Lower Algebra: Modules over the monoidal category of abelian groups

Question

Proposition 6.3.2.18 of Higher Algebra identifies $Mod_{Sp}(Pr^L)$, the symmetric monoidal category of right modules over the monoidal category $Sp$ of spectra in $Pr^L$ the category of presentable categories, with the full subcategory $Pr^L_{St}$ of stable presentable infinity categories. In particular Lurie proves that the functor $-\otimes Sp: Pr^L \to Pr^L_{St}$ is a localization functor.

Question 1: Is there an analogue for ordinary (non $\infty$) categories? In other words, is $-\otimes Ab: Pr^L \to Mod_{Ab}(Pr^L)\cong Pr^L_{Ab}$ a localization functor from the (ordinary) category of presentable categories to the (ordinary) category of presentable abelian categories?

Proof strategy: I believe that Lurie's argument can be modified to prove that $-\otimes Ab$ is a localization functor if I could prove that every presentable category that is a right $Ab$-module is also abelian.

I can verify that $C$ is pre-abelian:

• Since $C$ is presentable we know that that for any $a \in Ab$ the functor $- \otimes a$ has a right adjoint $[a, -]$. Thanks to the appendix of this paper (shared with me by Zhen Lin) we have that $C$ is enriched and tensored in $Ab$. Moreover, the $Ab$ action corresponds to the tensor.

• Since $C$ is presentable, it is complete and cocomplete. In particular it has a $0$ object, biproducts, kernels, and cokernels.

I cannot verify that $C$ is actually abelian which requires that:

• Every monomorphism is a kernel and every epimorphism is a cokernel.

Alternate strategy:

In Lurie's proof (summarized well in 5.3.1 of Groth's notes) he uses the description of $C \otimes Sp = Fun^R(C^{op}, Sp)$ and the fact that $Sp = lim(S \xrightarrow{\Omega} S \xrightarrow{\Omega} \ldots)$ to show that $C \otimes Sp$ is the stabilization $Sp(C)$. Then he shows stabilization of a stable category is itself.

Mimicking this, I can show that $C \otimes \mathbb{Z} \cong Fun^R(C^{op}, \mathbb{Z})$ which is obviously pre-abelian since it is presentable and $Ab$-enriched.

• Is $C \otimes \mathbb{Z}$ abelian? I think it might be because it is a subcategory of a presheaf category.

• If $C$ is already abelian does the tensor map $C \otimes \mathbb{Z} \to C$ give an equivalence?

Answer 1

A locally presentable category $\mathcal{C}$ has a (unique) structure of an $Ab$-module if and only if it is additive. Such a category need not be abelian.

This is one reason to prefer the setting of stable $\infty$-categories to the theory of abelian categories. The identification of presentable stable infty-categories with $Sp$-modules implies (among other things) that there is a robust theory of tensor products of presentable stable infty-categories. In the abelian setting, developing an analogous theory requires some fairly restrictive assumptions.

Edit: Actually, I think it's not as bad as I suggested; there's a well-behaved tensor product on Grothendieck abelian categories. But that's a smaller collection of categories than the $Ab$-modules.