# Topological category of topological monoids / operads

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?

There is an exactly analogous question for topological monoids (or topological groups). We could either construct the ordinary homotopy category that you describe, which is a strong invariant, or we could build the weak homotopy category where we invert maps $$M \to N$$ of topological monoids that are weak equivalences on the level of spaces. These are very different homotopy categories. As an example of this, if you restrict your attention to topological groups then the homotopy theory of topological groups is equivalent to the homotopy theory of pointed connected spaces, by the correspondence $$G \leftrightarrow BG$$. The space $$BG$$ doesn't remember a lot about the strict group-level structure, such as identities that are satisfied by the multiplication in $$G$$.