# Has the notion of a unique factorization category been defined and studied?

I am using a this notion in a paper that I am writing. The notion is that $(\mathcal{C},\otimes, I)$ (which we will just call $\mathcal{C}$) is a symmetric tensor category, with a unit $I$. Then a prime object of $\mathcal{C}$ is an object $x$ such that if $x\sim y\otimes z$ (where $\sim$ means is isomorphic to), then $I\sim y$ or $I\sim z$. We call a category $\mathcal{C}$ a unique factorization category if all (non-unit) objects may be written as a tensor product of prime objects uniquely up to reordering the factors. My question is: has this notion been defined in the literature? If so where?

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Since ~ is a restricted character, I changed it to '\cong' – M Turgeon Sep 26 '12 at 18:33
Thank you. I realized that after I posted, and tried to edit it. It wouldn't let me since you already edited it. That's very funny. – Baby Dragon Sep 26 '12 at 18:37
Certainly unique factorization monoids have been studied, and it seems to me that you just apply the decateorification process which takes $C$ to the monoid of iso-classes in $C$. A more interesting notion would remember all the morphisms (and I can think of one ...). Of course this is no answer to your question, just a remark. – Martin Brandenburg Jan 29 '13 at 23:14
@MartinBrandenburg I agree that their certainly is a low hanging notion of what to do with morphisms. But I have an example or two in mind where it is not necessarily true that I have unique factorization on morphisms. However the morphisms in the category are used in a strong way to show that I have a UFC. I was wondering if anything in a general sense has been done. – Baby Dragon Jan 30 '13 at 15:50
Sounds cool. I'd read about it. – Alexander Gruber Jan 30 '13 at 20:13