I am interested in developing intuition for when the monoidal unit in a closed monoidal abelian category is or isn't compact projective. As such, my question is not looking for a yes/no answer, but rather for classes of examples in which either a yes or no applies. I will repeat my question with more detail and examples:

**Definitions and remarks**

Recall that in an abelian category $\mathcal C$, an object $X$ is *compact* if $\hom(X,-) : \mathcal C \to \mathrm{AbGp}$ preserves filtered colimits. It is *projective* if $\hom(X,-)$ is right exact. Since all $\mathbb Z$-linear functors preserve finite direct sums and since (sm)all colimits are generated by finite direct sums, cokernels, and filtered colimits, $X$ is *compact projective* iff $\hom(X,-)$ is cocontinuous (=preserves all colimits). In a monoidal category with monoidal unit $\mathbf 1$, the functor $\hom(\mathbf 1,-)$ is the functor of *global sections* or *global elements* or *invariants*. Thus to say "$\mathbf 1$ is compact projective" is the same as to say "the functor of global sections is cocontinuous."

Recall that a monoidal structure on $\mathcal C$ is *closed* if $X\otimes : \mathcal C \to \mathcal C$ and $\otimes X : \mathcal C \to \mathcal C$ have left adjoints for all $X$; in particular, in a closed monoidal category, both $\otimes X$ and $X\otimes$ are cocontinuous. It is not hard to prove from this that in a closed monoidal category, if $\mathbf 1$ is compact projective, then so are all dualizable objects.

**Main questions**

What is a class of closed monoidal abelian categories "that appear in nature" for which the monoidal unit definitely is compact projective?

What is a class of closed monoidal abelian categories "that appear in nature" for which the monoidal unit definitely is not compact projective?

**Examples**

In any semisimple category, all objects are projective. In particular, let $G$ be a reductive algebraic group over a field of characteristic $0$. Then the category of algebraic $G$-modules is semisimple. This gives a class of examples where the answer to my question is "yes". I know essentially nothing about algebraic groups in characteristic $p$. I've been told that in general reductive groups in characteristic $p$ do not have semisimple representation theory. Do they nevertheless have $\mathbf 1$ compact projective? Under what conditions?

Consider the additive algebraic group $\mathbb G_a$, say over $\mathbb C$. Its category of algebraic modules is the category of vector spaces equipped with a locally nilpotent endomorphism. The only compact projective object is the $0$ object; in particular, the trivial module $\mathbf 1$ is not compact projective.

Let $R$ be a commutative algebra. The category of $R$-modules is symmetric monoidal with monoidal structure $\otimes_R$. The monoidal unit is $R$ acting on itself by multiplication. It is compact projective, for somewhat stupid reasons.

Let $A$ be an associative algebra. The category of $A$-$A$-bimodules is monoidal with monoidal unit $A$ acting on itself from both sides by multiplication. This is always compact. I believe, but could be mistaken, that it is projective iff $A$ is semisimple as an algebra.

For most schemes $X$, the functor of global sections $\mathrm{H}^0 : \operatorname{QuasiCoh}(X) \to \mathrm{AbGp}$ is not exact. Said another way, the monoidal unit $\mathcal O_X$ is not projective. But I don't have good intuition for naturally-appearing classes of schemes for which one can say definitively that $\mathcal O_X$ is/is not projective.

The finite-dimensional representation theory of a Drinfeld–Jimbo quantum group at generic $q$ is semisimple, so $\mathbf 1$ is compact projective. At roots of unity there are different possible choices for which representation theory to take, and I don't know all the answers.

The Temperley–Lieb category is the monoidal $\mathbb Z[q,q^{-1}]$ category freely-generated by a self-dual object of dimension $-q^2 - q^{-2}$. It is not abelian, but its abelian envelop is. Said abelian envelop has the property that the monoidal unit is compact projective. Deligne's category $\mathrm{GL}(t)$ is the abelian envelop of the free symmetric monoidal category generated by a dualizable object of dimension $t$. It also enjoys the property that the monoidal unit is compact projective. Indeed, this is a general property of categories presented by "string diagrams" with "skein relations".

There are many other categories that appear in nature. What about your favorite class?