nLab inner fibration

Idea

The notion of inner fibration of simplicial sets is one of the notions 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 $(\infty,1)$-profunctor comes from 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 $(\infty,1)$-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 $F : X \to Y$ 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

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

for $0 \lt i \lt n$ there exists a lift

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

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

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

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.