Smallest abelian braided monoidal subcategory containing an object $V$

Question

Let $\mathcal{C}$ be an abelian braided monoidal category with countable direct sums compatible with the tensor product (i.e. $X\otimes \bigoplus_{i \in \mathbb{N}} V_i \cong \bigoplus_{i \in \mathbb{N}}X\otimes V_i$) and let $V \in \mathcal{C}$ be an object. Is there a "smallest" abelian braided monoidal full subcategory $\mathcal{C}_V \subseteq \mathcal{C}$ with countable direct sums compatible with the tensor product containing $V$?

Answer 1

Of course, at least assuming you mean for your subcategory to inherit all its structure from that of $\mathcal{C}$ (so the braided monoidal structure, kernels, cokernels, and countable direct sums of the subcategory are required to be the same as those of $\mathcal{C}$) and you only care about the subcategory up to equivalence. Just take the intersection of all abelian braided monoidal full replete subcategories containing $V$ and closed under countable direct sums.

(There will probably not literally be a smallest subcategory, since that would involve making a choice of a single object from isomorphism classes and there are multiple ways to do so. But normally one only cares about the objects up to isomorphism.)

Or, more explicitly, $\mathcal{C}_V$ consists of all objects of $\mathcal{C}$ that can be obtained from $V$ by repeatedly taking kernels of maps, cokernels of maps, tensor products, and countable direct sums. (Note that this iteration may need to continue for $\omega_1$ steps because of the countable direct sums.)