REU Meeting - 2025-08-15

This following is a brief summary of our research meetings on 2025-08-14 and 2025-08-15.

Meeting summary

August 14

On August 14, Aaron carefully described the monoidal structure on ${Set}^{G}$ , which can be summed up as "inheriting the monoidal structure of $⟨ Set, \times, {∙} ⟩$ on the outputs of functors $F : G \to Set$ ." We then spent most of our time analyzing what information is contained in a monoid object in the monoidal category ${Set}^{G}$ . Our eventual conclusion was that such a monoidal object was equivalent to a functor $F^{'} : G \to Mon$ . More precisely, we made the conjecture:

Conjecture 1

There is an equivalence of (monoidal?) categories between the category of monoids in ${Set}^{G}$ and the category ${Mon}^{G}$ .

In the language of our representation theory, this amounts to saying that a monoid in the category of permutation representations of $G$ is effectively the same as a monoidal representation of $G$ .

Aaron also took a moment to posit that there was indeed a monoidal structure on ${Matr}_{F}$ , namely $⟨ {Matr}_{F}, \otimes, 1 ⟩$ .

We then moved on to consider the unusual monoidal category $⟨ Set, ∐, \emptyset ⟩$ , specifically analyzing how to interpret a monoid in that monoidal category. After analyzing the two commutative diagrams required for a monoid, we discovered that the "unit" diagram actually forced the monoid map $μ : X ∐ X \to X$ to be the "folding map", which simplify identifies the two copies of $X$ with the original set $X$ . Since the "unit map" $λ : \emptyset \to X$ was the empty map, this led us to conclude that a monoid in this monoidal category was simply a set (with no real additional information). More precisely, we made a second conjecture:

Conjecture 2

There is an equivalence of (monoidal?) categories between the category of monoids in $⟨ Set, ∐, \emptyset ⟩$ and the category $Set$ .

August 15

We spent a good chunk of this meeting going over the slides for the "Share-Out" happening later that day. After finishing that, we recapped our discoveries from the previous day and then brainstormed future directions of study. We decided that our next short-term goals are:

Translate the notations of submodules and direct sums of modules into the categories ${Vec}_{F}$ and ${Matr}_{F}$ , and then extend the notions of irreducible and indecomposable to representations of a group into those categories. Our long-term aim in this direction is to see how to extend/define these notions for arbitrary $C$ -representations of a group, and then state (and perhaps prove) a general version of Maschke's Theorem.
Study the general categorical concept of a Kan extension, and then see how induced representations can be viewed as nothing more than Kan extensions of restriction functors. The long-term aim in this direction is to extend the notion of induced representations (including their constructions) to the most general possible setting.

Tasks for next meeting

Translate the notations of submodules and direct sums of modules into the categories ${Vec}_{F}$ and ${Matr}_{F}$ , and then extend the notions of irreducible and indecomposable to representations of a group into those categories.
Read about the general categorical definitions of subobjects and direct sums.

References

Mac Lane, Categories for the Working Mathematician

Subobjects: See page 105 and onwards
Direct sums: See page 195 for the connection between direct sums and the more general notion of biproducts in additive categories