Discrete Opfibration

Definition

A discrete opfibration is an Opfibration $p:\mathcal{E}\to \mathcal{B}$ such that for every $e\in \mathcal{E}$ and $u:p(e)\to b^{\prime }$ in $\mathcal{B}$, there exists a unique cocartesian lift $\phi :e\to e^{\prime }$ with $p(\phi )=u$. Equivalently, all morphisms in $\mathcal{E}$ are cocartesian, the fibres ${\mathcal{E}}_b$ are discrete categories (only identities), and vertical morphisms are identities.

Characterisations

• Copresheaf viewpoint: discrete opfibrations over $\mathcal{B}$ are (up to isomorphism) exactly Grothendieck constructions $\int P\to \mathcal{B}$ of copresheaves $P:\mathcal{B}\to \mathbf{Set}$ (the category of elements).
• Duality: this is the dual notion to a Discrete Fibration (presheaves P : \mathcal B^{op} \to \mathbf{Set}}).

Examples

• For any copresheaf $P:\mathcal{B}\to \mathbf{Set}$, the projection $\int P\to \mathcal{B}$ is a split discrete opfibration.

Related Concepts

Opfibration Discrete Fibration Grothendieck Fibration Category of Elements Copresheaf