Filtered objects

Example 16.3.2 (Filtered objects). Under the same hypotheses on \(C\), regard \((\Z ,\leq )\) as a poset with its usual ordering and as a symmetric monoidal category under addition. The Day convolution on \(\Fun ((\Z ,\leq ),C)\) is the standard tensor product of increasing filtered objects: \[ (F \otimes _{\Day } G)_n \simeq \colim _{i+j\leq n} F_i \otimes G_j. \] The transition maps are induced by the order on \(\Z \). Replacing \((\Z ,\leq )\) by \((\N ,\leq )\) gives the analogous tensor product of nonnegatively filtered objects.