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Normally, one builds products in a category. Here I am asking about the inverse operation. Let me be precise.

Given a (monoidal) category $\mathcal{C}$ and an object $X$ of $\mathcal{C}$, does there exist objects $X_1$, $X_2$ of $\mathcal{C}$ such that $X \simeq X_1 \times X_2$, $\pi_1:X\rightarrow X_1$ and $\pi_2:X\rightarrow X_2$ satisfying the usual universal property of products.

So the question is: what are the additional conditions on $\mathcal{C}$ so that makes 'factoring' decidable: given an $X$, return either a product cone, or a witness that there are none. Naturally, this will mean first defining a notion of irreducibility, as the unit $\mathbf{I}$ always induces a 'trivial' factorization.

The motivation is a categorical study of pattern-matching, where one needs to be able to test if a (structural) property holds or not. So I am largely interested in syntactic categories where this testing is 'easy', but where I am missing the categorical infrastructure needed to make this generic.

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A brief search reveals this, which might be of relevance: – Tobias Fritz Dec 8 '12 at 15:35
Thanks @Tobias. Certainly if I was factoring in Groupoids rather than plain categories, that would be a good place to look. – Jacques Carette Dec 8 '12 at 21:34
What do you mean by factoring in Groupoids? Are you referring to the category of groupoids or about your $\mathcal{C}$ being a groupoid? I assume that you don't mean the latter, because the only groupoids that have products are the empty one and those equivalent to the trivial group... Also, your question may need to make a bit more precise. Even with fixed $\mathcal{C}$, the answer may depend on how you represent the objects as data structures. In many cases, even isomorphism is not decidable, e.g. in the category of finitely presented groups when a group is given in terms of a presentation. – Tobias Fritz Dec 10 '12 at 18:24
Your question seems worded for tensor product factorization in monoidal categories in general. But you refer to cones and a universal property, which suggests specifically cartesian product factorization. – Colin McLarty Dec 10 '12 at 18:26
@Tobias: I mean factoring in the category of groupoids. And I do realize that this will be representation-dependent: I am ok with starting with some sufficient conditions. @Colin: I originally asked for cartesian product, then realized (and edited my question) that this question made sense [albeit being vague] more generally, and did not fully edit my question to reflect that generalization. This is part of the question itself, in a sense: what question should I really be asking... This is why I added the underlying motivation. – Jacques Carette Dec 10 '12 at 19:48

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