Opfibration

Definition

An opfibration (Grothendieck opfibration) is a functor $p:\mathcal{E}\Rightarrow \mathcal{B}$ such that for every object $e\in \mathcal{E}$ and morphism $u\in \mathcal{B}[p(e),b^{\prime }]$, there exists a cocartesian morphism $\phi \in \mathcal{E}[e,e^{\prime }]$ with $p(\phi )=u$.

A morphism $\phi \in \mathcal{E}[e,e^{\prime }]$ over $u\in \mathcal{B}[b,b^{\prime }]$ is cocartesian when for every $h\in \mathcal{E}[e,y]$ and $v\in \mathcal{B}[b^{\prime },p(y)]$ with $p(h)=v\circ u$, there exists a unique $k\in \mathcal{E}[e^{\prime },y]$ such that $h=k\circ \phi$ and $p(k)=v$.

Equivalently, writing ${\mathcal{E}}_b=\{e\in \mathcal{E}\mid p(e)=b\}$ for the fibre over $b$, a choice of cocartesian lifts determines for each $u\in \mathcal{B}[b,b^{\prime }]$ a pushforward functor $u_{!}:{\mathcal{E}}_b\Rightarrow {\mathcal{E}}_{b^{\prime }}$ that is pseudofunctorial in $b$. Giving an opfibration over $\mathcal{B}$ is equivalent (up to equivalence) to giving a pseudofunctor $\mathcal{B}\Rightarrow \mathbf{Cat}$ via the Grothendieck construction.

Extra Structure

• Cloven opfibration: a specified choice of cocartesian lift for every $e\in \mathcal{E}$ and $u\in \mathcal{B}[p(e),b^{\prime }]$ (an op-cleavage).
• Split opfibration: a cloven opfibration whose chosen lifts are strictly functorial, i.e. $({\mathrm{id}}_b{)}_{!}={\mathrm{Id}}_{{\mathcal{E}}_b}$ and $(v\circ u{)}_{!}=v_{!}\circ u_{!}$ on the nose.

Properties

• Closed under pullback, composition, and retracts in $\mathbf{Cat}$.
• Pushforward along $u$ is left adjoint to reindexing along $u$ when the latter is defined in a fibred setting (Beck–Chevalley and projection formulas express compatibility with base change).
• A functor is a discrete opfibration iff cocartesian lifts exist and are unique; then each fibre is a discrete category and the structure is strictly functorial.

Variants

• Grothendieck Fibration: the dual notion using cartesian morphisms and reindexing $u^{\ast }$.
• Discrete Opfibration: unique cocartesian lifts; corresponds to copresheaves $\mathcal{B}\Rightarrow \mathbf{Set}$ via the Grothendieck construction.
• Cartesian Fibration and Cocartesian Fibration: higher-categorical generalisations in $\infty$-categorical settings.
• Street Fibration: 2-categorical generalisation; dually, Street opfibrations.

Examples

• Simple opfibration: For a pseudofunctor $Q:\mathcal{B}\Rightarrow \mathbf{Cat}$, the Grothendieck construction $\int Q\Rightarrow \mathcal{B}$ is a split opfibration; pushforward along $u\in \mathcal{B}[b,b^{\prime }]$ is given by the action $Q(u):Q(b)\Rightarrow Q(b^{\prime })$.
• Category of elements of a copresheaf: For $P:\mathcal{B}\Rightarrow \mathbf{Set}$, the projection $\int P\Rightarrow \mathcal{B}$ is a split Discrete Opfibration.
• Domain opfibration: If $\mathcal{C}$ has pushouts, the domain projection $\mathrm{dom}:{\mathcal{C}}^{\to }\Rightarrow \mathcal{C}$ is an opfibration; cocartesian morphisms are pushout squares over arrows $u$.
• Product projection: $p:\mathcal{B}\times \mathcal{D}\Rightarrow \mathcal{B}$ is both a fibration and an opfibration; cocartesian lifts are given by identities in the $\mathcal{D}$-component.

Related Concepts

Fibration Grothendieck Fibration Discrete Opfibration Cloven Fibration Split Fibration Street Fibration Cartesian Fibration Cocartesian Fibration Family Fibration Set-Indexed Family

References

jacobs1999-categorical-logic