REU Meeting - 2025-06-30

This following is a brief summary of our research meeting on 2025-06-30.

Meeting summary

We began by briefly chatting about the tasks set at the end of the last meeting. We noted the following:

• A matrix representation of $G$ is a group morphism $G\to {\mathrm{GL}}_n(F)={\mathrm{Aut}}_{{\mathbf{\text{Mat}}}_F}(n)$.
• An $F$-linear representation of $G$ is a group morphism $G\to \mathrm{GL}(V)={\mathrm{Aut}}_{{\mathbf{\text{Vec}}}_F}(V)$.
• A permutation representation of $G$ is a group morphism $G\to S_X={\mathrm{Aut}}_{\mathbf{\text{Set}}}(X)$.
• In general, a representation of $G$ by an object $c$ in a category $C$ is a group morphism $G\to {\mathrm{Aut}}_C(c)$. (
  Note that in any category $C$, the set of automorphisms of an object $c$ in $C$ is naturally a group under composition.)

Aaron noted that representations of $G$ can also be thought of as actions of $G$ (on the corresponding objects). This led to the slightly generalization to representations of monoids, as monoid morphisms $M\to {\mathrm{End}}_C(c)$.

We briefly recapped the "categorification" of a group $G$ as a one-object category, $\mathbf{\text{G}}$. We noted that this same construction could apply to any monoid.

We then roughly sketched out the equivalence of the following three categories:

1. the category of $F$-linear representations of a group $G$. In this category, the objects are pairs $(V,\rho )$ where $V$ is an $F$-vector space and $\varphi :G\to \mathrm{GL}(V)$ is a group morphism, and the arrows are intertwining maps;
2. the category of $F[G]$-modules, whose objects are $F[G]$-modules and arrows are module morphisms; and
3. the category ${\mathbf{\text{Vec}}}_F^{\mathbf{\text{G}}}$, whose objects are functors $\mathbf{\text{G}}\to {\mathbf{\text{Vec}}}_F$ and arrows are natural transformations.

Our plan is to translate everything (as much as we can) from the first two categories into statements in the third category.

Tasks for next meeting

1. Carefully describe the "categorification" construction.
   • Can you define a "categorization" functor $\mathbf{\text{Grp}}\to \mathbf{\text{Cat}}$? Make sure to describe what happens to arrows in $\mathbf{\text{Grp}}$, i.e., group morphisms.
   • What is the "image" of this functor?
   • Verify that functors $\mathbf{\text{G}}\to \mathbf{\text{H}}$ correspond to group morphisms $G\to H$.
   • Do you see how to similarly "categorify" any monoid?
2. Explicitly describe the functors between the three categories listed above.\
   • Make sure to verify that arrows in the third (functor) category correspond to intertwiners.
   • Convince yourselves the categories are equivalent.
   • Are any of the categories isomorphic?

References

Dummit & Foote: Section 18.1
Mac Lane: Chapter 1