nLab representable morphism

Idea

In the philosophy of the Grothendieck school, one starts with some category $C$ of “local models” of spaces, equips it with a subcanonical Grothendieck topology, $\tau$, and enlarges $C$ to some category of sheaves of sets on the site $(C,\tau)$ playing the role of spaces. There are further generalizations to stacks and so on.

When doing this, we often find that properties of “local model spaces” $X\in C$ have to be extended to properties of arbitrary spaces (i.e. sheaves on $(C,\tau)$). In fact it is most natural to do this in a relative situation, i.e. to talk about properties of morphisms rather than properties of objects, with an object $X$ regarded as the morphism $X\to 1$. Thus, one of the main steps in the construction of the theory is to extend good classes of morphisms of local models to the category of spaces. Grothendieck axiomatizes the situation, actually for general presheaves.

Representable morphisms are also important in algebraic set theory and appear implicitly in the notion of category with families.

Definition

Let $\mathcal{P}$ be a class of morphisms in a category $C$ which is closed under isomorphisms, i.e. it is replete when regarded as a full subcategory of the arrow category of $C$.

Definition

A morphism $\alpha : F\to G$ of presheaves of sets on $C$ is said to be representable by a morphism in $\mathcal{P}$ if for every morphism from a representable presheaf $h_X\to G$, the projection from the pullback $F\times_G h_X\to h_X$ is (the image under the Yoneda embedding of) a morphism in $\mathcal{P}$.

When $\mathcal{P}$ is the class of all morphisms in $C$, we simply say that $\alpha$ is representable.

In geometrical contexts, we usually assume that $\mathcal{P}$ is itself closed under pullbacks in $C$, i.e. if $f: X\to Y$ is in $\mathcal{P}$ and $g : V\to Y$ a morphism in $C$, then the pullback $X\times_Y V$ exists and the projection $X\times_Y V\to V$ is in $\mathcal{P}$. If $C$ has all pullbacks, then the class of all morphisms in $C$ satisfies this property.

If $\mathcal{P}$ is closed under pullback, then a morphism $h_X\to h_Y$ between representable presheaves is representable by a morphism in $\mathcal{P}$ if and only if it is itself (the image under the Yoneda embedding of) a morphism in $\mathcal{P}$. In this way, the class $\mathcal{P}$ of morphisms in $C$ is extended to a class $\hat{\mathcal{P}}$ of morphisms in the category of presheaves of sets $\hat{C} = Set^{C^{op}}$.

Examples

• representable morphism of stacks (extends this notion from presheaves to stacks)