Let $\mathcal C,\mathcal D,\mathcal E$ be monoidal categories, let $g$ be an oplax monoidal functor from $\mathcal C$ to $\mathcal D$ and let $G$ be a lax monoidal functor from $\mathcal D$ to $\mathcal E$.

Let $F$ be a lax monoidal functor from $\mathcal C$ to $\mathcal E$. Now let $\phi$ be a natural transformation from $F$ to $Gg$ making the following diagrams commute for any objects $x,y$ of $\mathcal C$:

$$ \require{AMScd} \begin{CD} F(x)\otimes F(y) @>{\phi_x\otimes\phi_y}>> Gg(x)\otimes Gg(y)\\ @V{m_F}VV @VV{m_G}V \\ @. G(g(x)\otimes g(y))\\ @V{m_F}VV @AA{G(m_g)}A \\ F(x\otimes y) @>>{\phi_{x\otimes y}}> Gg(x \otimes y) \end{CD} $$

$$ \begin{CD} I_{\mathcal E} @>{\epsilon_G}>> G(I_{\mathcal D}) \\ @V{\epsilon_F}VV @VV{G(\epsilon_g)}V \\ F(I_{\mathcal C}) @>>{\phi_{I_{\mathcal C}}}> Gg(I_{\mathcal C}) \end{CD} $$

Is there a name for such natural transformations? I'm tempted to call them bimonoidal natural transformations, since they seem to be analogous in some sense to the bifunctors $\mathcal C^{\text{op}}\times\mathcal C\to\mathcal D$ that are used in the definition of (co)ends.

Note: A special case of this is when $F$ is in fact strong monoidal. In that case, this is the 'obvious' idea of a monoidal natural transformation from a lax monoidal functor to an oplax monoidal functor.

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    $\begingroup$ dimonoidal, or extramonoidal, if anything :-) $\endgroup$ – Fosco Dec 22 '17 at 16:15
  • $\begingroup$ Can you maybe expand a little the analogy you see with dinatural transformations? $\endgroup$ – Fosco Dec 22 '17 at 18:21
  • $\begingroup$ @FoscoLoregian I tried to work out the details, and it didn't quite come out. The idea was to generalize from $\mathcal C^{\text{op}}\times\mathcal C\to\mathcal D$ to morphisms $A^*\otimes A\to B$ in a monoidal category with dual objects, then to generalize from there to the same thing in a bicategory rather than a monoidal category, and lastly to specialize to the case of the 2-category of monoidal categories and lax monoidal functors. The problem is that there doesn't seem to be a setting in which an oplax monoidal functor coule be regarded as the dual of a lax monoidal functor. $\endgroup$ – John Gowers Dec 22 '17 at 18:33
  • $\begingroup$ Laxity is not affected by covariance type. If $F : A \to B$ is lax monoidal, and you regard $A,B$ as one-object bicategories, then $F$ will be a lax functor; if $g$ is oplax, then it will be an oplax functor; this does not affect covariance, because if $F$ reverses 1-cells it's simply a lax functor $A^\text{op}\to B$ (you "reverse" 1-cells by redefining $\otimes : (X,Y)\mapsto Y\otimes X$; of course $A^\text{op}\cong A$ if $A$ was symmetric), if it reverses 2-cells it's a lax functor $A^\text{co}\to B$ (A^co now is A^op as 1-category, I hope you get without me fixing the abuse of notation ;) ) $\endgroup$ – Fosco Dec 22 '17 at 18:41

I don't know a good name for such transformations, but I can give you a reference and some more information about them. First of all note that more generally you can define a monoidal transformation $f F \to G g$ when $F$ and $G$ are lax monoidal and $f$ and $g$ are colax monoidal. The axioms then become hexagons (with one side degenerate in the unit case).

Such transformations may seem ad hoc, but they actually arise very naturally. They are the 2-cells in a double category whose objects are monoidal categories and whose two kinds of morphism are lax and colax monoidal functors. This double category was first defined in section 2.2 of Adjoint for double categories by Grandis and Paré, as a special case of an analogous double category whose objects are double categories.

In Comparing composites of left and right derived functors (example 4.8) I observed that an analogous double category can be defined for the algebras over any 2-monad, in which case the corresponding axiom on the 2-cells is that a certain cube commutes (the six sides of the hexagons corresponding to the six faces of the cube).

In fact this construction even sits inside something with a universal property: given a 2-category $K$, let $Q(K)$ denote the triple category of "quintets", whose objects are those of $K$, whose three kinds of morphisms are all those of $K$, whose three kinds of 2-cells are all 2-cells in $K$ ("quintets"), and whose 3-cells are commutative cubes in $K$. Any 2-monad on $K$ induces a "triple monad" $Q(T)$ on $Q(K)$, i.e. a monad in the 2-category of triple categories, triple functors, and "transversal" transformations. The triple category of algebras for $Q(T)$ (i.e. its Eilenberg-Moore object in this 2-category of triple categories) has as its objects $T$-algebras and as its three kinds of morphisms the strict, lax, and colax $T$-morphisms, while its 2-cells relating lax and colax $T$-morphisms are precisely these.

  • $\begingroup$ The nLab article for double categories calls them 'generalized natural transformations', though the Adjoints for Double Categories does not name them. $\endgroup$ – John Gowers Feb 26 at 11:04
  • $\begingroup$ @JohnGowers The nLab article says "There is a double category MonCat whose objects are monoidal categories, whose horizontal arrows are lax monoidal functors, whose vertical arrows are colax monoidal functors, and whose 2-cells are generalized monoidal natural transformations." [emphasis added] I don't think this is intended as a definition, merely an observation that they are a generalization of monoidal natural transformations. $\endgroup$ – Mike Shulman Feb 26 at 12:37

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