Discrete Fibration

Definition

A discrete fibration is a Grothendieck Fibration $p:\mathcal{E}\Rightarrow \mathcal{B}$ such that for every $e\in \mathcal{E}$ and morphism $u\in \mathcal{B}[b^{\prime },p(e)]$, there exists a unique arrow $\phi \in \mathcal{E}[e^{\prime },e]$ with $p(\phi )=u$.

Equivalently:

• All morphisms of $\mathcal{E}$ are cartesian, and
• The fibres ${\mathcal{E}}_b=\{e\in \mathcal{E}\mid p(e)=b\}$ are discrete categories (only identity morphisms).

By uniqueness of lifts, a discrete fibration is canonically cloven and the chosen lifts are strictly functorial, so every discrete fibration is a Split Fibration.

Characterisations

• Presheaf perspective: Discrete fibrations over $\mathcal{B}$ are (up to isomorphism) exactly the Grothendieck constructions of presheaves $P:{\mathcal{B}}^{op}\Rightarrow \mathbf{Set}$. Concretely, $p:\int P\Rightarrow \mathcal{B}$ is a split discrete fibration, and every discrete fibration arises this way.
• Reindexing as substitution: For $u\in \mathcal{B}[b^{\prime },b]$, the reindexing functor $u^{\ast }:{\mathcal{E}}_b\Rightarrow {\mathcal{E}}_{b^{\prime }}$ is determined on objects and is strictly functorial: $({\mathrm{id}}_b{)}^{\ast }=\mathrm{Id}$ and $(u\circ v{)}^{\ast }=v^{\ast }\circ u^{\ast }$.
• No nontrivial vertical maps: If $p(f)=\mathrm{id}$, then $f=\mathrm{id}$.

Properties

• Closed under pullback, composition, and retracts in Cat.
• Every functor $p:\mathcal{E}\Rightarrow \mathcal{B}$ that is a discrete fibration is automatically a Grothendieck Fibration.
• Discrete opfibrations are the dual notion (unique cocartesian lifts); they correspond to copresheaves $P:\mathcal{B}\Rightarrow \mathbf{Set}$ via the Grothendieck construction.

Examples

• Category of elements: For a presheaf $P:{\mathcal{B}}^{op}\Rightarrow \mathbf{Set}$, the projection $\int P\Rightarrow \mathcal{B}$ is a split discrete fibration.
• Slices: For any $b\in \mathcal{B}$, the domain projection $\mathrm{dom}:\mathcal{B}/b\Rightarrow \mathcal{B}$ is a discrete fibration (Yoneda corresponds to $P=\mathcal{B}(-,b)$).
• Product projection: $p:\mathcal{B}\times \mathbf{S}\Rightarrow \mathcal{B}$ (where $\mathbf{S}$ is a fixed discrete category) is a split discrete fibration.

Related Concepts

• Fibration
• Grothendieck Fibration
• Discrete Opfibration
• Split Fibration
• Cloven Fibration
• Category of Elements
• Presheaf
• Family Fibration
• Set-Indexed Family

References

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