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Recently, Jade Master asked whether the tensor product of chain complexes could be viewed as a special case of Day convolution. Noting that chain complexes may be viewed as $\mathsf{Ab}$-functors from a certain $\mathsf{Ab}$-category $\mathsf{C}$, Yuri Sulyma suggested¹ 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.

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    $\begingroup$ My current guess is that this Day convolution should be $(A^{\bullet}\otimes_\mathrm{Day}B^{\bullet})^n=\bigoplus_{p+q=n,n+1}A^p\otimes_R B^q$ in degree $n$. (This comes from a coend calculation: the above is the coequaliser of the diagram $A\rightrightarrows B$ with $A$ a direct sum of eight direct sums of $A^p\otimes_R A^q$, $B$ given by $\bigoplus_{p+q=n,n+1}A^p\otimes_R B^q$ as above, and the morphisms induced by the direct sums (for instance, one of them is, I think, $0\oplus0\oplus0\oplus0\oplus\mathrm{inj}_0\oplus\mathrm{inj}_0\oplus\mathrm{inj}_0\oplus\mathrm{inj}_1$.)) $\endgroup$ – Théo May 4 at 5:39
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    $\begingroup$ Let DF(A) denote the filtered derived category, Fun(Z, D(A)). Then, DF(A) admits a t-structure, known as the Beilinson t-structure, whose heart is equivalent to the abelian category of chain complexes in A. The category DF(A) admits a monoidal structure, by Day convolution, and this tensor product is compatible with the t-structure (i.e., the connective objects are closed under the tensor product). The identification of the heart of DF(A) with the category of chain complexes is a symmetric monoidal equivalence. $\endgroup$ – skd May 4 at 5:51
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    $\begingroup$ You might be interested in reading Im, Geun Bin, and G. Max Kelly. "A universal property of the convolution monoidal structure." Journal of Pure and Applied Algebra 43.1 (1986): 75-88. In particular, see page 10, §6. $\endgroup$ – Fosco May 4 at 10:42
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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 (see also his earlier paper 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 $$P(i,j;k) = \begin{cases} \mathbb{Z} & \mathrm{if\,\,} i+j=k, \\ \mathbb{Z} \oplus \mathbb{Z} & \mathrm{if\,\,} i+j=k+1, \\ \mathbb{Z} & \mathrm{if\,\,} i+j=k+2, \\ 0 & \mathrm{else}. \end{cases}$$

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    $\begingroup$ In the terminology of my answer, this profunctor is the bimodule. I didn't realize this was the original generality of Day convolution! This generality makes more sense because it is Morita invariant. $\endgroup$ – Phil Tosteson May 4 at 12:55
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    $\begingroup$ A chain complex in any $Ab$-enriched category is an $Ab$-enriched functor $\mathcal A$ $\mathsf C \to \mathcal A$. If $\mathcal A$ is additive, then this is equivalently an additive functor $\bar{\mathsf C} \to \mathcal A$, where $\bar{\mathsf C}$ is the additive envelope of $\mathsf C$ (i.e. its completion under direct sums). I haven't given this much thought, but I think that $\bar{\mathsf C}$ is closed under tensor product in $Ch(Ab)$. So by passing to a slightly larger base category, we get $\otimes$ as the Day convolution of a monoidal rather than a promonoidal structure. $\endgroup$ – Tim Campion May 4 at 17:47
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    $\begingroup$ This is great! Thanks, Alexander! (And everyone else!) $\endgroup$ – Théo May 5 at 6:45
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If you use the category $C$ to represent chain complexes and you mean day convolution using a functor $C \otimes C\to C$ it is not possible. This boils down to whether you can obtain the totalization functor from bi-complexes to chain complexes, as a left adjoint to restriction for some functor $m: C \otimes C \to C$.

You cannot do this because the left adjoint $m_!$ will always take representable projectives to representable projectives. I.e. we will have that $$ m_! (C \otimes C((i,j), -))(r) = C(m(i,j),r).$$

But the totalization of a representable projective of $C \otimes C$ is a direct sum of two different principal projectives of $C$, so no choice of $m$ will work.

What is going wrong is that the totalization functor is given by a unique $(C \otimes C, C)$ bi-module, and this bi-module cannot come from a homomorphism $C \otimes C \to C$, because in some sense it is "multi-valued." To fix this, one could change $C$ to a morita equivalent category, $C'$ for which the bi-module is in fact given by a homomorphism. To construct such a $C'$, we need to choose a collection of generating projectives of ${\rm Ab}^{C}$ which is closed under tensor product. I don't see a particularly nice choice. But skd's comment is that if we use derived Morita equivalence instead of ordinary Morita equivalence, there is a very nice choice of (non-projective) generators, where $C'$ becomes the category $\mathbb N, \leq$.

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