nLab
representable morphism of stacks

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Definition

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

X× YU X f U Y \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 ff 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)