Recall that a fuctor $T: \cal C \to \cal A$ from pointed small $\cal C$ with coproducts to additive and Karoubian — or, even better, abelian $\cal A$ is called quadratic if kernel of sum of obvious orthogonal idempotents in $\cal C(X, Y)$ a.k.a. cross-effect $T(X|Y) := \rm{ker} \, T(e_X) \oplus T(e_Y): T(X \sqcup Y) \to T(X) \oplus T(Y)$ is additive in each variable; "bilinear functor" seems to be a common name for such. They form full subcategory $\rm{Quad}(\cal C, \cal A)$ in category of all functors.
So, if one wants to define quadratic functor to a, generally speaking, triangulated — or a DG one, if that task seems easier — category $\cal B$, there are many different way to do so.
First one is "naive sheafy" approach — just use above definition replacing $\cal A$ with Karoubi envelope $\rm{Split}(\cal B)$, but if $\cal B = D(\cal A)$, it seems to me that this construction has little in common with category of simplicial objects in $\rm{Quad}(\cal C, \cal A)$ (homotopy category of which I'm interested in).
Second one is "a bit coherent": call a functor $T: \cal C \to \cal B$ quadratic if there's a natural transformation of bifunctors $s: T_{lin}(-) \oplus T_{lin}(-) \oplus T(-|-) \to T(- \sqcup -)$ with pointwise acyclic cone — and it probably should be monomorphic.
Are those two really different? General references to literature are welcome — Pirashvili, Dzhibladze et al definitely should've written something on this subject, but I couldn't find anything.
(Some background: I'm trying to find suitable formalism aiming at extending Dold-Kan correspondence from (bi)module categories to polynomial part of $\rm{Fun}(R\rm{-proj}, Q\rm{-mod})$ where $R, Q$ are some unital rings. Generally, I'm expecting to find "algebraic realization" of (nonconnective) $p$-local spectra category as suitable category of functors from something not very complicated into modules over Steenrod algebra, but good definition of DG-polynomial functors seems independently interesting.)
