How to generally characterize $\operatorname{Tor}_1$?

Question

There's this well known characterization of the first $\operatorname{Ext}$ group of two objects in a abelian category:

There is an isomorphism between $\operatorname{Ext}^1(B,A)$ and the group of equivalence classes of short exact sequences of the form $A \hookrightarrow \_\_ \twoheadrightarrow B$. If $A$ and $B$ are abelian groups, these are called the group extensions of $B$ by $A$.

For $\operatorname{Tor}_1$ I've seen a similar result for an $R$-module $M$,

$$\operatorname{Tor}_1(R/(r), M) \cong \{m \in M \mid rm = 0\} \,.$$

But this isomorphism is pretty particular compared to that characterization of $\operatorname{Ext}^1$. Can we say something about $\operatorname{Tor}_1$ that's more general and crisp like the corresponding statement about $\operatorname{Ext}^1$? Something about $\operatorname{Tor}_1(B,A)$ for $A$ and $B$ in any abelian tensor category perhaps? Or just anything more general than the above statement in $R$-mod? Looking at the page for this question here, I'm hoping that there actually is something nice to say since we're looking just at $\operatorname{Tor}_1$ instead of looking at $\operatorname{Tor}_n$ in general.

Answer 1

I don't think there's much hope for such a characterization. An important difference between Ext and Tor is that Ext makes sense in any abelian category, but Tor requires more structure to make sense, namely a bilinear tensor product. A more highbrow difference is that Ext involves maps in the derived category and so is related to representable functors in a derived sense while Tor is not.

Answer 2

In the case of abelian groups, $\operatorname{Tor}_1$ is the torsion product, presented by generators and relations in a similar fashion to the tensor product.

$\operatorname{Tor}(A,B)$ is presented by taking generators to be symbols of the form $(a, n, b)$ where $n$ is a positive integer, $a$ is an $n$-torsion element of $A$ and $b$ is an $n$-torsion element of $B$, and the families of relations are:

$$ (a+a', n, b) = (a, n, b) + (a', n, b) $$ $$ (a, n, b + b') = (a, n, b) + (a, n, b + b') $$ $$ (a, mn, b) = m(a, n, b)$$

whenever all of the terms in the identity are defined. The relation to the usual description is that if we have a free resolution $$ 0 \to A_1 \to A_0 \to A \to 0 $$ then the boundary map is

$$ \operatorname{Tor}(A,B) \to A_1 \otimes B : (a, n, b) \mapsto (n \tilde{a}) \otimes b $$

where $\tilde{a} \in A_0$ is any preimage of $a \in A$.

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For modules over an arbitrary ring (or maybe just commutative ring), I believe there analogous presentation obtained by also adding in relations resembling $R$-bilinearity. I also believe there to be similar presentations for higher tor groups.

It's really hard to find any references on this, though, so I'm not completely sure.