REU Meeting - 2025-08-06

This following is a brief summary of our research meeting on 2025-08-06.

Meeting summary

Aaron walked us through the solution to Exercise 4 in VII.1, hitting on the main details of the monoidal structure in $B^C$ when $B$ is a monoidal category and $C$ is an arbitrary category. He also described the equivalence between the category $B^{C\times D}$ and $(B^C{)}^D$.

Mark then revealed a possible new reference for categorical representation theory. He's requested from our library a physical copy of this new reference.

We also briefly examined a monoidal structure on ${\mathbf{\text{Set}}}^{\mathbf{\text{G}}}$, at least in the case $\mathbf{\text{G}}={\mathbf{\text{S}}}_3$. As we ran out of time, we noted that the "box product" of two permutation representations of $G$ was likely the representation on the Cartesian product of the sets (from the original two representations), where the group $G$ acts "component-wise."

Tasks for next meeting

• Describe a monoidal structure on ${\mathbf{\text{Set}}}^{\mathbf{\text{G}}}$.
• Write down what a means to have a monoid in ${\mathbf{\text{Set}}}^{\mathbf{\text{G}}}$. What does this mean in terms of representations?
• Is there a monoidal structure on ${\mathbf{\text{Matr}}}_F$? If so, write down what it means to have a monoid in that monoidal category and describe monoids in that monoidal category.

References

Mac Lane, Categories for the Working Mathematician: Chapter VII