The homotopy category of unitary operads as a full subcategory of the homotopy category of all operads

Benoit Fresse, Victor Turchin, Thomas Willwacher

Published 2017-03-17Version 1

We prove that the homotopy category of topological operads $P$ satisfying $P(0) = *$ forms a full subcategory of the homotopy category of all topological operads. We more precisely establish that we have a weak-equivalence of simplicial sets at the mapping space level which gives this embedding of homotopy categories when we pass to connected components. We also prove that an analogous result holds for the categories of $k$-truncated operads, which are operads defined up to arity $k$.