Question

There is a functor from bicomplexes to chain complexes sending a bicomplex to its associated total chain complex. Does this functor have a left adjoint, and if so, what is it?

Answer 1

It's not to hard to work out by hand what such an object would need to be if it existed. By considering the case that $D$ is a bicomplex supported in degree $(m,n)$, the adjunction condition $\operatorname{Hom}_{bicomplex}(L(C), D) = \operatorname{Hom}_{complex}(C, \operatorname{Tot}(D))$ imposes the condition that $L(C)$ have copies of $C_k$ in all degrees $(m,k-m)$. Similarly, by examining the case where $D$ has a single nonzero arrow, we find that up to signs, the differential $L(C)_{m,n} \to L(C)_{m-1,n}$ is given by $C_{m+n} \to C_{m+n-1}$, and the same holds for the vertical arrows.

As I understand it, there is a direct sum totalization, and a direct product totalization, and they have different adjunction properties. In particular, if $D$ has large support, then the left side of the adjunction condition involves a countably infinite product in each degree. While the direct product totalization yields an adjunction, the direct sum totalization will give you a dimension mismatch.

If you work with finite periodic complexes, the totalizations coincide, and the adjunction holds.

Since chain complexes are (up to some details about signs) representations of the $x \mapsto ax+b$ group $\mathbb{G}_a \rtimes \mathbb{G}_m$, the adjunction amounts to Frobenius reciprocity, and the problem here is quite similar to the problem of finding a well-behaved induction functor for passing from representations of this group to representations of its direct product with itself. Since the group has positive dimension, the usual induction ends up being some kind of direct integral. The direct product totalization seems to be some kind of completed restriction, but it is rather unclear to me.