Hardcore math tweet: the importance of knowing lots of small facts.

Working with Todd Trimble on category theory I'm both impressed and also annoyed by how often he can prove things I merely conjecture. Annoyed at myself, that is: why can't I do it?

(1/n)

One reason is that he's more persistent. Another is that he knows more techniques and small facts, which with persistence can be used to prove exciting things. He collects them. He really likes them.

This is how math always works. It's not just about big ideas!

(2/n)

One excuse: I did my PhD work in mathematical physics, while Todd actually studied category theory. I went into category theory much later, charmed by the big ideas. Now I'm slowly catching up on the techniques.

Collaborating with an expert is a great way to do this.

(3/n)

Here's a cute example - one of dozens. Fact: if

R: C → Set

is a right adjoint, it's representable. That is,

R(-) = hom(c, -)

for some object c ∈ C. Kind of amazing at first glance! Philosophically deep, too. But let's figure out how to prove it.

(4/n)

Let me give an example - one of dozens.

Fact: if

R: C → Set

is a right adjoint, it's representable. That is,

R(-) ≅ hom(c, -)

for some object c ∈ C.

Kind of amazing at first glance! Philosophically deep, too.

Okay - but let's figure out how to prove it.

(4/n)

How can we possibly get our hands on this magic object c ∈ C? Well, our hypotheses say R has a left adjoint

L: Set → C

That's our only way to get objects of C. So let's try

c = L(x)

for some x ∈ Set. Which x? Well, let's see....

(5/n)

We want an set x such that

hom(L(x), -) ≅ R(-)

What next? Well, L is the left adjoint of R so

hom(L(x), -) ≅ hom(x, R(-))

So we want

hom(x, R(-)) = R(-)

This is interesting. Can you see which set x makes this true?

(6/n)

Remember, if we put any object d ∈ C in the slot here

hom(x, R(-)) = R(-)

the left side is just the set of functions from x to R(d). If we take x = 1, the one-element set, then this is naturally isomorphic to R(d). So we're done!

(7/n)

So we can state our fact in a less mysterious way:

Fact: if

R: C → Set

is a right adjoint, then

R(-) ≅ hom(L(1), -)

where L: Set → C is left adjoint to R.

I must admit this would not interest me, had I not seen Todd put it to devastating use in our paper.

(8/n)

So: when you see theorems first read the statements, wrap your mind around them and see why all the hypotheses are needed. But if you want to get good at proving stuff, try to prove them! If you get stuck, read the proofs and learn the methods. Build your skills.

(9/n, n = 9) The social network of the future: No ads, no corporate surveillance, ethical design, and decentralization! Own your data with Mastodon!