Presheaf stabilization stable operad

Proof. Put \(A := \An _*^{\fin }\). The Day convolution structure on \(P\) identifies \(\Mm _P\) with \(\oDay (\Mm _{D\catop },\Mm _{\An })\). Thus, using the universal property of Day convolution twice, \(\oDay (\Mm _A,\Mm _P)\) identifies with \[ \oDay (\Mm _{D\catop },\oDay (\Mm _A,\Mm _{\An })). \] Under this identification, \(\oSp (\Mm _P)\) is obtained by restricting the inner \(\oDay (\Mm _A,\Mm _{\An })\) to reduced excisive functors, using Lemma 18.3.2. By Theorem 16.6.1 and the construction of the symmetric monoidal localization there, this restriction is precisely the multimorphism operad \(\Mm _{\Sp }\). Hence \[ \oSp (\Mm _P) \simeq \oDay (\Mm _{D\catop },\Mm _{\Sp }). \] The latter is the multimorphism operad of \(\Fun (D\catop ,\Sp )\) with Day convolution. Since \(\Sp \) is stable and its tensor product preserves colimits in each variable, so does the Day tensor product: for a fixed functor in all but one variable, it is computed as a left Kan extension of a pointwise tensor product, and both operations preserve colimits in the remaining variable. It follows that \(\oSp (\Mm _P)\) is stable. □