Recall that a **chain complex** is a (finite) diagram of the form
$$ V = \{ \dots \to V_3 \overset{d_3}\to V_2 \overset{d_2}\to V_1 \overset{d_1}\to V_0 \to 0 \} $$
where the $V_n$ are (finite-dimensional) vector spaces and for each $n$, $d_n \circ d_{n+1} = 0$. If $V$ and $W$ are chain complexes, a **chain map** $f: V \to W$ is a map $f_n : V_n \to W_n$ for each $n$ such that all the obvious squares commute — "$[d,f]=0$" — and the pair (chain complexes, chain maps) defines a category. In fact, it is a 2-category: the 2-morphisms between $f,g : V \rightrightarrows W$ are the **chain homotopies**, i.e. a system of maps $h_n: V_n \to W_{n+1}$ such that "$[d,h] = f-g$". The category of chain complexes has a **biproduct** (both a product and a coproduct) $\oplus$ given by the pointwise direct sum.

I thought I knew what the tensor product of chain complexes was. Namely, if $V$ and $W$ are chains, then the usual thing is to define $$ (V\otimes W)\_n = \bigoplus_{k=0}^n V_k \otimes W_{n-k} $$ and the chain maps are the sums of the obvious tensor products of differentials, decorated with signs.

But now I'm not sure why this is the tensor product picked. Namely, if I have a linear category, I think that a **tensor product** $V \otimes W$ should satisfy the following universal property: for any $X$, $\hom(V \otimes W,X)$ should be naturally isomorphic to the space of *bilinear* maps $V \times W \to X$. Now, I've never really known how to write down the word "bilinear" in a general category, without refering to individual points. But I think I do know what the "set" $V \times W$ is when $V$ and $W$ are chains — it's the set underlying $V \oplus W$ — and then I think I do know what bilinear maps should be.

In any case, then it's clear that the usual tensor product is not this. For example, if $V,W$ have no non-zero terms above degree $n$, then the bilinear maps $V \times W \to X$ I think cannot be interesting above degree $n$, whereas the above $\otimes$ has terms in degree $2n$.

In any case, in HDA6, Baez and Crans consider two-term chain complexes $V_1 \to V_0$ (they argue that these are the same as "2-vector-spaces"), and then construct a different tensor product, given by: $$ V\otimes W = \{ (V_1 \otimes W_1) \oplus (V_1 \otimes W_0) \oplus (V_0\otimes W_1) \to (V_0 \otimes W_0) \} $$ where the differential is the sum of the obvious tensor products of differentials and identity maps. They then assert that this tensor product satisfies the correct universal property, although they leave the details to the reader.

This leads naturally to:

Question:What is the precise universal property that $\otimes$oughtto have, and what "product" of chain maps satisfies this universal property?

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