# Adjoint functor theorem applied to a forgetful functor

Let $$\mathbf{Cat}$$ denote the category of small categories and $$\mathbf{MCat}$$ the category of small monoidal categories with monoidal functors. Consider the forgetful functor $$\operatorname{U}:\mathbf{MCat} \rightarrow \mathbf{Cat}$$. I want to show that this functor has a left adjoint by using a version of the adjoint functor theorem (and without giving an explicit construction of a left adjoint). I was able to verify that $$\mathbf{MCat}$$ is locally small, small complete, and that $$U$$ preserves small limits.

• If I want to apply GAFT, all that remains is to verify the solution set condition. Any ideas on how this can be done?
• If I want to apply SAFT I have to verify that $$\mathbf{MCat}$$ is well-powered and has a small cogenerating set. Is this true?

This is all self-taught, so that I am still struggling with the individual concepts. So please excuse if the questions are stupid.

• I don't know monoidal categories well. The forgetful functor from the category of monoids (commutative semi-group with $1$) has a left adjoint which sends a set to the monoids of words over the set, i.e. the "free" monoid. Commented Jul 29, 2022 at 11:00
• Thanks, but I wrote "without giving an explicit construction of a left adjoint" for a reason. I know what the left adjoint of $\operatorname{U}$ looks like. This question is rather intended as an excercise in applying "the" adjoint functor theorem. Commented Jul 29, 2022 at 11:04
• It's easy to check the conditions of adjoint functor theorem for monoids. I guess it's not that hard to transfer to small monoidal categories. Commented Jul 29, 2022 at 11:05

Let's say that a monoidal subcategory $$Z$$ of a monoidal category $$(X,\otimes,I)$$ is one for which

1. Morphisms $$f,g\in Z$$ imply $$f\otimes g\in Z$$
2. Objects $$A,B,C\in Z$$ imply the associators $$(A\otimes B)\otimes C\to A\otimes(B\otimes C)$$ are in $$Z$$
3. $$I\in Z$$
4. An object $$A\in Z$$ implies the unitors $$I\otimes A\cong A\cong A\otimes I$$ are in $$Z$$.

Evidently the intersection of families of monoidal subcategories is a monoidal subcategory. Therefore, given a functor $$F\colon Y\to X$$ there is a smallest monoidal subcategory $$MF(Y)$$ containing the image of $$F(Y)$$. In particular, $$F\colon Y\to X$$ factors as $$Y\to MF(Y)\hookrightarrow X$$. Moreover, $$MF(Y)\hookrightarrow X$$ is a monoidal functor, i.e. a functor in the category of small monoidal categories.

I now claim the morphisms in this subcategory are exactly the composites of morphisms of the form $$F(f_1)\otimes F(f_2)\otimes...\otimes F(f_n)$$ (wih various parenthesizations) for $$f_i$$ morphisms in $$X$$, and appropriate unitors and associators.

It follows that $$MF(Y)$$ has cardinality bounded by $$\kappa_Y$$, where $$\kappa_Y$$ is the smallest infinite cardinal bounding the cardinality of $$Y$$.

If $$\lambda$$ is its cardinal, then the isomorphism $$\lambda\cong UM(Y)$$ of $$\lambda$$ with the set of morphisms of $$MF(Y)$$ induces a monoidal category structure on $$\lambda$$ so that the resulting monoidal category is isomorphic to $$MF(Y)$$.

Thus the functor $$F\colon Y\to X$$ factors as $$Y\to M\to X$$ where $$M\to X$$ is a monoidal functor, and $$UM$$ is a cardinal bounded by $$\kappa_Y$$. Since the set of cardinals bounded by $$\kappa_Y$$ is a set, and since each set has a set of monoidal structures, and since between any two categories there is a set of functors between them, it follows that for every category $$Y$$ there is only a set of functors $$Y\to M$$ where $$M$$ is a monoidal category with $$UM$$ a cardinal bounded by $$\kappa_Y$$.

By the previous discussion, this is a solution set for the forgetful functor from small monoidal categories to small categories: any functor $$F\colon Y\to X$$ factors as $$Y\to M\to X$$ where $$M\to X$$ is a monoidal functor with $$UM$$ a cardinal bounded by $$\kappa_Y$$.

• I can't quite follow. Using the notation from the nLab, are you suggesting the following? Let $Y$ be a category and $X$ a monoidal category. Let $\kappa$ be the smallest infinite cardinal bounding the set $\operatorname{U}(X)$. Then a solution set $\{X_i\}_{i\in I}$is given by the set of monoidal categories which have a morphism set that is bounded by $\kappa$? What are the morphisms $f_i: Y \rightarrow U(X_i)$? Why does there exist an $i$ and a $t_i:X_i \rightarrow$ such that the solution set condition is satisfied? Commented Jul 30, 2022 at 11:35
• My original answer was imprecise: the solution is the set of functors from $Y$ to monoidal categories whose underlying set is a cardinal bounded by $\kappa$. I've edited my answer to provide more detail. Commented Jul 30, 2022 at 17:03
• @Margaret It might be a case that’s easier to understand if you write out the details on your own. Basically, a functor from a fixed category $C$ to the underlying category of any monoidal category must factor through some monoidal category not much bigger than $C,$ essentially since the operations of a monoidal category have only finitely many inputs—you can only combine finitely many morphisms of $C$ at a time in a small set of different ways. Commented Jul 30, 2022 at 21:33
• Thank you, your edit made things clearer. A few things I am still unsure about: When you speak of "the cardinality of a (small) category“, you mean the cardinality of the set of morphisms in that category, right? In other words, it seems like from "It follows that $MF(Y)$ has cardinality …" onwards you are switching to the " arrows-only definition" of a category, right? Commented Jul 31, 2022 at 13:33
• Yes: cardinality of a category is the cardinality of its set of morphisms. This style of argument ic certainly standard: see the end of V.7 in Mac Lane's Categories for the Working Mathematician. Commented Jul 31, 2022 at 16:55