Street Fibration

Definition

A Street fibration is the 2-categorical analogue of a Grothendieck Fibration. Concretely, it is a 2-functor (or more generally a 1-cell in a 2-category) $p:\mathcal{E}\Rightarrow \mathcal{B}$ equipped with cartesian liftings of 1-cells and 2-cells in the base. A 1-cell $\phi \in \mathcal{E}[e^{\prime },e]$ over $u\in \mathcal{B}[b^{\prime },b]$ is cartesian when it satisfies the expected 2-dimensional universal property: any map and 2-cell into $e$ lying over $u$ factors essentially uniquely through $\phi$, with the factorisation strictly preserved by $p$ on 2-cells.

Equivalently, giving a Street fibration over $\mathcal{B}$ is (up to biequivalence) the same as giving a pseudofunctor ${\mathcal{B}}^{op}\Rightarrow \mathbf{Cat}$ via the 2-categorical Grothendieck construction.

Extra Structure

• Cloven Street fibration: a specified choice of cartesian lifts (a 2-dimensional cleavage).
• Split Street fibration: a cloven Street fibration whose chosen lifts are strictly 2-functorial (identities and composition hold on the nose).

Variants

• Street opfibration: defined dually using cocartesian 1-cells and 2-cells.
• Cartesian Fibration and Cocartesian Fibration: higher-categorical analogues in $(\infty ,1)$-settings.

Examples

• Ordinary Grothendieck Fibration: viewing categories as 2-categories with only identity 2-cells, every Grothendieck fibration $p:\mathcal{E}\Rightarrow \mathcal{B}$ is a Street fibration in $2\text{-}\mathbf{Cat}$.
• Family/Indexed examples: for a pseudofunctor $P:{\mathcal{B}}^{op}\Rightarrow \mathbf{Cat}$, its Grothendieck construction $\int P\Rightarrow \mathcal{B}$ is a split Street fibration.

Related Concepts

• Fibration
• Grothendieck Fibration
• Opfibration
• Discrete Fibration
• Cartesian Fibration
• Cocartesian Fibration

References