When is/isn't the monoidal unit compact projective?

Question

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?

Answer 1

Regarding schemes: on a scheme $X$, the functor of global sections is exact iff no quasicoherent sheaves have higher cohomology. By Serre's criterion for affineness, for reasonable schemes this is equivalent to $X$ being affine, essentially because in this case global sections is a functor $\text{QCoh}(X) \to \Gamma(X, \mathcal{O}_X)\text{-Mod}$ and exactness is one of the only obstructions to it being an equivalence.

Answer 2

I think Corollary 1.13.7 from Etingof's lectures (www.math.mit.edu/~etingof/tenscat.pdf) might be useful:

Given a multiring category (= locally finite $k$-linear abelian monoidal category with biexact tensor) with right duals, then the unit object is projective iff the category is semisimple. This is a continuation of what you said yourself in the beginning.

Also, $Rep(GL_t)$ is the Karoubian (not abelian) envelope of the free rigid symmetric monoidal category with one generator of dimension $t$. It is abelian when $t $ is not an integer, in which case it is also semisimple, so it's not very interesting in term of your question. When $t$ is an integer, the situation is more complicated - the category is not abelian.

An interesting example is the Deligne category $Rep(S_t)$. It is the Karoubian envelope of the free rigid symmetric monoidal category generated by a Frobenius-algebra-object of dimension $t$. This category is not abelian when $t$ is a non-negative integer, but it has an abelian envelope, which is a tensor category. It is not semisimple, and of course $1$ is not projective.