Cartesian Fibration

Definition

A cartesian fibration is a functor of ∞-categories (e.g. quasicategories) $p:\mathcal{C}\to \mathcal{S}$ such that for every edge $f:s\to t$ in $\mathcal{S}$ and object $x\in \mathcal{C}$ with $p(x)=t$, there exists a $p$-cartesian edge $\phi :x^{\prime }\to x$ in $\mathcal{C}$ with $p(\phi )=f$. An edge $\phi :x^{\prime }\to x$ over $f:s\to t$ is $p$-cartesian if it satisfies the expected lifting/universal property on mapping spaces (or equivalently, induces a homotopy pullback on over-categories).¹²

Equivalently, cartesian fibrations over $\mathcal{S}$ classify contravariant functors ${\mathcal{S}}^{op}\to {\mathbf{Cat}}_{\infty }$ via straightening/unstraightening, generalising the 1-categorical correspondence between Grothendieck Fibrations and pseudofunctors ${\mathcal{B}}^{op}\to \mathbf{Cat}$.¹

Properties

• Stability: closed under pullback, composition, and equivalence over the base.
• Fibres and reindexing: the fibre ${\mathcal{C}}_s$ over $s\in \mathcal{S}$ is an ∞-category, and a choice of cartesian lifts along $f:s^{\prime }\to s$ determines a reindexing functor $f^{\ast }:{\mathcal{C}}_s\to {\mathcal{C}}_{s^{\prime }}$, functorial up to equivalence.
• Straightening/unstraightening: cartesian fibrations over $\mathcal{S}$ are equivalent to functors ${\mathcal{S}}^{op}\to {\mathbf{Cat}}_{\infty }$; dually, Cocartesian Fibrations correspond to functors $\mathcal{S}\to {\mathbf{Cat}}_{\infty }$.¹
• 1-categorical case: if $p:\mathcal{E}\to \mathcal{B}$ is a Grothendieck Fibration, then $N(p):N(\mathcal{E})\to N(\mathcal{B})$ is a cartesian fibration; conversely, a cartesian fibration of nerves comes from a Grothendieck fibration up to equivalence.

Examples

• Unstraightening of a functor: For $F:{\mathcal{S}}^{op}\to {\mathbf{Cat}}_{\infty }$, the unstraightening $\mathrm{Un}(F)\to \mathcal{S}$ is a cartesian fibration with fibre $\mathrm{Un}(F{)}_s\simeq F(s)$ and reindexing $f^{\ast }\simeq F(f)$.
• Arrow category: Over an ∞-category $\mathcal{S}$, the target projection ${\mathrm{ev}}_1:{\mathcal{S}}^{{\Delta }^1}\to \mathcal{S}$ is a cartesian fibration; cartesian edges are given by pullback squares in $\mathcal{S}$ (when defined internally).²
• Nerve lift of classical examples: The Codomain Fibration and Family Fibration become cartesian fibrations after applying nerves and passing to the ∞-categorical setting.

Related Concepts

• Cocartesian Fibration: dual notion with cocartesian lifts and pushforward along base morphisms.
• Grothendieck Fibration: 1-categorical precursor whose nerve yields a cartesian fibration.
• Street Fibration: 2-categorical generalisation; cartesian edges internal to a 2-category.
• Fibration: disambiguation/index of fibration notions across fields.
• Kan Fibration: simplicial-set notion with horn fillers (not the same as cartesian fibrations).
• Topological Fibration: overview for fibrations in topology.

References

Footnotes

1. https://www.math.ias.edu/~lurie/papers/HTT.pdf ↩ ↩² ↩³

2. https://ncatlab.org/nlab/show/cartesian+fibration ↩ ↩²