Recently, Jade Master [asked](https://twitter.com/JadeMasterMath/status/1252045020804177921) whether the tensor product of chain complexes could be viewed as a special case of Day convolution. Noting that chain complexes [may be viewed](https://math.stackexchange.com/questions/2139250/can-the-category-of-chain-complexes-be-realized-as-a-functor-category) as $\mathsf{Ab}$-functors from a certain $\mathsf{Ab}$-category $\mathsf{C}$, Yuri Sulyma [suggested](https://twitter.com/YuriSulyma/status/1252380708980453378)¹ that *maybe* we could obtain the tensor product of two chain complexes as a Day convolution by endowing $\mathsf{C}$ with the monoidal structure given by $[n]\otimes\_\mathsf{C}[m]\overset{\mathrm{def}}{=}[n+m-1]$.


**Questions:** Is this affirmation true? More precisely:


1. Given two chain complexes $X\_\bullet$ and $Y\_\bullet$ on an abelian category $\mathcal{A}$, is their Day convolution as $\mathsf{Ab}$-functors from $(\mathsf{C},\otimes\_\mathsf{C})$ the usual tensor product of chain complexes $\otimes\_{\mathsf{Ch}(\mathcal{A})}$?
2. If not, is there some other monoidal structure on $\mathsf{C}$ for which Day convolution gives $\otimes\_{\mathsf{Ch}(\mathcal{A})}$?
3. If this fails too, is there perhaps another way to view $\otimes\_{\mathsf{Ch}(\mathcal{A})}$ as a special case of some general construction in enriched category theory?




---


¹Note that his account is protected and hence his reply is not public.



Is the tensor product of chain complexes a Day convolution?
The answer to the question posed in the title of your post is **yes**, the tensor product of chain complexes is a Day convolution product. The important thing to note is that, to define a Day convolution monoidal structure on the $\mathcal{V}$-enriched functor category $[\mathcal{C},\mathcal{V}]$ (where $\mathcal{V}$ is a complete and cocomplete symmetric monoidal closed category, e.g. $\mathbf{Ab}$), we needn't demand $\mathcal{C}$ to be a monoidal $\mathcal{V}$-category: it suffices for $\mathcal{C}$ to be a **promonoidal** $\mathcal{V}$-category. This is the generality at which Day convolution was originally defined in Day's thesis, [which may be found here](http://web.science.mq.edu.au/~street/DayPhD.pdf) (see also [his earlier paper](https://mathscinet.ams.org/mathscinet-getitem?mr=0272852) in the Reports of the Midwest Category Seminar IV, where the word "premonoidal" was used).


A promonoidal structure on a small $\mathcal{V}$-category $\mathcal{C}$ consists of tensor product and unit "profunctors", i.e. $\mathcal{V}$-functors $P \colon \mathcal{C}^\mathrm{op}\times\mathcal{C}^\mathrm{op} \times \mathcal{C} \to \mathcal{V}$ and $J \colon \mathcal{C} \to \mathcal{V}$, together with associativity and unit constraints subject to the usual two 
"pentagon" and "triangle" axioms. Given a promonoidal structure on $\mathcal{C}$, we may construct the **Day convolution** monoidal structure on $[\mathcal{C},\mathcal{V}]$, whose tensor product is given at a pair of $\mathcal{V}$-functors $F,G \in [\mathcal{C},\mathcal{V}]$ by the coend
$$F\ast G = \int^{A,B \in \mathcal{C}} P(A,B;-) \otimes FA \otimes GB$$ in $\mathcal{V}$, 
and whose unit object is the $\mathcal{V}$-functor $J \in [\mathcal{C},\mathcal{V}]$, and so on. This monoidal structure on $[\mathcal{C},\mathcal{V}]$ is biclosed (i.e. the tensor product $\mathcal{V}$-functor has a right $\mathcal{V}$-adjoint -- equivalently, preserves (weighted) colimits -- in each variable). In fact, every biclosed monoidal structure on $[\mathcal{C},\mathcal{V}]$ arises in this way from some promonoidal structure on $\mathcal{C}$. (For instance, one recovers the $\mathcal{V}$-functor $P$ from the tensor product $\ast$ by $P(A,B;C) = (\mathcal{C}(A,-) \ast \mathcal{C}(B,-))C$.)


So, since the $\mathbf{Ab}$-category $\mathbf{Ch}$ of chain complexes is (equivalent to) an $\mathbf{Ab}$-enriched functor category $[\mathcal{C},\mathbf{Ab}]$ (for the $\mathbf{Ab}$-category $\mathcal{C}$ described in the question to which you linked), and since the standard monoidal structure on $\mathbf{Ch}$ is $\mathbf{Ab}$-enriched and biclosed, this monoidal structure must be the Day convolution monoidal structure for some promonoidal structure on $\mathcal{C}$. And it isn't too hard to describe that promonoidal structure. For instance, (presuming I haven't bungled the calculation) the functor $P$ is defined on objects by $$P(i,j;k) = \begin{cases} \mathbb{Z} & \mathrm{if\,\,} i+j=k, \\
\mathbb{Z} \oplus \mathbb{Z} & \mathrm{if\,\,} i+j=k+1, \\
\mathbb{Z} & \mathrm{if\,\,} i+j=k+2, \\
0 & \mathrm{else}.
\end{cases}$$


