The homotopy theory of operad subcategories

Abstract

We study the subcategory of topological operads \({\mathsf {P}}\) such that \({\mathsf {P}}(0) = *\) (the category of unitary operads in our terminology). We use that this category inherits a model structure, like the category of all operads in topological spaces, and that the embedding functor of this subcategory of unitary operads into the category of all operads admits a left Quillen adjoint. We prove that the derived functor of this left Quillen adjoint functor induces a left inverse of the derived functor of our category embedding at the homotopy category level. We deduce from this result that the derived mapping spaces associated to our model category of unitary operads are homotopy equivalent to the standard derived operad mapping spaces, which we form in the model category of all operads in topological spaces. We prove that analogous statements hold for the subcategory of k-truncated unitary operads within the model category of all k-truncated operads, for any fixed arity bound \(k\ge 1\), where a k-truncated operad denotes an operad that is defined up to arity k.