Pushforward

Definition

Let $p:\mathcal{E}\Rightarrow \mathcal{B}$ be an opfibration. For a morphism $u\in \mathcal{B}[b,b^{\prime }]$, a pushforward along $u$ is the functor

$$u_{!}:{\mathcal{E}}_b\Rightarrow {\mathcal{E}}_{b^{\prime }}$$

determined by a choice of cocartesian lifts over $u$.

Explicitly, for an object $x\in {\mathcal{E}}_b$, choose a coCartesian morphism

$${\phi }_x:x\to u_{!}x$$

with $p({\phi }_x)=u$. The codomain $u_{!}x$ lies in the fibre ${\mathcal{E}}_{b^{\prime }}$ and is called the pushforward of $x$ along $u$.

On Morphisms

Given a vertical morphism

$$f:x\to y$$

in the fibre ${\mathcal{E}}_b$, the pushforward morphism

$$u_{!}f:u_{!}x\to u_{!}y$$

is the unique vertical morphism in ${\mathcal{E}}_{b^{\prime }}$ such that

$$u_{!}f\circ {\phi }_x={\phi }_y\circ f.$$

Its existence and uniqueness follow from the cocartesian universal property of ${\phi }_x$.

Interpretation

Pushforward is the covariant transport of objects and morphisms along a base morphism in an opfibration. It is dual to Reindexing, which transports objects contravariantly in a Grothendieck Fibration.

With a chosen op-cleavage, the assignments $b\mapsto {\mathcal{E}}_b$ and $u\mapsto u_{!}$ assemble into a pseudofunctor

$$\mathcal{B}\Rightarrow \mathbf{Cat}.$$

If the opfibration is split, this functoriality is strict.

Examples

• In the Grothendieck construction of a pseudofunctor $Q:\mathcal{B}\Rightarrow \mathbf{Cat}$, pushforward along $u$ is exactly $Q(u)$.
• In a domain opfibration, pushforward along a map is given by pushout.
• In a product projection $\mathcal{B}\times \mathcal{D}\Rightarrow \mathcal{B}$, pushforward along $u$ leaves the $\mathcal{D}$-component unchanged.

Related Concepts

Opfibration
Cartesian Lift
Grothendieck Fibration
Reindexing
Pseudofunctor