# category theory – Tor for modules over categories


$$M otimes_A N = int^{a in A} Ma otimes_k Na,$$

which lies in $$Vect$$. This should be enough to define $$mathrm{Tor}^A_i(M, N)$$ by the long exact sequence. I have questions about this functor:

• Is it balanced? I assume yes. Does this already follow from the Embedding Theorem and balancedness for modules?
• Is it computable in terms of free modules? What is a free module, even? I know about the free-forgetful adjunction, but is there a more hands-on description of free modules? Is $$Hom_A(a,-)$$ for $$a in A$$ a free module? Are all free modules direct sums of this?
• Are projective modules flat?

I’d guess that all answers to these questions are affirmative. Where can I find more about that?