equivalences in/of $(\infty,1)$-categories
For ordinary categories there is the notion of
Grothendieck fibration between two categories.
and the special case of a category fibered in groupoids.
The analog of this for quasi-categories are
left fibrations/right fibrations of quasi-categories.
There are more types of fibrations between the simplicial sets underlying the quasi-category
inner fibrations – these correspond to bundles of quasi-categories : an inner fibration $E \to \Delta[1]$ over the interval is the quasi-categorical analog of a cograph of a profunctor: it characterizes the fibers $C, D$ over the endpoints $0,1 \in \Delta[1]$ as quasi-categories. Notably having an inner fibration $C \to \Delta[0]$ over the point says precisely that $C$ is a quasi-category.
categorical fibrations – these appear as the fibrations in the sense of model category theory in the Joyal model structure for quasi-categories $sSet_{Joyal}$ . But they have no particular intrinsic meaning in higher category theory. In fact, there is also the model structure on marked simplicial sets which is Quillen equivalent to $sSet_{Joyal}$ and in which the model-theoretic fibrations coincide precisely with the Cartesian fibrations that do have an intrinsic category theoretic meaning.
We list the different definitions in the order of their generality. The examples of each definition are also examples of the following definitions.
All morphisms in the following are morphisms of simplicial sets.
A trivial fibration (trivial Kan fibration) is a morphism that has the right lifting property with respect to the boundary inclusions $\partial \Delta[n] \hookrightarrow \Delta[n], n \geq 1$.
A morphism with left lifting property against all Kan fibrations is called anodyne.
A morphism of simplicial sets $f : X \to S$ is a left fibration or left Kan fibration if it has the right lifting property with respect to all horn inclusions except the right outer ones. It is a right fibration or right Kan fibration if its extends against all horns except the left outer ones.
Morphisms with the left lifting property against all left/right fibrations are called left/right anodyne morphism maps.
Write
for the full SSet-subcategory of the overcategory of SSet over $S$ on those morphisms that are right fibrations.
This is a Kan complex-enriched category and as such an incarnation of the (∞,1)-category of right fibrations. It is modeled by the model structure for right fibrations. For details on this see the discussion at (∞,1)-Grothendieck construction.
A Cartesian fibration is an inner fibration $p : X \to S$ such that
for every edge $f : X \to Y$ of $S$
and every lift $\tilde{y}$ of $y$ (that is, $p(\tilde{y})=y$),
there is a Cartesian edge $\tilde{f} : \tilde{x} \to \tilde{y}$ in $X$ lifting $f$.
(HTT, def 2.4.2.1)
see also
A categorical fibration is a fibration in the model structure for quasi-categories: morphism $f : X \to S$ with the right lifting property against all monomorphic categorical equivalences .
(HTT, p. 81).
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.
The morphisms with the left lifting property against all inner fibrations are called inner anodyne.
By the small object argument we have that every morphism $f : X \to Y$ of simplicial sets may be factored as
with $X \to Z$ a left/right/inner anodyne cofibration and $Z \to Y$ accordingly a left/right/inner Kan fibration.
…
…
Under the operation of forming the opposite quasi-category, left fibrations turn into right fibrations, and vice versa: if $p : C \to D$ is a left fibration then $p^{op} : C^{op} \to D^{op}$ is a right fibration.
Therefore it is sufficient to list properties of only one type of these fib rations, that for the other follows.
In classical homotopy theory, a continuous map $p : E \to B$ of topological spaces is said to have the homotopy lifting property if it has the right lifting property with respect to all morphisms $Y \stackrel{(Id, 0)}{\to} Y \times I$ for $I = [0,1]$ the standard interval and every commuting diagram
there exists a lift $\sigma : Y \times I \to E$ making the two triangles
commute. For $Y = *$ the point this can be rephrased as saying that the universal morphism $E^I \to B^I \times_B E$ induced by the commuting square commuting square
is an epimorphism. If it is even an isomorphism then the lift $\sigma$ exists uniquely . This is the situation that the following proposition generalizes:
A morphism $p : X \to S$ of simplicial sets is a left fibration precisely if the canonical morphism
is a trivial Kan fibration.
This is a corollary of the characterization of left anodyne morphisms in Properties of left anodyne maps by Andre Joyal, recalled in HTT, corollary 2.1.2.10.
The notion of right fibration of quasi-categories generalizes the notion of category fibered in groupoids. This follows from the following properties.
For $C \to *$ a right (left) fibration over the point, $C$ is a Kan complex, i.e. an ∞-groupoid.
Due to Andre Joyal. Recalled at HTT, prop. 1.2.5.1.
It follows that the fiber $X_c$ of every right fibration $X \to C$ over every point $c \in C$, i.e. the pullback
is a Kan complex.
For $C$ and $D$ quasi-categories that are ordinary categories (i.e. simplicial sets that are nerves of ordinary categories), a morphism $C \to D$ is a right fibration precisely if the correspunding ordinary functor exhibits $C$ as a category fibered in groupoids over $D$.
This is HTT, prop. 2.1.1.3.
A canonical class of examples of a fibered category is the codomain fibration. This is actually a bifibration. For an ordinary category, a bifiber of this is just a set. For an $(\infty,1)$-category it is an $\infty$-groupoid. Hence fixing only one fiber of the bifibration should yield a fibration in $\infty$-groupoids. This is asserted by the following statement.
Let $p : K \to C$ be an arbitrary morphism to a quasi-category $C$ and let $C_{p/}$ be the corresponding under quasi-category. Then the canonical propjection $C_{p/} \to C$ is a left fibration.
Due to Andre Joyal. Recalled as HTT, prop 2.1.2.2.
The collection of left anodyne morphisms (those with left lifting property against left fibrations) is equivalently $LAn = LLP(RLP(LAn_0))$ for the following choices of $LAn_0$:
$LAn_0 =$
the collection of all left horn inclusions
$\{ \Lambda[n]_{i} \to \Delta[n] | 0 \leq i \lt n \}$;
blah-blah
blah-blah
This is due to Andre Joyal, recalled as HTT, prop 2.1.2.6.
A simplicial set $K$ is the nerve of an ordinary category $C$, $K \simeq_{iso} N(C)$ precisely if the terminal morphism $K \to \Delta[0]$ is an inner fibration with unique inner horn fillers, i.e. precisely if for all morphisms
with $n \in \mathbb{N}$ and $0 \lt i \lt n$ there is a unique morphism $\Delta[n] \to K$ making the diagram
commute.
This is HTT, prop. 1.1.2.2
It follows that under the nerve every functor $f : C \to D$ between ordinary categories is an inner fibration.
This is immediate, but let’s spell it out:
In any commutative diagram
by the above the bottom morphism is already unqiely specified by the remaining diagram.
By the above there exists a unique lift into $N(C)$
and by uniqueness of lifts into $N(D)$ this must also make the lower square commute