I am currently reading Gelfand and Manin's Methods of Homological Algebra, and at some point (IV. 10. Exercise 2 I guess? The numbering in the book is quite inconvenient to get around, so I'm mentioning it for completeness, but I'll re-explain my question so that there's no need to look into the book) they say the following:
Let $\mathcal{D}$ be a triangulated category, $X^\bullet$ a finite chain complex of objects in $\mathcal{D}$, and $T \in \text{Tot} X^\bullet$ be its convolution.
I do not understand what they mean by the word convolution. In the previous pages of the book, the letter $T$ usually denotes the translation operator of a triangulated category ($T^k$ would map a complex $X^\bullet$ to $X^\bullet[k]$, which is the same complex shifted by an index of $k$).
I was guessing that it meant taking the convolution of the complex $X^\bullet$ with itself in the "standard" sense of convolution (the one we know for power series for example), which would give another chain complex. So the $n$-th term of that convolution would look something like:
\begin{equation} \coprod_{k \in \mathbb{Z}} X^k \otimes X^{n-k} \end{equation}
Where maybe it's a product $\prod$ instead of a coproduct $\coprod$, and maybe we have another operation instead of $\otimes$. That's just a guess based on what discrete convolutions look like; but maybe it's not exactly these operations.
But even the above guess raises a few questions:
- What are the natural notions of products/coproducts on the category of chain complex $\text{Ch}(\mathcal{A})$, given $\mathcal{A}$ an arbitrary abelian category? I have found the following thread; which seems to point that the natural notion of coproduct is the usual direct sum (defined componentwise). That is indeed pretty convenient. But:
Is there a natural notion of product on Ch$(\mathcal{A})$?
Though I'm not expecting it to be necessarily as nice as the coproduct, because taking componentwise products doesn't sound as good.
- Is my guess for the definition of convolution correct? Or am I not using the right operations/the definition is not as elementary?
In any case, why is the convolution of $X^\bullet$ an element of $\text{Tot} X^\bullet$ ?
That's it for my question. For context, the statement they then prove with these definitions is that there exists a spectral sequence
\begin{equation} E_1^{p,q} = H^q(X^p) \implies H^n(T) \end{equation}