Koszul symmetry

Definition 16.6.6 (Koszul symmetry). Keeping the same underlying Day convolution monoidal structure on \(\Ab ^{\Z }\), we replace its ordinary symmetry by the Koszul symmetry, which on homogeneous elements is given by \[ A_n\otimes B_m\longrightarrow B_m\otimes A_n, \qquad a\otimes b\longmapsto (-1)^{nm}b\otimes a. \] We refer to the resulting symmetric monoidal category as the category of graded abelian groups with the Koszul symmetry.