Hardcore math tweet:

The "projection formula" or "Frobenius law" shows up in many branches of math, from logic to group representation theory to the study of sheaves.

Let& #39;s see what it means in an example!

(1/n)

Given any group homomorphism f: G → H you can "restrict" representations of H along f and get representations of G.

But you can also take reps of G and freely turn them into reps of H: this is "induction".

Induction is the left adjoint of restriction.

(2/n)

You can get a lot of stuff just knowing that induction is the left adjoint of restriction: this is called "Frobenius reciprocity".

But you can also take tensor products of representations, and then a new fact shows up: the projection formula!

(3/n)

Where does this mysterious fact come from? It comes from the fact that the category of representations of a group is a symmetric monoidal *closed* category, and restriction preserves not only the tensor product of representations, but also the internal hom!

(4/n)

You get the projection formula whenever you have adjoint functors between symmetric monoidal closed categories and the right adjoint is symmetric monoidal and also preserves the internal hom! The 6-line proof uses the Yoneda lemma:

#InWirthmuellerContexts">https://ncatlab.org/nlab/show/Frobenius+reciprocity #InWirthmuellerContexts

(5/n)">https://ncatlab.org/nlab/show...

This may not help you much unless you know some category theory and/or you& #39;ve wrestled with the projection formula and wondered where it comes from. But to me it& #39;s a tremendous relief. I thank Todd Trimble for getting to the bottom of this and explaining it to me.

(6/n, n=6)