I hope this question is suitable to be posted here on MO.

I wonder if there is a systematic relation between the notation of a classifying space, and the notion of a moduli space. I don't consider myself as someone with an expert level knowledge in any of these objects, but let me explain what I think.

The classifying space of a group/H-group, or a monoial category, most of the time is ought to be a representing object for a cohomology theory. The word classifying here, is mainly referring to the fact that $BG$ does parametrise certain ``bundles'' over paracompact spaces with a $G$-structure. The moduli space, is also ought to be a space which somehow parametrises the class of objects in a certain category - hope this analogy is not too vague. It somehow puts the totality of certain object in a `topological space'. Hence, both of these spaces are meant to parametrise certain objects.

Of course, I have not declared which definitions I take for these object, and I hope there is a common understanding about these spaces when an example is given. My main question is this. If $C$ is a category for which I can associated a classifying space $BC$ as well as a moduli space to the class of its objects, say $\mathcal{M}_C$. Then, is there a map

$$BC\to\mathcal{M}_C$$

or the way around?

Let me add that the work of Madsen and Weiss, on the stable Mumford conjecture, provides an example of the relation that I have asked above. They show that the infinite loop space associated to a certain spectrum $\mathbb{C} P_{-1}$ is related to the classifying space of the cobordism category of oriented surfaces, say $BC_2$ (in fact there is a weak homotopy equivalence) which is meant to solve Mumford's conjecture as $\Omega^{\infty-1}\mathbb{C} P_{-1}$ and the moduli space of Riemannian surfaces are ought to be the same?! But, I still don't know where a map such as $\Omega^\infty\mathbb{C}P_{-1}\to \mathcal{M}_\infty$ or a map $BC_2\to \mathcal{M}_\infty$ is constructed where $\mathcal{M}_\infty$ is the moduli space of Riemannian surfaces of arbitrary genus. I am aware of the map Madsen-Tillmann map $\mathbb{Z}\times B\Gamma_\infty^+\to\Omega^\infty\mathbb{C}P_{-1}$ with $\Gamma_\infty$ is the stable mapping class group and $(-)^+$ is Qullien's plus construction. The work of Galatius-Madsen-Tillmann-Weiss also provides $\alpha_\infty:BC_2\to\Omega^{\infty-1}\mathbb{C}P_{-1}$. But, I don't think of the source nor the target as a moduli space.

I will be very grateful for any advise. Please let me know if I am mixing some objects.

Thanks.