nLab
representable morphism

Contents

Idea

In the philosophy of the Grothendieck school, one starts with some category CC of β€œlocal models” of spaces, equips it with a subcanonical Grothendieck topology, Ο„\tau, and enlarges CC to some category of sheaves of sets on the site (C,Ο„)(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.

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 Ξ±:Fβ†’G\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 Xβ†’Gh_X\to G, the projection from the pullback FΓ— Gh Xβ†’h 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 Ξ±\alpha is representable.

In geometrical contexts, we usually assume that 𝒫\mathcal{P} is itself closed under pullbacks in CC, i.e. if f:Xβ†’Yf: X\to Y is in 𝒫\mathcal{P} and g:Vβ†’Yg : V\to Y a morphism in CC, then the pullback XΓ— YVX\times_Y V exists and the projection XΓ— YVβ†’VX\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 Xβ†’h 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

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