nLab representable morphism of stacks

Context

$(\infty,1)$-Topos Theory

(∞,1)-topos theory

Constructions

structures in a cohesive (∞,1)-topos

Contents

Definition

A morphism $f : X \to Y$ of stacks over a site $C$ is called representable if for all representable objects $U \in C \stackrel{Y}{\hookrightarrow} Stacks(C)$ and all morphisms $U \to Y$ the homotopy pullback $X \times_Y U$ in

$\array{ X \times_Y U &\to& X \\ \downarrow &{}^{\simeq}\swArrow& \downarrow^f \\ U &\to& Y }$

is again representable.

Properties

Push-forward in generalized cohomology

Along representable morphisms $f$ of stacks over smooth manifolds (smooth infinity-groupoids) is induced a push-forward in generalized cohomology operation.

References

The general definition appears for instance as def. 38.5 in

(there with stacks perceived equivalently and dually under the Grothendieck construction as fibered categories).

Applications of push-forward in generalized cohomology along representable morphisms appear for instance in

Revised on November 7, 2012 21:53:48 by Urs Schreiber (82.169.65.155)