REU Meeting - 2025-08-04

This following is a brief summary of our research meeting on 2025-08-04.

Meeting summary

Mark first gave a short summary of the definition of a monoidal category, with the first example of a category of $(R,R)$-bimodules (with tensor product as the operation and $R$ as the identity object).

We then recapped how the category $\mathbf{\text{Set}}$ with Cartesian product (and identity object a singleton set) is a strict monoidal category, and how a monoid object in such a category is a conventional monoid.

We then investigated the "dual" situation in the category $\mathbf{\text{Set}}$, in which we instead use disjoint union (and identity object the empty set) for the monoidal structure. We briefly looked into monoidal objects in that monoidal category, which seemed strange.

We noted how in a category of endofunctors of a fixed category $C$, using the operation of composition (and the identity functor as the identity object), the notion of a monoid in such a monoidal category exactly gave the definition of a monad. We noted that monads naturally arise from adjoint pairs of functors, and how apparently all monads arise in such a way.

Finally, we observed that Exercise 4 in VII.1 might be useful in our situation (of categorical representation theory).

Tasks for next meeting

• (Optional) Give a nice description of monoid objects in $(\mathbf{\text{Set}},⨆,\mathrm{\varnothing })$.
• Complete Exercise 4 in Section VII.1
• Apply the result of the above exercise to the case of representations of a group $G$ in some of our favorite categories
• Write down what a means to have a monoid in ${\mathbf{\text{Set}}}^{\mathbf{\text{G}}}$. What does this mean in terms of representations?

References

Mac Lane, Categories for the Working Mathematician: Chapter VII