nLab representable morphism of stacks

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.

Related concepts

• representable morphism.

References

The general definition appears for instance as def. 38.5 in

• Aise Johan de Jong, The Stacks Project – Categories (pdf)

(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

• Johannes Ebert, Jeffrey Giansiracusa, Pontrjagin–Thom maps and the homology of the moduli stack of stable curves, Math. Ann. (2011) 349:543–575 (arXiv:0712.0702)