nLab
representable morphism

Contents

Contents

Idea

In the philosophy of the Grothendieck school, one starts with some category CC of “local models” of spaces, equips it with a subcanonical Grothendieck topology, τ\tau, and enlarges CC to some category of sheaves of sets on the site (C,τ)(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” XCX\in C have to be extended to properties of arbitrary spaces (i.e. sheaves on (C,τ)(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 XX regarded as the morphism X1X\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 𝒫\mathcal{P} be a class of morphisms in a category CC which is closed under isomorphisms, i.e. it is replete when regarded as a full subcategory of the arrow category of CC.

Definition

A morphism α:FG\alpha : F\to G of presheaves of sets on CC is said to be representable by a morphism in 𝒫\mathcal{P} if for every morphism from a representable presheaf h XGh_X\to G, the projection from the pullback F× Gh Xh XF\times_G h_X\to h_X is (the image under the Yoneda embedding of) a morphism in 𝒫\mathcal{P}.

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

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

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

Examples

Last revised on June 12, 2019 at 02:49:36. See the history of this page for a list of all contributions to it.