representable morphism

In the philosophy of the Grothendieck school, one starts with some category $C$ of βlocal modelsβ 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.

The main role of properties of spaces have to be done in a relative setup, that is, the emphasis is on **properties of morphisms**. 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.

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 $C$.

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 $\hat{C} = Set^{C^{op}}$.

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

Revised on October 5, 2012 19:56:39
by Mike Shulman
(192.16.204.218)