The homotopy theory of bialgebras over pairs of operads (Q393510)

Let \(P\) be an operad in the category of \(\mathbb{K}\)-vector spaces such that \(P(0)=0\) and \(P(1)=\mathbb{K}\) and such that the vector spaces \(P(n)\) are finite-dimensional for all \(n>1\). Let \(\mathrm{Ch}^+_{\mathbb{K}}\) be the category of positively graded chain complexes. It is proved that the category of \(P\)-coalgebras \({}^P\mathrm{Ch}^+_{\mathbb{K}}\) can be equipped with a cofibrantly generated model category such that a morphism is a weak equivalence (a cofibration resp.) if it is a weak equivalence (a cofibration resp.) between the underlying chain complexes, the fibrations being the maps having the right lifting property with respect to trivial cofibrations. Then, using an adjunction \(P:{}^Q\mathrm{Ch}^+_{\mathbb{K}} \rightleftarrows {}^Q_P\mathrm{Ch}^+_{\mathbb{K}}:U\), the above model structure is transferred to the category \({}^Q_P\mathrm{Ch}^+_{\mathbb{K}}\) of \((P,Q)\)-bialgebras. The author obtains on \({}^Q_P\mathrm{Ch}^+_{\mathbb{K}}\) a cofibrantly generated model structure such that a map \(f\) of \({}^Q_P\mathrm{Ch}^+_{\mathbb{K}}\) is a weak equivalence (a fibration resp.) if and only if \(U(f)\) is a weak equivalence (a fibration resp.) of \({}^Q\mathrm{Ch}^+_{\mathbb{K}}\), a cofibration being a map satisfying the left lifting property with respect to the trivial fibrations.