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\begin{document}

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\section*{moduli space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{yoneda_lemma}{}\paragraph*{{Yoneda lemma}}\label{yoneda_lemma}

[[!include Yoneda lemma - contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{caveat}{Caveat}\dotfill \pageref*{caveat} \linebreak
\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{coarse_moduli_space}{Coarse moduli space}\dotfill \pageref*{coarse_moduli_space} \linebreak
\noindent\hyperlink{because}{``\ldots{} because they have automorphisms.''}\dotfill \pageref*{because} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{rough idea} is (but see the \hyperlink{caveat}{caveat} below) that the term \textbf{moduli space} is essentially a synonym for [[representable functor|representing object]] and for [[classifying space]]. People tend to say ``classifying space'' when in the context of [[topology]], and they tend to say \emph{moduli space} when in a context of [[complex geometry]] or [[algebraic geometry]].

More precisely, when a moduli space actually does exist as an ordinary space (or [[scheme]]), it is called for emphasis a \textbf{fine moduli space}. Fine here refers to the completeness of the description, not shared by coarse moduli below.

Those classifying ``spaces'' that are called moduli spaces are typically [[orbifold]]s, hence modeled not just as spaces but as [[groupoid]]s with extra structure. These are typically conceived as [[stack]]s: these are then called \textbf{[[moduli stack]]}s. Typically these are demanded to be [[Deligne-Mumford stack]]s.

So the term \emph{fine moduli space} mainly indicates that a given object that might be a [[Deligne-Mumford stack]] is actually just a plain [[scheme]]. But there is also the notion of \textbf{coarse moduli space}, which is a kind of conceptual hack designed to be able to keep thinking about what really wants to be a [[stack]] still as a plain [[sheaf]].

A coarse moduli space is one that at least has the right underlying [[set]] of points as the \emph{right} [[moduli stack]] has: as long as we don't look at families but just at single things, it does give the right information. From the point of view of [[derived algebraic geometry]], the coarse moduli spaces are 0-truncations of derived [[moduli stacks]] when they exist.

\hypertarget{caveat}{}\subsubsection*{{Caveat}}\label{caveat}

But one has to be \textbf{careful with this rough idea}: as there is usually some \textbf{implicit fine print} in the notion of moduli space:

while [[classifying space]] is the term typically used for a [[representable functor|representing object]] in a [[homotopy category]], a representing object is usually called a \emph{moduli space} only if it lives in the ``original'' category. For more on this see the chapter

\begin{itemize}%
\item \hyperlink{because}{\ldots{} because they have automorphisms}

\end{itemize}
below.

\hypertarget{history}{}\subsection*{{History}}\label{history}

The term possibly originates with Riemann, who was the first to study what are now called moduli spaces of (compact) [[Riemann surface]]s. A ``modulus'' here is meant to be a \emph{parameter} that parameterizes isomorphism classes of Riemann surfaces.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\ldots{}

\hypertarget{coarse_moduli_space}{}\subsubsection*{{Coarse moduli space}}\label{coarse_moduli_space}

A \textbf{coarse moduli space} for a [[presheaf]] $(Sch/\mathbb{C})^{op} \to Set$ on complex [[scheme]]s is a [[scheme]] $M \in Sch/\mathbb{C}$ equipped with a morphism

\begin{displaymath}
\Psi_M : F \to h_M
  \,,
\end{displaymath}
where $h$ denotes the [[Yoneda embedding]], such that

a) $F(Spec(\mathbb{C})) \to h_M(Spec \mathbb{C}) = hom(Spec \mathbb{C}, M)$ is a [[bijection]]

b) given $M'$ and $\Psi_{M'} : F \to h_{M'}$ then there exists unique $M \to M'$ such that $\itexarray{ F && \stackrel{\Psi_{M'}}{\to}&& h_{M'} \\ & {}_{\Psi_M}\searrow && \nearrow \\ && h_M}$

So a coarse moduli space is one that at least has the right underlying set of points as the \emph{right} [[moduli stack]] has: as long as we don't look at families but just at single things, it does give the right information. From the point of view of [[derived algebraic geometry]], the coarse moduli spaces are 0-truncations of derived [[moduli stacks]] when they exist.

\hypertarget{because}{}\subsection*{{``\ldots{} because they have automorphisms.''}}\label{because}

A widespread slogan is

\begin{quote}%
\textbf{Slogan}: A type of objects that has nontrivial [[automorphism]]s cannot have a fine moduli space.


\end{quote}
The typical example motivating and illustrating this slogan is the example of [[elliptic curve]]s: from a single [[elliptic curve]] $E$ with a non-trivial automorphism $\alpha : E \to E$ one may build, for instance, the family $\mathcal{E} \to S^1 \times S^1$ of elliptic curves over the torus which is obtained by starting with the trivial family $E \times S^1 \times [0,1] \to S^1 \times [0,1]$ over the cylinder and then by gluing the two ends by identifying the fiber using the automorphism $\alpha$. The resulting family is \emph{locally trivial} but globally nontrivial.

Such locally trivial but globally non-trivial families can in principle not be classified by a representing object in the category of spaces that we are working with, as long as the [[Grothendieck topology]] which we use to say what a locally trivial family is is [[subcanonical coverage|subcanonical]]:

for assume that a classifying object $K$ exists in our category and assume that at least one globally nontrivial but locally trivial family $\mathcal{E} \to X$ exists in the category.

