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.

  • $\begingroup$ 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. $\endgroup$ Jul 29, 2022 at 11:00
  • $\begingroup$ 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. $\endgroup$
    – Margaret
    Jul 29, 2022 at 11:04
  • $\begingroup$ 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. $\endgroup$ Jul 29, 2022 at 11:05

1 Answer 1


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 $\kappa_Y$.

  • $\begingroup$ 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? $\endgroup$
    – Margaret
    Jul 30, 2022 at 11:35
  • $\begingroup$ 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. $\endgroup$ Jul 30, 2022 at 17:03
  • 2
    $\begingroup$ @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. $\endgroup$ Jul 30, 2022 at 21:33
  • $\begingroup$ 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? $\endgroup$
    – Margaret
    Jul 31, 2022 at 13:33
  • 1
    $\begingroup$ 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. $\endgroup$ Jul 31, 2022 at 16:55

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