Projective and Reedy model category structures for (infinitesimal) bimodules over an operad

Julien Ducoulombier (1) , Benoit Fresse (2) , Victor Turchin (3)

1 D-MATH - Departement Mathematik ETHZ
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 University of Kansas - Department of Mathematics

We study projective and Reedy model category structures for bimodules and infinitesimal bimodules over a topological operad. In both cases, we build explicit cofibrant and fibrant replacements. We show that these categories are right proper and, under some conditions, left proper. We study the Extension/Restriction adjunctions. We give also a characterisation of Reedy cofibrations and we check that the two model structures produce compatible homotopy categories. In the case of bimodules the homotopy category induced by the Reedy model structure is a subcategory of the projective one. In the case of infinitesimal bimodules the Reedy and projective homotopy categories are the same.

Domaines: Mathématiques [math]; Topologie algébrique [math.AT]

Julien Ducoulombier, Benoit Fresse, Victor Turchin. Projective and Reedy model category structures for (infinitesimal) bimodules over an operad. Applied Categorical Structures, 2022, 30 (5), pp.825-920. ⟨10.1007/s10485-022-09675-z⟩. ⟨hal-02382918⟩