Free enriched monoidal categories

Question

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!

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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).

Answer 1

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?