Homotopy Theory of Modules over Operads and Non-ς Operads in Monoidal Model Categories

There are many interesting situations in which algebraic structure can be described by operads [1, 12, 13, 14, 17, 20, 27, 32, 33, 34, 35]. Let (C,⊗, k) be a symmetric monoidal closed category (Section 2) with all small limits and colimits. It is possible to define two types of operads (Definition 6.1) in this setting, as well as algebras and modules over these operads. One type, called Σ-operad, is based on finite sets and incorporates symmetric group actions; the other type, called non-Σ operad, is based on ordered sets and has no symmetric group contribution. (In this paper we use the term Ω-operad for non-Σ operad, where Ω = O for “Ordered.”) Given an operad O, we are interested in the possibility of doing homotopy theory in the categories of O-modules and O-algebras, which in practice means putting a Quillen model structure on these categories of modules and algebras. In this setting, O-algebras are left O-modules concentrated at 0 (Section 6.3). Of course, to get started we need some kind of homotopy theoretic structure on C itself; this structure should mesh appropriately with the monoidal structure on C. The basic assumption is this.