Some constructions can be seen as objects representing a functor.
For example,
- Consider a topological group $G$ and a functor $\mathcal{F}:\text{Top}\rightarrow \text{Gpd}$ defined as $M\mapsto \mathcal{F}(M)$ where $\mathcal{F}(M)$ is the category of principal $G$-bundles over $M$ (whose objects are principal $G$-bundles over $M$ and morphisms are $G$-equivariant maps that induce identity on base spaces). Here, $\text{Top}$ is with morphisms as homotopy classes of continuous maps. In case this functor $\mathcal{F}$ is representable, then, we should get an object $\mathcal{F}_G$ of $\text{Top}$, that is a topological space, such that $\mathcal{F}(M)\cong \text{Hom}(M,\mathcal{F}_G)$ for each $M$ in $\text{Top}$. Then it is understood that this is a representable functor and the object of $\text{Top}$ representing this functor is denoted by $BG$, called the classifying space of $G$. I see many times that same can be said about $G$ being Lie group, but, I am not very confident about this, so, not saying anything more.
- Consider a commutative ring $R$ and $R$-modules $M,N$ with a functor $\mathcal{F}:R\text{-Mod}\rightarrow \text{Set}$ defined as $Q\mapsto \mathcal{F}(Q)$ where $\mathcal{F}(Q)$ is the set of bilinear maps $M\times N\rightarrow Q$. In case this functor $\mathcal{F}$ is representable, then, we we should get an object get an $R$-module $\mathcal{F}_{M,N}$ such that $\mathcal{F}(Q)\cong \text{Hom}(\mathcal{F}_{M,N},Q)$ for each $Q$ in $R\text{-Mod}$. Then, it is understood that this functor is rep. and the object of $R\text{-Mod}$ representing this functor is denoted by $M\otimes_RN$, called the tensor product of $M$ and $N$.
- Consider the category $\mathcal{C}$ of CW complexes with morphisms as homotopy classes of continuous maps and a functor $\mathcal{F}^n:\mathcal{C}\rightarrow \text{Set}$ defined as $M\mapsto \mathcal{F}^n(M)$, where $\mathcal{F}^n(M)$ is the set underlying the cohomology group $H^n(M,\mathbb{Z})$. In case this functor $\mathcal{F}^n$ is representable, then, we should get an object $\mathcal{F}^n_n$ of $\mathcal{C}$, that is a CW comple, such that $\mathcal{F}^n(M)=\text{Hom}(M,\mathcal{F}^n_n)$ for each $M$ in $\mathcal{C}$. Then, it is understood that this is a representable functor and the object of $\mathcal{C}$ representing this functor is denoted by $K(\mathbb{Z},n)$, called the Eilenberg-Maclane space of degree $n$.
Are there any other such constructions that can be introduced using the notion of representable functor. There is a restriction of maximum 5 tags, but constructions from any field are welcome.
As mentioned by Laurent Moret-Bailly, Dmitri Pavlov most constructions in category theory can be expressed in terms of Representable functors. I am looking for some non trivial examples, I do not have any precise definition of "non trivial" yet.
