If $A,B$ and $C$ are compact objects in some category $\mathcal C$ is it true that $A \times B \approx A \times C$ implies $B \approx C$? It is the case for compact sets (finite sets) and finite groups although I don't know if it is true for finitely presented groups (compact groups).
If this is not true in general for compact objects then for which class of objects is it true for?
I'm also interested in the case where $A,B$ and $C$ are compact topological spaces. I realize that compact space does not imply compact object in $\textbf{Top}$ but is it still true nonetheless that $A \times B \approx A \times C$ implies $B \approx C$?
I hope you find this problem interesting!
$[0,1] \times [0,1]^{\mathbb{N}} \simeq [0,1]^2 \times [0,1]^{\mathbb{N}}$ but $[0,1] \not\simeq [0,1]^2$.