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I’ve been inactive here for some months now; I hope this will significantly change soon.
I have written a stubby beginning of iterated monoidal category, with what is admittedly a conjectural definition that aims to be slick. I am curious whether anyone can help me with the following questions:
Is the definition correct (i.e., does it unpack to the usual definition)? If so, is there a good reference for that fact?
Assuming the definition is correct, it hinges on the notion of normal lax homomorphism (between pseudomonoids in a 2-category with 2-products). Why the normality?
In other words (again assuming throughout that the definition is correct), it would seem natural to consider the following type of iteration. Start with any 2-category with 2-products $C$, and form a new 2-category with 2-products $Mon(C)$ whose 0-cells are pseudomonoids in $C$, whose 1-cells are lax homomorphisms (with no normality condition, viz. the condition that the lax constraint connecting the units is an isomorphism), and whose 2-cells are lax transformations between lax homomorphisms. Then iterate $Mon(-)$, starting with $C = Cat$. Why isn’t this the “right” notion of iterated monoidal category, or in other words, why do Balteanu, Fiedorowicz, Schwänzel, and Vogt in essence replace $Mon(-)$ with $Mon_{norm}(-)$ (where all the units are forced to coincide up to isomorphism)?
Apologies if these are naive questions; I am not very familiar with the literature.
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