Then by definition of local triviality there exists a covering [[sieve]] $\{U_i \to X\}$ such that the [[pullback]] of $\mathcal{E}$ to each $U_i$ is equivalent to the trivial family over $U_i$. Accordingly, it will be represented by the trivial element in $Hom(U_i, K)$. But if $\{U_i \to X\}$ is a covering [[sieve]] in a subcanonical topology, then it follows from the [[sheaf]] property of the [[representable functor|representable presheaf]] $Hom(-,K)$ that there is a unique element in $Hom(X,K)$ whose restriction to all the $U_i$ represents the trivial family. But the trivial family over $X$ has this property and hence must be that unique element. Accordingly, it would follow that there is no non-trivial family classified by $K$ over $X$, which is a contradiction and shows that the ``fine moduli space'' $K$ cannot exist under the given assumptions.

Notice well the two assumptions that were made to make this argument work:

\begin{enumerate}%
\item the topology with which we glue families must be subcanonical: this is the assumption that fails for the situations in which one would speak of [[classifying space]]s instead of moduli spaces:

For instance [[vector space]]s do certainly have nontrivial [[automorphism]]s. A (topological, say) family of vector space is a [[vector bundle]]. So a naive application of the above argument might lead one to conclude that there cannot be a classifying space of vector bundles!

But of course it is a standard fact that there there are [[topological space]]s $\mathcal{B} U(n)$ such that [[homotopy]] classes of continuous maps $X \to \mathcal{B} U(n)$ classify isomorphism classes of rank-$n$ [[vector bundle]]s. But this means that the functor that assigns families of vector bundles to topological space is -- while not [[representable functor|representable]] in [[Top]] -- representable in the [[homotopy category]] $Ho(Top)$

\begin{displaymath}
Rank n VectBund(X)/iso  \simeq Hom_{Ho(Top)}(X, \mathcal{B} U(n))
  \,.
\end{displaymath}
This might make the [[classifying space]] $\mathcal{B} U(n)$ look like something that deserves to be called a fine moduli space. But one should beware that the standard [[Grothendieck topology]] of [[Top]], which is [[subcanonical coverage|subcanonical]] in [[Top]] is not so in $Ho(Top)$: the functor $Hom_{Ho(Top)}(-, \mathcal{B} U(m))$ has a non-standard opinion about what it means to glue two topological spaces to a third one. That's how in $Ho(Top)$ the above argument that ``automorphisms prevent a classifying object'' breaks down.

In fact, there is a theorem by [[Danny Stevenson]] and [[David Roberts]], extending a theorem by [[John Baez]] and Danny Stevenson that shows that large classes of [[principal ∞-bundle]]s, even, do have classifying topological spaces in this sense. These are objects that not only have automorphisms, but have automorphisms of automorphisms, etc, and hence seem to contradict the above slogan in a maximal way.

But these examples also indicate why it may be misguided to think of the ``classifying spaces'' such as $\mathcal{B} U(n)$ as being moduli spaces in the first place: the [[homotopy category]] $Ho(Top)$ in fact regards its objects -- even though they are presented as [[topological space]]s -- not really as spaces but as the [[∞-groupoid]]s that they are equivalent to under the [[homotopy hypothesis]]. So in that sense objects such as $\mathcal{B} U(n)$ are not classifying spaces, but are classifying [[groupoid]]s after all.


\item Another situation that makes the above argument break down is if the category of ``spaces'' that one works with is such that gluing trivial families by nontrivial automorphisms never produces globally nontrivial families. For instance because the category of ``spaces'' is such that no nontrivial families exist at all. This happens for instance in the case that we take a ``space'' to be simply a (finite, say) [[set]]. In the category [[Set]] of (finite) sets every fiber bundle $\mathcal{E} \to X$ (i.e. every surjection such that all fibers are isomorphic to a typical fiber $F$ ) are necessarily globally trivial in that they are isomorphic to $F \times X \to X$. And in fact, there \emph{is} a ``fine moduli space'' in this context (inside the category of non-necessarily finite sets), namely the set $\mathbb{N}$ of natural numbers: the family $(\mathcal{E} \to X) \in Set$ with typical fiber the $n$-element set is classified by the map that sends all $x \in X$ to $n \in \mathbb{N}$.



\end{enumerate}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

[[!include moduli spaces -- contents]]

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[moduli]], [[moduli problem]], [[moduli stack]]


\item [[classifying space]]. [[moduli stack]]


\item [[classifying morphism]], [[modulating morphism]]


\item [[representable functor]]


\item [[classifying space]], [[classifying stack]], [[classifying topos]]


\item [[moduli stack]], [[derived moduli space]], [[Deligne-Mumford stack]]


\item [[geometric invariant theory]], [[Moduli Problems and DG-Lie Algebras]],


\item [[universal principal bundle]], [[universal principal infinity-bundle]], * [[moduli space of curves]],


\item [[Hilbert scheme]], [[Quot scheme]], [[Picard scheme]], [[moduli stabilization]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

\begin{itemize}%
\item [[David Ben-Zvi]], \emph{Moduli spaces} (\href{http://www.math.utexas.edu/users/benzvi/math/pcm0178.pdf}{pdf})


\item [[Ib Madsen]], \emph{Moduli spaces from a topological viewpoint}, ICM 2006 (\href{http://www.icm2006.org/proceedings/Vol_I/20.pdf}{pdf})



\end{itemize}
A bit of elementary exposition of these ideas is at

\begin{itemize}%
\item [[basic ideas of moduli stacks of curves and Gromov-Witten theory]]

\end{itemize}
[[!redirects moduli spaces]] [[!redirects space of moduli]]

[[!redirects fine moduli space]] [[!redirects coarse moduli space]]

[[!redirects fine moduli spaces]] [[!redirects coarse moduli spaces]]



\end{document}