earlier. So in categorical language, this identity
composed with x should be the same as x, and the way we
have defined composition here is going to be the monoid
composition and hopefully we can see that the category
laws map onto the Monoid laws appropriately.
Here I started with an existing monoid, and then sort of
shown how that translates to categories, but we can also
take a category of one object and map that back to a Monoid
because the requirements for being a category are going to
be the same as the requirements for being a monoid. @adelbertchang
Simplicity in Composition
Adelbert Chang explains how monoids
map onto category theory
Category Theory
Monoid
1
2
3
4 5
6
monoid
composition
arrow
composition
identity arrow
monoid
identity