Any object $A$ in any topos decomposes as $A\times1$ and $1\times A$, and in $FinSet$ objects with no other product decompositions are "prime numbers". Is there any extension of the theory of product-indecomposable objects to toposes in general?

Edit: I would also be interested in any kind of theory of $\otimes$-indecomposable objects for any kind of monoidal categories.