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Suppose $M$ and $N$ are monoidal categories and let $M\times{N}$ denote the associated product category. $M\times{N}$ comes equipped with two natural projection functors $\pi\_{M}:M\times{N}\rightarrow{M}$ and $\pi\_{N}:M\times{N}\rightarrow{N}$.
I am interested in the pushout of the diagram of $\pi\_{M}$ along $\pi\_{N}$.
A pushout of the aforementioned diagram is a monoidal category $P$ along with two monoidal Functors $\phi\_{M}:M\rightarrow{P}$ and $\phi\_{N}:N\rightarrow{P}$ such that the diagram commutes.
I am familiar with push-outs in set theory but I'm not sure how to go about finding the push-out of one monoidal functor along another.
Any elucidation of what monoidal category would be the push-out of this diagram would be very helpful.



Pushout of the diagram of a monoidal product category along its projection functors
The answer is that it depends (immensely) on what you mean by "category of monoidal categories".


Let me start with a general statement:


Suppose $C$ is a pointed category.


Suppose further that $C$ has finite products, and let $M,N\in C$; and $P$ with maps $p: M\to P, q: N\to P$ such that $p\circ \pi\_M = q\circ \pi\_N$.


Compose with the canonical map $i\_M : M\simeq M\times \* \to M\times N$ to get $p = q\circ 0 = 0$, where $0 : M\to N$ is the $0$ morphism, i.e. the unique morphism $M\to \*\to N$.


Similarly, $q = 0$.


It follows that the pushout is trivial.


Now, whether or not the category of monoidal categories is pointed sort of depends on your definition of the latter; you have to specify a few things:


1- are you really considering the $1$-category of monoidal categories or some higher version of it where you include monoidal natural isomorphims (or even monoidal natural transformations ? - although in this case I suppose you'll have to specify what kind of "2-pushout" you mean);


2- are you considering lax monoidal functors or strong monoidal functors ? In the latter case, and if you're really interested in the $1$-category, does "strong" require the unit to be mapped to the unit strictly, or do you only require an isomorphism $1\to f(1)$ (as part of the data) ?


To see how the answers might differ:


* if you're considering the $1$-category of monoidal categories with strictly unit-preserving strong monoidal functors, then it is pointed and the above argument applies, and shows that the pushout is the trivial monoidal category.
* If you're considering "not-necessarily-strictly unit preserving strong monoidal functors" (but still as a $1$-category), then it is no longer pointed and you have to adapt the argument. It is easy to see that $p,q$ are constant, by the same argument, but they must be mapped to the same thing ($p\circ\pi\_M(1\_M,1\_N) = p(1\_M)$).


But now it's not too hard to get from there that a pushout simply does not exist: if it existed, it would have a distinguished object ($p(1\_M)=q(1\_N)$) which *isn't* the unit, and $p,q$ are constant at that object. It follows that there can in general be several functors out of $P$ with the appropriate value on $p(1\_M)=q(1\_N)$, simply by altering the value of the unit or something. So in this second case, there is no pushout. This is because we've used a notion of morphisms that is sensible to isomorphisms *within* the categories, but our $1$-category isn't.


I'll leave the "lax monoidal functors" case as an exercise to work out.


* Now what happens if we change and go to the $(2,1)$-category of monoidal categories ? Well, if we take strong monoidal functors as our morphisms, and monoidal natural isomorphisms as our $2$-morphisms, then this is exactly the category of $E\_1$-algebras in the $(2,1)$-category of categories, and in particular it *is* pointed (in the $(2,1)$-categorical sense).


But does the argument above work for "higher" categories ? Not quite as written, we've only proved that $p,q$ are trivial but there might be some higher homotopical stuff hidden. And in fact, there is: in the $(2,1)$-category of monoidal categories, you don't impose $p\circ\pi\_M = q\circ\pi\_N$, but you add *the data* of an equivalence $p\circ\pi\_M\simeq q\circ\pi\_N$. If they're both trivial, this is imposing the data of a self equivalence of the trivial functor $M\times N\to P$, i.e. a monoidal morphism $M\times N\to Aut(1\_P)$, i.e. a monoid map $\pi\_0(M)\times\pi\_0(N)\to Aut(1\_P)$, where by $\pi\_0$ I mean here something stronger than usual, namely the quotient of the object set by the equivalence relation generated by "there exists an arrow $x\to y$".


By the Eckmann-Hilton argument, $Aut(1\_P)$ is commutative, so this factors through the abelianization $(\pi\_0(M)\times\pi\_0(N))^{ab}\cong \pi\_0(M)^{ab}\times\pi\_0(N)^{ab}$, and in fact the group-completion thereof.


So the homotopy pushout should have a single object (the unit), but as automorphisms of the unit, it has $\pi\_0(M)^{ab,grp}\times \pi\_0(N)^{ab,grp}$ (this does have a monoidal structure). Let me call this group $A$ and the associated monoidal category $BA$. Then the structure maps are given by the trivial morphism $M\to BA$, the trivial morphism $N\to BA$, and the equivalence between $M\times N\to M\to BA$ and $M\times N\to N\to BA$ is given, on the object $(m,n)$, by the corresponding class of $(m,n)$ in $A = Aut(1\_{BA})$.


The details are not super important, but what this discussion shows is that the specifics of what you mean by "category of monoidal categories" will impact the answer a lot !


