nLab inner fibration




The notion of inner fibration of simplicial sets is one of the notion of fibrations of quasi-categories.

When the notion of (∞,1)-category is incarnated in terms of the notion of quasi-category, an inner fibration is a morphism of simplicial sets CDC \to D such that each fiber is a quasi-category and such that over each morphism f:d 1d 2f : d_1 \to d_2 of DD, CC may be thought of as the cograph of an (∞,1)-profunctor C d 1C d 2C_{d_1} ⇸ C_{d_2}.

So when D=*D = {*} is the point, an inner fibration C*C \to {*} is precisely a quasi-category CC.

And when D=N(Δ[1])D = N(\Delta[1]) is the nerve of the interval category, an inner fibration CΔ[1]C \to \Delta[1] may be thought of as the cograph of an (∞,1)-profunctor CDC ⇸ D.

This (,1)(\infty,1)-profunctor comes from an ordinary (∞,1)-functor F:CDF : C \to D precisely if the inner fibration KΔ[1]K \to \Delta[1] is even a coCartesian fibration. And it comes from a functor G:DCG : D \to C precisely if the fibration is even a Cartesian fibration. This is the content of the (∞,1)-Grothendieck construction.

And precisely if the inner fibration/cograph of an (,1)(\infty,1)-profunctor KΔ[1]K \to \Delta[1] is both a Cartesian as well as a coCartesian fibration does it exhibit a pair of adjoint (∞,1)-functors.


A morphism of simplicial sets F:XYF : X \to Y is an inner fibration or inner Kan fibration if its has the right lifting property with respect to all inner horn inclusions, i.e. if for all commuting diagrams

Λ[n] i X F Δ[n] Y \array{ \Lambda[n]_i &\to& X \\ \downarrow && \downarrow^{\mathrlap{F}} \\ \Delta[n] &\to& Y }

for 0<i<n0 \lt i \lt n there exists a lift

Λ[n] i X F Δ[n] Y. \array{ \Lambda[n]_i &\to& X \\ \downarrow &\nearrow& \downarrow^{\mathrlap{F}} \\ \Delta[n] &\to& Y } \,.

The morphisms with the left lifting property against all inner fibrations are called inner anodyne.


General properties


By the small object argument we have that every morphism f:XYf : X \to Y of simplicial sets may be factored as

f:XZY f : X \to Z \to Y

with XZX \to Z a left/right/inner anodyne cofibration and ZYZ \to Y accordingly a left/right/inner Kan fibration.


Inner fibrations were introduced by Andre Joyal. A comprehensive account is in section 2.3 of

Their relation to cographs/correspondence is discussed in section 2.3.1 there.

Last revised on June 1, 2021 at 14:25:16. See the history of this page for a list of all contributions to it.