Codomain Fibration

Definition

Let $\mathcal{C}$ be a category with pullbacks. The arrow category ${\mathcal{C}}^{\to }$ has as objects the morphisms $f\in \mathcal{C}[X,B]$ of $\mathcal{C}$, and as morphisms the commutative squares in $\mathcal{C}$. The codomain functor $\mathrm{cod}:{\mathcal{C}}^{\to }\Rightarrow \mathcal{C},(f\in \mathcal{C}[X,B])\mapsto B$ is a Grothendieck Fibration.

A morphism in ${\mathcal{C}}^{\to }$ over $u\in \mathcal{C}[B^{\prime },B]$ is cartesian if and only if the corresponding square in $\mathcal{C}$ is a pullback. Thus, given $f\in \mathcal{C}[X,B]$ and $u\in \mathcal{C}[B^{\prime },B]$, a cartesian lift of $u$ at $f$ is the pullback $u^{\ast }f\in \mathcal{C}[X^{\prime },B^{\prime }]$ together with the pullback square. Existence of pullbacks guarantees the cartesian lifting property.

Fibres and reindexing

• Fibre over $B\in \mathcal{C}$: the slice category $\mathcal{C}/B$.
• For $u\in \mathcal{C}[B^{\prime },B]$, reindexing $u^{\ast }:\mathcal{C}/B\Rightarrow \mathcal{C}/B^{\prime }$ is given by pullback along $u$.
• A morphism in ${\mathcal{C}}^{\to }$ lying over ${\mathrm{id}}_B$ is cartesian iff it is an isomorphism in the slice $\mathcal{C}/B$.

Cloven and split structure

If $\mathcal{C}$ has a chosen pullback operation that is strictly functorial in the base (i.e. pullbacks of identities are identities and pullbacks compose on the nose), then the cleavage picking $u^{\ast }f$ makes $\mathrm{cod}:{\mathcal{C}}^{\to }\Rightarrow \mathcal{C}$ a Cloven Fibration, and in fact a Split Fibration: $({\mathrm{id}}_B{)}^{\ast }={\mathrm{Id}}_{\mathcal{C}/B},(u\circ v{)}^{\ast }=v^{\ast }\circ u^{\ast }.$

Properties

• Cartesian morphisms in ${\mathcal{C}}^{\to }$ are precisely pullback squares in $\mathcal{C}$.
• Reindexing $u^{\ast }$ preserves limits that are stable under pullback (in particular, pullbacks and monomorphisms).
• Base change isomorphisms (Beck–Chevalley) hold for pullback squares in $\mathcal{C}$.
• Dually, if $\mathcal{C}$ has pushouts, the domain projection $\mathrm{dom}:{\mathcal{C}}^{\to }\Rightarrow \mathcal{C}$ is an Opfibration with cocartesian morphisms the pushout squares.

Examples

• Set: Fibre over $B$ is $\mathbf{Set}/B$; $u^{\ast }$ is precomposition/pullback of functions.
• Top: Fibre over $B$ is $\mathbf{Top}/B$; $u^{\ast }$ takes pullbacks of continuous maps.
• Grp: Fibre over $B$ is $\mathbf{Grp}/B$; pullbacks exist and give reindexing.
• Any topos (or locally cartesian closed category): slices are cartesian closed and reindexing is strictly functorial.

Related Concepts

• Fibration
• Grothendieck Fibration
• Cloven Fibration
• Split Fibration
• Opfibration
• Set-Indexed Family

References

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