Grothendieck Fibration

Definition

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

A morphism $\phi \in \mathcal{E}[e^{\prime },e]$ over $u\in \mathcal{B}[b^{\prime },b]$ is cartesian when for every $h\in \mathcal{E}[x,e]$ and $v\in \mathcal{B}[p(x),b^{\prime }]$ with $p(h)=u\circ v$, there exists a unique $k\in \mathcal{E}[x,e^{\prime }]$ such that $h=\phi \circ k$ 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 cartesian lifts determines for each $u\in \mathcal{B}[b^{\prime },b]$ a reindexing functor $u^{\ast }:{\mathcal{E}}_b\Rightarrow {\mathcal{E}}_{b^{\prime }}$, pseudofunctorial in $b$. This presentation is equivalent to giving a pseudofunctor ${\mathcal{B}}^{op}\Rightarrow \mathbf{Cat}$ via the Grothendieck construction.

Extra Structure

• Cloven Fibration: a Grothendieck fibration equipped with a specified choice of cartesian lift for every $e\in \mathcal{E}$ and $u\in \mathcal{B}[b^{\prime },p(e)]$ (a cleavage).
• Split Fibration: a cloven fibration whose chosen lifts are strictly functorial, i.e. $({\mathrm{id}}_b{)}^{\ast }={\mathrm{Id}}_{{\mathcal{E}}_b}$ and $(u\circ v{)}^{\ast }=v^{\ast }\circ u^{\ast }$ on the nose.

Variants

• Opfibration: the dual notion using cocartesian morphisms and pushforward along $u\in \mathcal{B}[b,b^{\prime }]$.
• Discrete Fibration: a fibration in which cartesian lifts (hence reindexing) are unique; fibres embed as discrete opfibrations.
• Cartesian Fibration: higher-categorical generalisation (e.g. in $\infty$-categories/quasicategories) using cartesian edges.
• Cocartesian Fibration: the corresponding higher-categorical op-variant using cocartesian edges.
• Street Fibration: 2-categorical generalisation defined using 2-cells and cartesian liftings in a 2-category.

Examples

• Codomain Fibration: for a category $\mathcal{C}$ with pullbacks, $p:{\mathcal{C}}^{\to }\Rightarrow \mathcal{C}$ sending $f\in \mathcal{C}[X,B]$ to $B$; cartesian morphisms are pullback squares.
• Simple Fibration: the Grothendieck construction $\int P\Rightarrow \mathcal{B}$ of a pseudofunctor $P:{\mathcal{B}}^{op}\Rightarrow \mathbf{Cat}$; canonical example of a split fibration.
• Family Fibration: over Set, objects are families $(B,(E_b{)}_{b\in B})$ and $p$ forgets to $B$; reindexing is substitution along functions.

References

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