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There are two principal ways to define a monoidal category:

  • The biased definition includes a unit object $I$, a binary tensor product $A\otimes B$, and a ternary associativity isomorphism $(A\otimes B)\otimes C\cong A\otimes (B\otimes C)$ and unit isomorphisms satisfying appropriate axioms.
  • The unbiased definition includes an $n$-ary tensor product $(A_1\otimes\cdots \otimes A_n)$ for all $n\ge 0$ (where $n=0$ gives the unit $I = ()$), with associativity isomorphisms such as $((A\otimes B) \otimes () \otimes (C)) \cong (A\otimes B\otimes C)$ satisfying appropriate axioms.

The two definitions are equivalent in an appropriate sense (though this is a nontrivial coherence theorem). However, this is no longer true for "lax" kinds of monoidal category, where the associativity and unit isomorphisms are replaced by not-necessarily-invertible transformations. In the lax case, the unbiased definition seems to be more-studied, and is usually what people mean by a "lax monoidal category". There are good reasons for this, but "biased-lax" monoidal categories, and more general biased-lax structures, do occasionally pop up.

In the unbiased case, there are only two consistent choices of direction for the transformations: $((A\otimes B) \otimes () \otimes (C)) \to (A\otimes B\otimes C)$ gives a lax monoidal category, while the opposite direction gives a colax one. In the biased case, there are more choices: in addition to choosing $(A\otimes B)\otimes C\to A\otimes (B\otimes C)$ or the opposite, we can choose how to orient the two unit morphisms: either $A \otimes I \to A$ or $A \to A\otimes I$, and also either $I\otimes A \to A$ or $A\to I\otimes A$. For instance, a skew monoidal category pairs $A\to I\otimes A$ with $A\otimes I\to A$. (Thanks Maxime for pointing this out in the comments.)

In this question I am interested in biased-lax monoidal categories where the unit transformations go in the same direction, say $A\otimes I\to A$ and $I\otimes A\to A$. It seems that it should be possible to identify a biased-lax monoidal category of this sort with a particular kind of unbiased-lax one, by defining the $n$-ary tensor product in terms of the binary one by right-associativity: $(A_1\otimes\cdots \otimes A_n) = (A_1 \otimes (A_2 \otimes \cdots \otimes (A_{n-1}\otimes A_n)\cdots ))$ (or perhaps left associativity, depending on which direction the biased-lax associativity map goes). I have seen this claimed in print, and have even claimed it myself, but I have not seen a proof written out. So my questions are:

  1. Has anyone studied biased-lax monoidal categories of this sort (or related structures like biased-lax bicategories, biased-lax monoids in a monoidal bicategory, etc.) in detail?
  2. In particular, is there a better name for them? (The only reference I know of is the paper "$T$-categories" by Albert Burroni, who called "biased-colax" bicategories of this sort "pseudo-categories" — clearly not a good name in light of modern terminological conventions.)
  3. (The main question) Has anyone written out a proof that biased-lax monoidal categories of this sort can be identified with certain unbiased-lax ones?
  4. What is an intrinsic characterization of the unbiased-lax monoidal categories that arise in this way? (I expect they should be the ones such that certain of the associativity maps happen to be isomorphisms.)
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    $\begingroup$ Re 2: Skew monoidal categories are an example of biased-monoidal categories which seem to 'ping' a bit more when searching google, but it depends on how you want to orient the unit morphisms. $\endgroup$ – Maxime Lucas May 30 '18 at 10:05
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    $\begingroup$ @MaximeLucas Thanks! I should have mentioned skew-monoidal categories and specified the direction of unit morphisms; I'll edit the question. I think both unit morphisms have to go in the same direction for my question (3) to be true, which is not the case for skew-monoidal categories. (The actual direction is then arbitrary, up to passage to opposite categories.) $\endgroup$ – Mike Shulman May 30 '18 at 14:45
  • $\begingroup$ There seems to be an answer to questions 3 and 4 for skew monoidal categories in arxiv.org/abs/1708.06087: in that case, unbiased-(co)lax monoidal categories have to be replaced by (co)lax algebras over a slightly fancier Cat-operad, and there is a "certain of the associativity maps are identities" condition called being an "LBC-algebra" that characterizes the lax algebras arising from skew-monoidal categories in this way. $\endgroup$ – Mike Shulman May 31 '18 at 20:33
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    $\begingroup$ That is right. You can restrict this to obtain an equivalence between left normal skew monoidal categories (those for which $l:ia \to a$ is invertible) and normal colax monoidal categories satisfying the LBC condition (when viewed as colax $L$-algebras). (The corresponding multicategory result was described in Theorem 6.3 of arxiv.org/abs/1708.06088) $\endgroup$ – john Jun 1 '18 at 7:47
  • $\begingroup$ This special case about left normal skew monoidal categories would, I suspect, also be a special case of the result you are after about biased (co)lax monoidal categories. $\endgroup$ – john Jun 1 '18 at 7:54

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