Toën derived Hall multiplication is associative and unital

Fix $\alpha ,\beta ,\gamma \in {\mathbb{Q}}_c(X^{(0)})$ and write $p=s\times t:X^{(1)}\to X^{(0)}\times X^{(0)}$. For any map $a$ and finite-support functions $\mu ,\nu$, one has $(a_{!}\mu )\otimes \nu =(a\times \mathrm{i}\mathrm{d}{)}_{!}(\mu \otimes \nu )$: the fiber of $a\times \mathrm{i}\mathrm{d}$ at $(y,z)$ is $F_y\times \{z\}$, with the same homotopy groups and weight as $F_y$, and its component sum is the defining one for $a_{!}$. The first square in [F4], transposed so that its lower map is $p$, is homotopy cartesian and $p$ is proper by [F2]. Base change therefore gives $(\alpha \ast \beta )\ast \gamma =c_{!}{p}^{\ast }(c\times \mathrm{i}\mathrm{d}{)}_{!}((p\times \mathrm{i}\mathrm{d}{)}^{\ast }(\alpha \otimes \beta \otimes \gamma ))=(c\circ g{)}_{!}((s\circ f,t\circ f,t\circ k{)}^{\ast }(\alpha \otimes \beta \otimes \gamma ))$. Here $f,g,h,k$ are the maps on $X^{(2)}$ from Toën derived Hall algebra by homotopy-cofiber convolution, and the three entries are source of $f$, cofiber of $f$, and cofiber of $k$; [F4] identifies the last entry with the cofiber of $g$.

Let ${\delta }_0={\chi }_0$. For ${\delta }_0\ast {\chi }_x$, arrows with source zero have target equivalent to their cofiber; the target map on this restricted homotopy type has contractible fibers over the component of $x$. Its pushforward is therefore ${\chi }_x$, and is zero on other components.

The second square in [F4], again transposed with lower map $p$, is homotopy cartesian and gives $\alpha \ast (\beta \ast \gamma )=c_{!}{p}^{\ast }(\mathrm{i}\mathrm{d}\times c{)}_{!}((\mathrm{i}\mathrm{d}\times p{)}^{\ast }(\alpha \otimes \beta \otimes \gamma ))=(c\circ h{)}_{!}((s\circ f,s\circ k,t\circ k{)}^{\ast }(\alpha \otimes \beta \otimes \gamma ))$. By definition of the induced cofiber map, $s\circ k=t\circ f$; also $c\circ h=c\circ g$ because $h$ is the composite of $f$ and $g$. Thus the last expression is the same as in step 1.1. This proves associativity on finite-support functions.

For ${\chi }_x\ast {\delta }_0$, the selected arrows have source in the component of $x$ and zero cofiber, hence are quasi-isomorphisms. Their core is the path homotopy type of the component of $x$; the target map is an equivalence onto that component with contractible homotopy fibers. The pushforward is again ${\chi }_x$ and vanishes elsewhere. Thus ${\delta }_0$ is a two-sided unit.

Steps 1.1–1.2 prove associativity on arbitrary finite-support functions, and steps 1.2–2.2 prove both unit identities. Finiteness of each convolution is supplied by [F2].