Graded objects

Example 16.3.1 (Graded objects). Regard \(\Z \) as a discrete symmetric monoidal category under addition, and let \(C\) be a cocomplete symmetric monoidal \(\infty \)-category whose tensor product preserves colimits separately in each variable. Day convolution equips the \(\infty \)-category \(\Fun (\Z ,C)\) of \(\Z \)-graded objects with a symmetric monoidal structure. Explicitly, \[ (F \otimes _{\Day } G)_n \simeq \coprod _{i+j=n} F_i \otimes G_j. \] For \(C=\Ab \) this is the usual tensor product of graded abelian groups, and for \(C=\Sp \) it is the corresponding tensor product of graded spectra.