Cocartesian Fibration

Definition

A cocartesian 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 $p(x)=s$, there exists a $p$-cocartesian edge $\phi :x\to x^{\prime }$ in $\mathcal{C}$ with $p(\phi )=f$. An edge $\phi$ over $f$ is $p$-cocartesian if it satisfies the expected universal property on mapping spaces (equivalently, induces a homotopy pushforward on under-categories).¹²

Equivalently, cocartesian fibrations over $\mathcal{S}$ classify covariant functors $\mathcal{S}\to {\mathbf{Cat}}_{\infty }$ via straightening/unstraightening, generalising the 1-categorical correspondence between Opfibrations and pseudofunctors $\mathcal{B}\to \mathbf{Cat}$.¹

Properties

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

Examples

• Unstraightening: for $F:\mathcal{S}\to {\mathbf{Cat}}_{\infty }$, the unstraightening $\mathrm{Un}(F)\to \mathcal{S}$ is a cocartesian fibration with fibre $\mathrm{Un}(F{)}_s\simeq F(s)$ and pushforward $f_{!}\simeq F(f)$.
• Arrow category: for an ∞-category $\mathcal{S}$, the source evaluation ${\mathrm{ev}}_0:{\mathcal{S}}^{{\Delta }^1}\to \mathcal{S}$ is a cocartesian fibration; cocartesian edges correspond to (homotopy) pushout squares over the base edge.
• Nerve of classical opfibrations: if $\mathcal{C}$ has pushouts, the domain projection $\mathrm{dom}:{\mathcal{C}}^{\to }\to \mathcal{C}$ is an opfibration; applying the nerve yields a cocartesian fibration.

Related Concepts

• Cartesian Fibration
• Grothendieck Fibration
• Opfibration
• Street Fibration
• Fibration
• Topological Fibration
• Kan Fibration

References

Footnotes

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

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