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From what I have read so far, a classifying space is a representing object of some (co)representable functor. For example, the $n^\text{th}$ Eilenberg–MacLane space is the classifying space for the $n^\text{th}$ singular cohomology functor since $$H^n_{\text{sing}}(X;G)\cong[X, K(G,n)]_{\text{Hotop}}.$$ Also principal $G$-bundles over a manifold $X$ are classified by the classifying space $BG,$ where $G$ is a Lie group. This is written as $$G\text{Bun}(X)\cong[X, BG]_{\text{Hotop}}.$$ So, maps in to (or from) the classifying space classify some data over associates to our object $X$ up to isomorphisms. On the other hand, in my mind, a moduli space is a space whose points are (isomorphism classes of) geometric structures/objects associated to $X.$ This is very intuitive as many sources say the term "modulus" is used synonymously with "parameter" and so a moduli space parametrizes the associated geometric structures/objects. The easiest example being the real projective plane $G(1,\mathbb{R}^3,\mathbb{R})=\mathbb{R}P^2$ whose each point represent a $1$-dimensional vector subspace of $\mathbb{R}^3.$ Next, moduli spaces on this line are general Grassmannians $G(k, V,\mathbb{F}).$ Another example is the moduli space $\mathcal{M}_g$ whose points are Riemann surfaces of genus $g$ up to biholomorphisms.

Please correct me if I am wrong at some point so far. However it seems like in literature people use the words moduli space and classifying space as synonyms. I would like to clarify this confusion, and know the precise difference and relationship between them.

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    $\begingroup$ Not a complete answer, but for some $G$ the standard classifying spaces really are moduli spaces. For example a model for the classifying space $BO(n)$ for vector bundles is the grassmannian $Gr_n(\mathbb{R}^\infty)$ of $n$-dimentional linear subspaces of $\mathbb{R}^\infty$. Another example is $BDiff(M)$ for a smooth manifold $M$: the space of embeddings $E = Emb(M, \mathbb{R}^\infty)$ is contractible and has a nice enough free action by $Diff(M)$ so $\mathcal{M}(M) = E/Diff(M)$, the moduli space of submanifolds of $\mathbb{R}^\infty$ diffeomorphic to $M$, is a model for $BDiff(M)$. $\endgroup$ – William May 20 at 23:52
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This is only a partial answer but too long for a comment.

At least for principal $G$-bundles, any model for the classifying space $BG$ is a "space of $G$-torsors". By "$G$-torsor" I mean a topological space with a free and transitive $G$-action, for example the fibres of a principal $G$-bundle.

There is a topological characterization of $BG$ as follows:

Suppose $E$ is a contractible space with a free $G$ action such that the quotient map $E\to E/G$ is a fibre bundle. Then $E \to E/G$ is a model for the universal principal $G$ bundle. (In particular $E/G$ is a model for $BG$.)

Moreover every universal bundle $EG \to BG$ arises in this way.

But what is the space $E/G$? Each point in $E/G$ is a $G$-orbit in $E$, which is already a $G$-torsor. Any continuous function $f\colon X \to BG$ picks for each $x\in X$ a $G$-torsor $f(x)\in BG$, each already equipped with a $G$-action from $E$, and because $f$ is continuous these actions also vary continuously from fibre to fibre resulting in a principal $G$-bundle over $X$. Varying $f$ by a homotopy results in a different but isomorphic principal bundle.


In certain cases our group $G$ is the structure group of a different type of bundle we're studying: for example $O(n)$ is the structure group for rank $n$ vector bundles, and if $M$ is a smooth manifold $Diff(M)$ is the structure group for $M$-bundles. In special cases the classifying space can be modelled using moduli spaces of these fibre types: $BO(n)$ can be described as the Grassmannian $Gr_n(\mathbb{R}^\infty)$ of all $n$-dimensional linear subspaces of $\mathbb{R}^\infty$, where $O(n)$ has a free transitive action on the contractible Stiefel manifold $St_n(\mathbb{R}^\infty)$ of $n$-frames, and $BDiff(M)$ the moduli space of submanifolds of $\mathbb{R}^\infty$ diffeomorphic to $M$, where $Diff(M)$ acts on the space of embeddings $Emb(M, \mathbb{R}^\infty)$. (Note that these are only really classifying spaces for bundles over paracompact spaces.) In these cases we are able to identify each $G$-orbit with the fibre type we're interested in.

I have often wondered if for any $G$ and any $G$-space $F$ whether we can model $BG$ as a moduli space of objects of "type" $F$ as in the case of vector and manifold bundles, but I do not know.

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  • $\begingroup$ Thank you for your answer. I have another question kind of related to this. I am learning about stacks these days and, have read that they are also solving moduli problems. Do you know how stacks appears as classifying objects? $\endgroup$ – Bumblebee May 23 at 18:30
  • $\begingroup$ Probably $\uparrow$ must be a separate question. But I am not sure, since it might be trivial to other people. $\endgroup$ – Bumblebee May 23 at 18:32
  • $\begingroup$ @Bumblebee I don't know the answer to that question, I think it would be worth asking a separate question about that. (Make sure to refer to this question since it is related) $\endgroup$ – William May 23 at 18:53
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The main difference is that maps to moduli spaces represents certain classes of maps to an object in the same category, while in classifying spaces they are in different categories: maps to the classifying space are defined only up to homotopy, i.e. maps in the homotopy category, while vector bundles are defined in the category of topolgical spaces - any vector bundle is homotopy equivalent to the original space. See here for more details: https://ncatlab.org/nlab/show/moduli+space#because

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  • $\begingroup$ Thank you. I think, I see your point :). Could you please explain the phrase "maps to moduli spaces represents certain classes of maps" little bit more. $\endgroup$ – Bumblebee May 23 at 18:41
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    $\begingroup$ The typical example you should have in find is that of a vector bundle on topological spaces. Vector bundles over a point are just vector spaces, and then a vector bundle is a "continuous family of vector spaces" when by continuous we mean that we assign to this family a topological structure and a projection map to the original space. We say that there is a moduli space that classifies such families if there is a universal example of such a family - for v.b. this is the canonical bundle over $BG$ - such that any other example is just a pullback of this universal bundle along some map to $BG$ $\endgroup$ – E. KOW May 24 at 12:05
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    $\begingroup$ Now essentially any moduli problem has this form, using the "category of elements" that assign to any functor $F:\mathcal{C} \to Set$ (AKA a "moduli problem") a category $\operatorname{El\left(F\right)}$ with a functor to $\mathcal{C}$. If the functor is representable (i.e. there is a moduli space $X$) then this category is equivalent to the slice category $\mathcal{C}_{/X}$. In that sense, we think of the category $El\left(F\right)$ as combining the kind of objects we want to assign to a point, such as vector spaces, with the right notion of being a continuous family with a map to an object. $\endgroup$ – E. KOW May 24 at 12:30

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