# Free enriched monoidal categories

Suppose $$(\mathcal{V},\otimes,1)$$ is a symmetric monoidal category and $$\mathbb{C}$$ is a $$\mathcal{V}$$-category. I will deliberately avoid usual powerful assumptions (eg completeness/cocompleteness) on $$\mathcal{V}$$.

If $$\mathcal{V}$$ contains an object $$0$$ such that there is a family of morphisms $$X\otimes 0 \to 0$$ natural in $$X$$, it is easy to construct an enriched monoidal $$\mathcal{V}$$-category $$\mathcal{M}(\mathbb{C})$$ over $$\mathbb{C}$$ analogous to the construction of the free monoidal category on an ordinary category, as follows. Its objects are finite tuples of objects of $$\mathbb{C}$$, while the hom-objects are:

$$\mathcal{M}(\mathbb{C})((c_1,\dotsc,c_m),(c'_1,\dotsc,c'_n)) = \begin{cases} \mathbb{C}(c_1,c'_1) \otimes \cdots \otimes \mathbb{C}(c_m,c'_m) & \text{if }n=m; \\ 0 & \text{otherwise} \end{cases}$$

Composition is defined component-wise after applying symmetries in $$\mathcal{V}$$ to line things up. If $$0$$ is initial, we get a canonical factorization of any $$\mathcal{V}$$-functor from $$\mathbb{C}$$ to a monoidal $$\mathcal{V}$$-category $$(\mathcal{E},\oplus,I)$$ through a (strong) monoidal $$\mathcal{V}$$-functor $$\mathcal{M}(\mathbb{C}) \to \mathcal{E}$$. If I'm not mistaken, this factorization is unique and this really is the free monoidal $$\mathcal{V}$$-category over $$\mathbb{C}$$.

Question: Can free monoidal $$\mathcal{V}$$-categories exist under other circumstances?

It seems to me that the above hypothesis has at least some necessary components. Let $$\mathbf{I}$$ denote the unit $$\mathcal{V}$$-category, with one object $$*$$ and $$\mathbf{I}(*,*) = 1$$, the unit in $$\mathcal{V}$$. Suppose the free monoidal $$\mathcal{V}$$-category on $$\mathbf{I}$$ exists, say $$(\mathcal{F},\oplus,I)$$, and consider $$Z := \mathcal{F}(I,*)$$.

If given any object $$H$$ of $$\mathcal{V}$$ it is possible to construct a monoidal $$\mathcal{V}$$-category $$(\mathcal{E},\oplus,I)$$ containing an object $$X$$ such that $$\mathcal{E}(I,X) = H$$, we can deduce that $$Z$$ is at least weakly initial. For the time being I can only easily do this for objects in $$\mathcal{V}$$ which are idempotent with respect to $$\otimes$$.

Meanwhile, assuming free monoidal $$\mathcal{V}$$-categories exist on more general $$\mathcal{V}$$-categories, we should recover the existence of the claimed morphisms $$H \otimes 0 \to 0$$.

It feels like I might be missing counterexamples where more interesting things happen or a slick proof via a suitable Yoneda embedding. Any references where these ideas are discussed would be appreciated!

Update: I found a counterexample (see answers) which actually shows that the construction above need not even be a valid definition of a $$\mathcal{V}$$-category for general $$\mathcal{V}$$! I am now curious when free monoidal $$\mathcal{V}$$-categories exist under the additional hypothesis that every object of $$\mathcal{V}$$ can appear as a hom-object of some $$\mathcal{V}$$-category (which avoids that issue).

A silly counterexample can be obtained by considering $$\mathcal{V} = \mathrm{FinSet}_{\mathrm{bij}}$$, the category of finite sets and bijections, with product of sets as the monoidal operation. Formally, the only enriched categories for this choice of $$\mathcal{V}$$ are trivial groupoids: a collection of objects where every hom-object is a singleton set $$1$$. This is because we require an enriched category to be equipped with 'identity morphisms' $$1 \to \mathbb{C}(c,c)$$ (which forces the latter to be isomorphic to $$1$$) and composition morphisms $$\mathbb{C}(c',c) \otimes \mathbb{C}(c,c') \to \mathbb{C}(c,c)$$, which forces each of the hom-objects in the domain to be isomorphic to $$1$$ also.

