In the article "Factorization homology of topological manifolds" by Ayala and Francis, a symmetric monoidal $\infty$-category $\mathcal{V}$ is fixed as the target or coefficient category. This category is absolutely key because it massively influences the behaviour of the resulting homology theory.


In Definition 3.4 of said article there are some assumptions on $\mathcal{V}$, namely $\otimes$-presentability. This is needed for the main Theorems of the article to work.


Examples include spaces and differential graded modules over a ring.


Can someone give more examples and non-examples?


More concretely, will cdgas or dgas (or adapted version thereof?) satisfy the hypotheses? What about, more generally, algebras over operads?


Thanks for any hints.



Coefficient (or target) category for factorization homology
One rich source of examples is that every combinatorial symmetric
monoidal model category gives rise to such a category V.


In particular, this covers all the examples in the main post,
including cdgas, which do form such a model category with
tensor product as the monoidal structure.


