# Strongly unbiased symmetric monoidal category

Let $\mathcal{C}$ be a category. Define a strongly unbiased symmetric monoidal structure on $\mathcal{C}$ to be a rule which associates to every finite set $I$ a functor $\mathcal{C}^I \to \mathcal{C}$, written $(A_i)_{i \in I} \mapsto \bigotimes_{i \in I} A_i$ on objects (and likewise on morphisms), equipped with natural isomorphisms $$\bigotimes_{i \in \{j\}} A_i \cong A_j$$ $$\forall \sigma : I \xrightarrow{\cong} J :~~ \bigotimes_{\large i \in I} A_{\sigma(i)} \cong \bigotimes_{\large j \in J} A_j$$ $$\forall (J_i)_{i \in I} : ~~\bigotimes_{\large i \in I} \bigotimes_{\large j \in J_i} A_j \cong \bigotimes_{\large j \in \coprod_{i \in I} J_i} A_j$$ subject to some coherence conditions (which should be easy to work out).

Question 1. Where does this definition appear in the literature? Perhaps with another name?

Question 2. There is an obvious notion of morphism between strongly unbiased symmetric monoidal structures; and likewise for the usual biased version. What is a reference for the result that there is an equivalence of categories between (unbiased symmetric monoidal structures on $\mathcal{C}$) and (symmetric monoidal structures on $\mathcal{C}$)?

I do not ask how to prove this (probably it is merely a restatement of the Coherence Theorem for symmetric monoidal categories); instead, I would like to ask for a detailed reference which is at best citable.

Question 3. There is a $2$-category of strongly unbiased symmetric monoidal categories. For example, a lax morphism between strongly unbiased symmetric monoidal categories $(\mathcal{C},\otimes) \to (\mathcal{D},\otimes)$ is a functor $F : \mathcal{C} \to \mathcal{D}$ equipped with natural morphisms $\bigotimes_{i \in I} F(A_i) \to F(\bigotimes_{i \in I} A_i)$, subject to some coherence conditions. Is there a reference for the result that this $2$-category is equivalent to the $2$-category of (biased) symmetric monoidal categories?

Notice that this was already asked here, but the answers are not satisfactory. Therefore let me also remark that, in this case, I am not interested in "related" or "similar" questions. Notice that Questions 1 and 2 are partially answered by Proposition 1.5 in Deligne-Milne's Tannakian categories, but they only indicate the results without proper reasoning and without giving the strongly unbiased structures a proper name.

• In the definition of a strongly unbiased symmetric monoidal category, there are 7 coherence diagrams. – Martin Brandenburg Mar 9 '16 at 15:57
• We'll see if any answers turn up, but I wouldn't be surprised at all if this were unwritten folklore. – Omar Antolín-Camarena Mar 14 '16 at 19:19
• I'm not sure that for Question 2 you really need to use the coherence theorem; it feels more like proving that the category of finite sets is equivalent to it's skeleton. Pick for each finite set $I$ a bijection $\beta_I : I \to \{1, 2, \cdots, |I|\}$ so that $\beta_{\{1,2,\cdots,n\}}$ is the identity and use those $\beta_I$'s to construct a strongly unbiased symmetric monoidal category given an unbiased one. I haven't thought it through but I wouldn't expect you need coherence to prove this is an equivalence (I could easily be wrong). – Omar Antolín-Camarena Mar 14 '16 at 19:25
• Did this ever get worked out? I need it for a lecture I'm giving tomorrow. – John Baez Jan 17 '17 at 6:59
• @JohnBaez: A year ago I started to write up what I had proven so far (hence, this is far from being complete or proofread!): dropbox.com/s/seffmb47ipj9so5/sym-mon-cat-unbiased.pdf?dl=0 – Martin Brandenburg Apr 12 '17 at 13:56