Monoidal categories and Generators

Question

Let $\mathcal{C}$ be a Cocomplete Cowellpowered Monoidal category. Does $\mathcal{C}$ need to have a generator? I think it does not, but it seems hard to get a counter example.

Answer 1

Let $\mathcal{C}$ be the following category:

• The objects are pairs $(I, X, p)$ where $I$ and $X$ are sets and $p : X \to I$ is a surjection.
• The morphisms $(I, X, p) \to (J, Y, q)$ are maps $f : X \to Y$ such that $q \circ f = p$. (That means we must have $I \subseteq J$ to have a morphism.)
• Composition and identities are inherited from $\mathbf{Set}$.

I leave it to you to verify that $\mathcal{C}$ is cocomplete and cowellpowered. (Hint: There is a fully faithful embedding $\mathcal{C} \to [\operatorname{ob} \mathbf{Set}, \mathbf{Set}]$.) Any cocomplete category is monoidal under coproduct, of course, but if you insist on an interesting monoidal structure, let me also point out that $\mathcal{C}$ is cartesian closed.

On the other hand, $\mathcal{C}$ has no generator. Indeed, given any set $\{ (I_a, X_a, p_a) : a \in A \}$ of objects in $\mathcal{C}$, we can find an object $(J, Y, q)$ such that $I_a \cap J = \emptyset$ for any $a \in A$, which implies that there is no morphism $(I_a, X_a, p_a) \to (J, Y, q)$ for any $a \in A$.

Perhaps it should also be pointed out that $\mathcal{C}$ is even an infinitary pretopos, i.e. satisfies all of Giraud's axioms except for the existence of a generator. So $\mathcal{C}$ has many good properties.