# Relation/Difference between moduli spaces and classifying spaces.

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.

• 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 $BDiff(M)$. – William May 20 at 23:52

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.

• 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? – Bumblebee May 23 at 18:30
• Probably $\uparrow$ must be a separate question. But I am not sure, since it might be trivial to other people. – Bumblebee May 23 at 18:32
• @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) – William May 23 at 18:53

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

• 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. – Bumblebee May 23 at 18:41
• 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 $BG$ – E. KOW May 24 at 12:05
• 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. – E. KOW May 24 at 12:30