REU Meeting - 2025-08-20

This following is a brief summary of our research meetings on 2025-08-20.

Meeting summary

We first talked about the notion of submodules in the category $F [G] -Mod$ and then showed how the notion translated to the equivalent categories ${Vec}_{F}^{G}$ and $F {-Lin}_{G}$ ^[1] There were no surprises, fortunately.

When then talked about the general notion of subobjects in categories. We saw that some categories, like $S e t$ have a special object $Ω$ called a subobject classifier (equipped with an arrow from the terminal object of the category) such that arrows to the subobject classifier are in natural bijection with subobjects. We went through the example of $S e t$ , in which $Ω = {0, 1}$ , the terminal object is ${1}$ , the map $true : {1} \to Ω$ is the set map that sends $1 \mapsto 1$ , and how for each set $X$ and subset $S \subseteq X$ , there is a unique map $f : X \to Ω$ such that the inclusion $S ↪ X$ is the pullback of the "truth" map $t : {1} \to Ω$ :

This sounds fancy, but the map $f : X \to Ω$ is simply the map

f (x) = {\begin{cases} 0, & if x \notin S \\ 1, & if x \in S \end{cases}

Then we wondered whether most categories had subobject classifiers, and I quick search of the internet (mainly nLab) told us the answer: no. In particular, categories like $G r p$ and ${V e c}_{F}$ do not have subobject classifiers.

Without such a classifier, the fallback is to viewing subobjects as "equivalence classes of monics".

For us, the takeway is that we will probably have a hard time extending the notions of "submodules" and "direct sums" to the landscape of $C$ -valued representations of a group $G$ , i.e., to categories like $C^{G}$ . So for now we'll stick to representations with values in $S e t$ , ${V e c}_{F}$ and $T o p$ .

Tasks for next meeting

Investigate the notions of subobjects in the familiar categories $S e t$ , ${V e c}_{F}$ and $T o p$ , and then extend the notions of reducible and irreducible representations to these new settings.
Recall the definition of the biproduct in a preadditive category, and also scan the general info about the notion of direct sum over at Wikipedia. Then try to extend the notions of decomposable and indecomposable to ${V e c}_{F}$ and $T o p$ . What goes wrong with $S e t$ ?

References

Preadditive categories
Direct sum - Wikipedia

I can't remember if this is the notation we settled on for $F$ -linear representations. ↩︎