These are all equivalent, but as regards my question, the free symmetric monoidal $$\mathcal{V}$$-category on $$\mathbb{C}$$ exists: it is the trivial groupoid whose objects form the free monoid on the collection of objects of $$\mathbb{C}$$.

I say this example is 'silly' because the axioms of enriched categories mean that we're "actually" enriching over the trivial $$1$$-object category here, and if I enriched over that instead then my original observation becomes valid again (we just have $$0=1$$ in this case). To eliminate this case, let's generalize the above argument about which objects of $$\mathcal{V}$$ can actually arise as hom-objects.

Definition: An object $$Z$$ in $$\mathcal{V}$$ is a monoid mediator if there exist monoids $$M,N$$ in $$\mathcal{V}$$ and an object $$Y$$ such that $$Z$$ can be equipped with a right-$$M$$-action $$Z \otimes M \to Z$$ and a left-$$M$$-action $$N \otimes Z \to Z$$ and dually for $$Y$$, and there exist morphisms $$Y \otimes Z \to M$$ and $$Z \otimes Y \to N$$ respectively coequalizing (not universally) the maps $$Y \otimes N \otimes Z \rightrightarrows Y \otimes Z$$ and $$Z \otimes M \otimes Y \rightrightarrows Z \otimes Y$$.

Every hom-object $$\mathbb{C}(c,c')$$ in a $$\mathcal{V}$$-category is a monoid mediator, with $$M = \mathbb{C}(c,c)$$, $$N = \mathbb{C}(c',c')$$ and $$Y = \mathbb{C}(c',c)$$, their monoid structures and actions given by identities and composition; conversely given any monoid mediator we can construct a $$2$$-object $$\mathcal{V}$$ category in which is appears as a hom-object.

With this definition, we can replace $$\mathcal{V}$$ with its full subcategory on the monoid mediators. Using the fact that $$\mathcal{V}$$ was symmetric monoidal, we can deduce that this is a monoidal subcategory (and this operation is idempotent, since all monoids are monoid mediators). Note that if $$\mathcal{V}$$ was monoidal closed, then every object was already a monoid mediator of quite a special form!

Conjecture: If $$Z$$ is a monoid mediator in $$\mathcal{V}$$ then there exists a monoidal $$\mathcal{V}$$-category $$(\mathcal{E},\oplus,I)$$ and an object $$X$$ such that $$\mathcal{E}(I,X) \cong Z$$.

I believe I can prove this in the case that $$\mathcal{V}$$ is cartesian monoidal, since in that case we can assume that $$M = N$$ in the definition of monoid mediator by replacing both with $$M \times N$$ and projecting out the redundant component or inserting a unit in each of the morphisms demanded. Then we can construct a monoidal category in which every endomorphism monoid is $$M \times N$$ by gluing together an $$\mathbb{N}$$-indexed number of copies of the two-object category alluded to above. But I do not know if the above conjecture is realistic in general.

If the above conjecture holds, we can deduce (using the notation from the original question) that if the free category $$(\mathcal{F},\oplus,I)$$ on $$\mathbb{I}$$ exists, then $$\mathcal{F}(I,*)$$ must be weakly initial in the subcategory of monoid mediators. It seems that the morphism should be unique, but without a $$\mathcal{V}$$-indexed functorial family of witnesses we cannot deduce that $$\mathcal{F}(I,*)$$ forms a cone over the identity...

Modified question:

• Is there a symmetric monoidal category $$\mathcal{V}$$ in which every object is a monoid mediator but some object $$X$$ fails to appear as a hom-object of the form $$\mathcal{E}(I,X)$$ in a monoidal $$\mathcal{V}$$-category $$(\mathcal{E},\oplus,I)$$?
• Can $$\mathcal{F}(I,*)$$ fail to be initial even when the conjecture holds?