I would like to create a big list of Grothendieck topoi (or Grothendieck $\infty$-topoi) which do / do not satisfy the law of excluded middle. That is, let’s list some examples of topoi whose internal logic does / does not validate that inference rule

$$ \overline{P \vee \neg P}$$

IIRC, there is a subtle distinction between double-negation elimination and LEM, but that if we’re talking about the entire topos validating the inference rule, they are equivalent, so we are equivalently asking whether these topoi do / do not validate the inference rule

$$\overline{\neg \neg P \Rightarrow P}$$

So for example, $Set$ (or $Spaces$) does validate this rule, but beyond that I am very fuzzy:

• Which presheaf topoi validate LEM?

• The sheaf topoi on which topological spaces validate LEM?

• $G$-sets for which (topological) groups validate LEM?

• …