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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.

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Why restrict to toposes? If you allow arbitrary categories with finite products then you get the rich theory of irreducible elements in (semi-)lattices. –  François G. Dorais Oct 8 '13 at 10:59
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This definition does not give irreducible elements in semilattices. Only 1 has this property in a semilattice since any other $A$ factors at least as $A\times A$. If you weaken it to say the only factors of $A$ are 1 and $A$, then it picks out elements maximal among those $\neq 1$. In a topos every $A$ equals $A\times U$ where $U$ is the support of $A$. So primes in this sense must be globally supported. There is no reason you could not study such objects but I wonder if a different definition of prime would not be more interesting. –  Colin McLarty Oct 8 '13 at 13:59
    
François, here's part of my motivation: I'd like to at least restrict my attention to categories with two monoidal structures (like product & coproduct, or tensor & biproduct), so I can think about generalizing the kinds of questions about the relationship between addition and multiplication raised by the Collatz and Goldbach conjectures. I thought toposes might be a relatively easy place to start. –  Mark Gomer Oct 8 '13 at 17:19
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If you allow the monoidal product to be biproduct then this question is certainly extensively studied for module categories and more generally abelian categories. –  Qiaochu Yuan Oct 10 '13 at 5:16
    
$\mathbb{A}^1$ is irreducible in the category of varieties (I've asked this here on MO, cannot find it right now). –  Martin Brandenburg Oct 10 '13 at 10:26

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