Toën derived Hall algebra by homotopy-cofiber convolution

Let $\mathcal{T}$ be a locally finite dg-enhancement over the finite field $\mathbb{k}={\mathbb{F}}_q$ in the sense of Locally finite dg enhancements for derived Hall theory. Let $X^{(0)}$ be the core homotopy type of its perfect-module category: points are perfect modules, paths are quasi-isomorphisms, and higher paths are the derived mapping spaces. Let $X^{(1)}$ be the core homotopy type of arrows between perfect modules and $X^{(2)}$ the core homotopy type of composable pairs. These are the cores of the corresponding objectwise homotopy theories, so their equivalences are objectwise quasi-isomorphisms.

An arrow $u:x\to y$ determines three maps

where $\mathrm{Cone}(u)$ is the functorial homotopy cofiber in the supplied dg enhancement. Thus $s\times t:X^{(1)}\to X^{(0)}\times X^{(0)}$ remembers the source and cofiber, and $c:X^{(1)}\to X^{(0)}$ remembers the target. For a composable pair $x\to y\to z$, write $f$ for the first arrow, $g$ for the second, $h$ for the composite, and $k:\mathrm{Cone}(f)\to \mathrm{Cone}(h)$ for the induced map of cofibers.

Set

The product is the convolution

where $(\alpha \otimes \beta )(x,y)=\alpha (x)\beta (y)$ on $X^{(0)}\times X^{(0)}$, and $c_{!}$ and pullback are those of Locally finite homotopy types and rational cardinality. For an isomorphism class $x$ of perfect modules, let ${\chi }_x$ be its characteristic function. The ${\chi }_x$ form a basis, and

where $g_{x,y}^z$ is the weighted homotopy cardinality of the fiber of $c$ over $z$ restricted to arrows with source isomorphic to $x$ and cofiber isomorphic to $y$.

The zero module gives the candidate unit ${\chi }_0$. The definition is well-defined: local finiteness and finite support of the convolution follow from Derived Hall moduli types are locally finite and the input span is proper ↗.

The horizontal map sends an arrow to its target; the vertical map sends it to its source and homotopy cofiber. The product pulls two input functions back along the vertical map and pushes them forward along the horizontal map. The diagram records this correspondence; it supplies no proof of associativity.