The category of topological monoids can be made into a topological category in a naive way. Namely, the space of all continuous homomorphisms between two topological monoids is a closed subspace of the space of all continuous maps between the underlying topological spaces.
My question is that, regarding homotopy theory, is this a "good" construction? More precisely, if we take the homotopy category of this topological category, which is a category enriched over the category of homotopy types, does it give the "correct" homotopy theory?
More generally, consider all the topological operads (with a single color). There is also a naive way of making the category of all topological operads into a topological category, similar to the above one. Does the homotopy category of this topological category behave as we would expect?