Cloven Fibration

Definition

A cloven fibration is a Grothendieck Fibration $p:\mathcal{E}\Rightarrow \mathcal{B}$ equipped with a specified choice of cartesian lift (a cleavage). Concretely, for every object $e\in \mathcal{E}$ and morphism $u\in \mathcal{B}[b^{\prime },p(e)]$ in $\mathcal{B}$, a cloven fibration picks a cartesian arrow $\chi (u,e):u^{\ast }e⟶e\text{with}p(\chi (u,e))=u.$

These chosen lifts determine for each $u\in \mathcal{B}[b^{\prime },b]$ a reindexing functor on fibres $u^{\ast }:{\mathcal{E}}_b\Rightarrow {\mathcal{E}}_{b^{\prime }},$ well-defined up to coherent isomorphism and pseudofunctorial in $b$.

Notation

• Chosen cartesian lift of $u\in \mathcal{B}[b^{\prime },p(e)]$ at $e$: $\chi (u,e):u^{\ast }e\to e$ with $p(\chi (u,e))=u$.
• Reindexing on objects: $u^{\ast }:{\mathcal{E}}_b\Rightarrow {\mathcal{E}}_{b^{\prime }}$, with coherence isomorphisms
  • $({\mathrm{id}}_b{)}^{\ast }\cong {\mathrm{Id}}_{{\mathcal{E}}_b}$
  • $(u\circ v{)}^{\ast }\cong v^{\ast }\circ u^{\ast }$.

Properties

• The cleavage specifies a concrete choice of cartesian lifts, turning the abstract existence in a Grothendieck Fibration into explicit data.
• Reindexing functors $u^{\ast }$ are determined by the chosen lifts and assemble pseudofunctorially ${\mathcal{B}}^{op}\Rightarrow \mathbf{Cat}$.
• Base change along $u$ preserves cartesian arrows and interacts coherently with composition via the cleavage isomorphisms.

Relation to Split Fibration (stub)

A split fibration is a cloven fibration whose chosen reindexing is strictly functorial: $({\mathrm{id}}_b{)}^{\ast }={\mathrm{Id}}_{{\mathcal{E}}_b}\text{and}(u\circ v{)}^{\ast }=v^{\ast }\circ u^{\ast }$ on the nose (no coherence isomorphisms needed). Every Discrete Fibration is canonically split. See Split Fibration for details.

Examples

• Simple Fibration: For a pseudofunctor $P:{\mathcal{B}}^{op}\Rightarrow \mathbf{Cat}$, the Grothendieck construction $\int P\Rightarrow \mathcal{B}$ is naturally a split (hence cloven) fibration.
• Discrete Fibration: Unique cartesian lifts give a canonical cleavage that is strictly functorial, so it is split.
• Codomain Fibration: If $\mathcal{C}$ has chosen pullbacks, then $p:{\mathcal{C}}^{\to }\Rightarrow \mathcal{C}$ is cloven; with strictly functorial chosen pullbacks it becomes split.
• Family Fibration over Set: Reindexing is given by substitution along functions; with strictly functorial substitution it is split.

Related Concepts

• Fibration
• Grothendieck Fibration
• Split Fibration
• Opfibration
• Discrete Fibration
• Street Fibration
• Cartesian Fibration
• Cocartesian Fibration

References

jacobs1999-categorical-logic riehl2014-categorical-homotopy-theory