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Gabriel-Morita theory for excisive model categories. (English) Zbl 1428.18041
Let \(\mathcal{E}\) be a pointed monoidal model category which is tractable (i.e. cofibrantly generated and the monoidal unit \(I\) is cofibrant). Let \(T\) be a strong monad on \(\mathcal{E}\), i.e. a monad equipped with a “strength” \(\sigma_{X,Y} : X \otimes T(Y) \to T(X \otimes Y)\) satisfying several axioms. Then the strenght defines a “linear approximation” \(\lambda : - \otimes T(I) \to T\) which induces an adjunction between the category \(\mathsf{Mod}_{T(I)}\) of \(T(I)\)-modules and the category \(\mathcal{E}^T\) of \(T\)-algebras. The authors’ main theorem gives a criterion for this adjunction to be a Quillen equivalence. If \(\mathcal{E}\) is the category of \(\Gamma\)-spaces and \(T\) is induced by a pointed simplicial Lawvere theory, this allows them to recover a theorem of S. Schwede [Topology 40, No. 1, 1–41 (2001; Zbl 0964.55017)].
This criterion can be interpreted as a kind of Gabriel-Morita theory as follows. Thanks to Kock’s correspondence, strong monads correspond to enriched monads. When the enrichment is over the category of abelian groups, then the authors’ theorem is a derived version of Morita’s criterion for the equivalence of two categories of modules over two rings, which can itself be deduced from Gabriel recognition theorem for module categories among abelian categories.
Roughly speaking, the authors’ criterion is summarized by the following. In addition to being a tractable monoidal model category, \(\mathcal{E}\) is equipped with a standard system of simplices and is required to be excisive, i.e. the derived suspension functor should be conservative (excisive categories have the useful property that homotopy pushouts are characterized by weakly equivalent parallel homotopy cofibres). The monad \(T\) should satisfy several technical hypotheses: it should preserve filtered colimits and reflexive coequalizers, the unit \(X \to T(X)\) should be a cofibration for each cofibrant object \(X\), and \(\mathcal{E}^T\) should admit a transferred model structure. It should also be homotopically right exact, i.e. behave well with respect to cell extensions. Under theses hypotheses, the linear approximation \(\lambda\) induces a Quillen equivalence if and only if the monad \(T\) satisfies a derived version of linearity: the strength \(\sigma : X \otimes T(Y) \to T(X \otimes Y)\) should be a weak equivalence for all cofibrant \(X,Y\).
18N40 Homotopical algebra, Quillen model categories, derivators
18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
18D25 Actions of a monoidal category, tensorial strength
55P42 Stable homotopy theory, spectra
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