# Do flat resolutions guarantee the existence of Tor (without enough projectives)?

Let $\mathcal A$ be an abelian category with a symmetric monoidal structure $\otimes$. Suppose that $\mathcal A$ does not have enough projectives, but every object has a flat resolution Then, is the existence of the Tor guaranteed?

• Are you asking about Tor in the sense of universal $\delta$-functors? If so, can't you use the effaceability criterion? (I suppose one might have to ask for a functorial flat resolution to ensure that one really gets a functor...) – Zhen Lin Jun 24 '15 at 18:34