Initial strict monoidal category

Question

nlab provides a universal property of the cube category $\Box$.

Definition. The cube category is the initial strict monoidal category $(M,\otimes,I)$ equipped with an object $int$ together with two maps $i_0,i_1 : I \to int$ and a map $p:int→I$ such that $pi_0=1_I = p i_1$.

I will assume all categories to be small.

The existence and properties of the object int may be described as a functor $F : C \to M$, where $C$ is the category on two objects $c,x$, freely generated on two arrows $j_0,j_1 : c \to x$, and one arrow $q : x \to c$ under the relation $qj_0 = 1_c = qj_1$. Requiring $F(c) = I$ gives the same characterization of the object $int$ inside the strict monoidal category $M$.

However, I am not sure in what category one considers $(M,\otimes,I)$ (together with $int$) initial. Is it simply in the category of strict monoidal categories (under lax/strong/strict monoidal functors)? Or is it in the category of strict monoidal categories such that int-objects are mapped to int-objects, and arrows $i_0,i_1,p$ to the corresponding such arrows (under lax/strong/strict monoidal functors)?

I am assuming the last one, under strict monoidal functors.

In any case, I was wondering whether this concept has been expanded upon. In general, one may consider any pointed category $(C,c \in C)$, any strict monoidal category $(M,\otimes,I)$, and functors $F : C \to M$ such that $F(c) = I$. The category of such pairs $(F,(M,\otimes,I))$, under a suitable sense of what the morphisms should be, may contain an initial object $(F,(\Box^C,\otimes,I))$.

So the question is: does $(\Box^C,\otimes,I)$ exist, and is it possible to give it a nice description depending on $C$?

For C = *, the one-point category, one should simply retrieve $C$, with the trivial monoidal structure. For $C = (c \to x)$, the category on two objects and an arrow between them pointed at $c$, one should get $(\mathbb{N},+,0)$, where $F : C \to \mathbb{N}$ maps $c$ to $0$ and $x$ to $1$.

For $C$ the category on two objects $c,x$ and two arrows $i_0,i_1 : c \to x$ ($q$ is removed) one should get the cube category $\Box'$ without degeneracy maps, only face maps.

This seems to be a natural way of generating strict monoidal categories with a pointed category as basis. And I would expect that it has a right adjoint $U$, for a suitible sense of what categories we are working in are, such that $U$ forgets the monoidal structure of $(M,\otimes,I)$ and gives the pointed category $(M,I \in M)$.

Answer 1

Under the forgetful 2-functor $U$ from strict monoidal categories to categories, we get a description of the cube category as an initial object in the comma category $C/U$, whose objects are strict monoidal categories with a functor from your categories $C$ and whose morphisms are strict monoidal functors which commute with the structure functors from $C$ after applying $U$.

Decategorifying by one level for intuition's sake, this is like asking for an initial object in the category of monoids under a set. But this is just the free monoid on that set! More generally, initial objects of the kind described above arise in the wild as the unit $C\to UFC$ of an adjunction.

So we see that this is really a question about the existence of free strict monoidal ategories. These can be constructed in basically the same way as free monoids: take as objects, finite strings of objects from the base category, and as morphisms, sequences of morphisms between each object in a string. The monoidal unit is the empty string, etc. It's more or less obvious that this gives an adjoint (technically, 2-adjoint) to the forgetful 2-functor $U$, with unit the inclusion of length-1 strings, so that your proposed initial objects will all exist.

EDIT: The above isn't quite right for this situation, since the cube category is not free: the one object of $C$ has to be mapped to the monoidal unit $I$. One should handle this by thinking not just about free objects, but about imposing relations on them, in this case the relation $c=I$. This is now an issue of colimits within strict monoidal categories, for which I think the best approach is to generalize the notion of algebraic theory to a 2-algebraic theory, and show that the 2-categories of models of such are cocomplete. In this particular case, it's easy to describe the colimit: $c$s just disappear from strings of objects, so you have objets $x^{\otimes n}$ for every $n\geq 0$, representing the $n$-cube.