The classification of objects (by isomorphism classes) is an important problem in mathematics. These objects can be representations of a quiver, subspaces $V \subseteq \mathbb{C}^n$ of dimension $k$, plane curves, and so on. We classify by studying the relevant moduli. Moduli have their earliest roots in Riemann's 1857 paper. Over time, it has influenced many areas of mathematics through the work of invariant theorists like Hilbert, Weil, Mumford (and his Geometric Invariant Theory), and, of course, Grothendieck, with his formalization of analytic moduli theory. The study of moduli has advanced significantly since Grothendieck's definition of representability and the introduction of (pre)stacks and descent theory.
We will define the moduli functor in this post. So, a moduli space is a space (a scheme or a variety) whose points are in natural bijection with the isomorphism classes of families of objects (algebro-geometric objects such as those mentioned above). The meaning of 'space' depends on the context. Let us understand the meaning of the natural bijection. We need a functorial viewpoint here, and the Yoneda lemma would be handy. In algebraic geometry, we usually study objects over a base scheme $S$. But what about the objects under a base change? Hence, the language of functors.
A moduli functor is a contravariant functor $$F: \text{Sch}^{\text{op}} \to \text{Set}$$ which assigns to each scheme $S$ the set of isomorphism classes of families of objects over $S$, where for a map $f: S \to T$, we have $F(f): F(T) \to F(S)$ which is the pullback map from a family of objects over $T$ to $S$. So the moduli functor will retain the classification problem under a base change.
A moduli functor is a contravariant functor $$F: \text{Sch}^{\text{op}} \to \text{Set}$$ which assigns to each scheme $S$ the set of isomorphism classes of families of objects over $S$, where for a map $f: S \to T$, we have $F(f): F(T) \to F(S)$ which is the pullback map from a family of objects over $T$ to $S$. So the moduli functor will retain the classification problem under a base change.
Recall that we say that a contravariant functor $F: \mathcal{C}^{\text{op}}$ is representable by an object $X$ if there is a natural isomorphism $$\xi: F \to h_X$$ where $h_X$ is the functor of points associated to $X$. Now, whenever a functor $F: \text{Sch}^{\text{op}} \to \text{Set}$ is representable by a scheme $M$, such that $\xi: F \to h_M$ and $h_M(S) = Hom(S,M)$, then we call $M$ a fine moduli space. There is also a universal family on the scheme $S$ such that its pullback recovers all the families in $\mathcal{C}$. But we do not always get a fine moduli space and a universal family (the family of objects in $M$ for $id: M \to M$). In other cases, we may get a coarse moduli space. Furthermore, representability by a scheme can fail if the objects have nontrivial automorphism groups; then we move to a moduli stack (hopefully in the next post).
A very good example of representability in moduli theory is the Grassmannian. Grassmannian is used to classify the k-dimensional linear subspaces of an n-dimensional vector space. But in algebraic geometry, we are looking for families of objects over a base scheme $S$. We have a functor $$G(k,n): \text{Sch}^{\text{op}} \to \text{Set}$$ such that $G(k,n)$ associates to scheme $S$ the set $G(k,n)(S)$ of isomorphism classes of surjections $q: \mathcal{O}^{\oplus n}_S \to \mathcal{Q}$ where $Q$ is locally free $\mathcal{O}_S$-module of rank $n-k$. So, $\mathcal{Q}$ is locally free of rank $n$, but the quotient must be locally free of rank $n-k$. There is also a notion pullback here. So it is a moduli functor. In fact, it is representable by the Grassmannian scheme (making it a fine moduli space). There is a generalization of the Grassmannian moduli functor by Grothendieck (see here) in the construction of the Hilbert and Quot schemes.
