  It has been proven by Serre, Larsen-Pink and Chin, that over a smooth curve
over a finite field, the monodromy groups of compatible semi-simple pure lisse
sheaves have "the same" $\pi_0$ and neutral component. We generalize their
results to compatible systems of semi-simple lisse sheaves and overconvergent
$F$-isocrystals over arbitrary smooth varieties. For this purpose, we extend
the theorem of Serre and Chin on Frobenius tori to overconvergent
$F$-isocrystals. To put our results into perspective, we briefly survey recent
developments of the theory of lisse sheaves and overconvergent $F$-isocrystals.
We use the Tannakian formalism to make explicit the similarities between the
two types of coefficient objects.
