nLab
inner fibration

Idea

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 $C\to D$ such that each fiber is a quasi-category and such that over each morphism $f:d_1\to d_2$ of $D$, $C$ may be thought of as the cograph of an (∞,1)-profunctor $C_{d_1}⇸C_{d_2}$.

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

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

This $(\mathrm{\infty },1)$-profunctor comes form an ordinary (∞,1)-functor $F:C\to D$ precisely if the inner fibration $K\to \Delta [1]$ is even a coCartesian fibration. And it comes from a functor $G:D\to C$ 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 $(\mathrm{\infty },1)$-profunctor $K\to \Delta [1]$ is both a Cartesian as well as a coCartesian fibration does it exhibit a pair of adjoint (∞,1)-functors.

Definition

A morphism of simplicial sets $f:X\to S$ 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

$$\begin{array}{ccc}\Lambda [n{]}_i& \to & X\\ \downarrow & & {\downarrow }^F\\ \Delta [n]& \to & Y\end{array}$$\array{ \Lambda[n]_i &\to& X \\ \downarrow && \downarrow^{\mathrlap{F}} \\ \Delta[n] &\to& Y }

for $0<i<n$ there exists a left

$$\begin{array}{ccc}\Lambda [n{]}_i& \to & X\\ \downarrow & \nearrow & {\downarrow }^F\\ \Delta [n]& \to & Y\end{array}.$$\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.

Properties

General properties

Remark

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

$$f:X\to Z\to Y$$f : X \to Z \to Y

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

Related concepts

• Kan fibration, anodyne morphism

• right/left Kan fibration, right/left anodyne map

• inner fibration

• Cartesian fibration

• fibration in the model structure for quasi-categories

• fibration of (infinity,1)-operads

References

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

• Jacob Lurie, Higher Topos Theory

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