Let $(\cal{C},\otimes)$ and $(\cal{D},\odot)$ be two monoidal categories. Moreover, assume that $\cal{C}$ and $\cal{D}$ are abelian and semisimple. Let $X,Y$ be two simple objects in $\cal{C}$, and let $$ X \otimes Y \cong Z_1 \oplus \cdots \oplus Z_k, $$ be their decomposition into simple objects. Is there a name for a (not necessarily monoidal) functor $f:\cal{C} \to \cal{D}$ which satisfies, for all simple $X,Y$, $$ f(X) \otimes f(Y) \cong f(Z_1) \oplus \cdots \oplus f(Z_k)? $$
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$\begingroup$ Trivially every additive $f$ has that property. Do you know of any non-additive functor for which this holds? $\endgroup$– Tobias FritzCommented Jan 15, 2018 at 18:20
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$\begingroup$ At Tobias: Sorry, there was a typo in question. It's now fixed, and the property in non longer implied by additivity. $\endgroup$– Abo Kutis-FelanCommented Jan 15, 2018 at 19:32
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$\begingroup$ Ah, okay! So then you probably want $f$ to be additive as well? If so, then your property is equivalent to requiring $f(A\otimes B)\cong f(A)\otimes f(B)$ for all objects $A$ and $B$, but without postulating any coherences, right? $\endgroup$– Tobias FritzCommented Jan 15, 2018 at 20:35
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$\begingroup$ Yes, I guess additivity should be assumed, and then the condition is as you put it. $\endgroup$– Abo Kutis-FelanCommented Jan 16, 2018 at 7:58
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I don’t think there’s a well-established name, but they’re called “quasi monoidal functors” in https://arxiv.org/abs/1711.00645
answered Jan 15, 2018 at 23:07