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 fibration fibered in groupoids.
The analog of this for quasi-categories are
the special case of left/right (Kan-) fibrations of quasi-categories,
rspectively.
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 $\Lambda[n]_k \to \Delta[n]$ except possibly the right outer ones: $0 \leq k \lt n$.
It is a right fibration or right Kan fibration if its extends against all horns except possibly the left outer ones: $0 \lt k \leq n$.
So $X \to S$ is a left fibration precisely if for all commuting squares
for $n \in \mathbb{N}$ and $0 \leq k \lt n$, a diagonal lift exists as indicated.
Morphisms with the left lifting property against all left/right fibrations are called left/right anodyne 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.
Ordinary categories fibered in groupoids have a simple characterization in terms of their nerves. Let $N : Cat \to sSet$ be the nerve functor and for $p : E \to B$ a morphism in Cat (a functor), let $N(p) : N(E) \to N(B)$ be the corresponding morphism in sSet.
Then
The functor $p : E \to B$ is an fibration in groupoids precisely if the morphism $N(p) : N(E) \to N(B)$ is a right Kan fibration of simplicial sets
To see this, first notice the following facts:
For $C$ a category, the nerve $N(C)$ is 2-coskeletal. In particular all $n$-spheres for $n \geq 3$ have unique fillers
and (implied by that) all $n$-horns for $n \gt 3$ have fillers
This is discussed at nerve.
If $p : E \to B$ is an ordinary functor, then $N(f) : N(E) \to N(B)$ is an inner fibration, meaning that its has the right lifting property with respect to all inner horn inclusions $\Lambda[n]_i \hookrightarrow \Delta[n]$ for $0 \lt i \lt n$.
This is discussed at inner fibration.
From the above lemmas it follows that $N(p) : N(E) \to N(B)$ is a right fibration already precisely if it has the right lifting property with respect only to the three horn inclusions
So we check explicitly what these three conditions amount to
$n=1$ – The existence of all fillers
means that for all objects $e \in E$ and morphism $f : b \to p(e)$ in $B$, there exists a morphism $\hat f : \hat b \to e$ in $E$ such that $p(\hat f) = f$.
$n=2$ – The existence of fillers
means that for all diagrams
in $E$ and commuting triangles
in $B$, there is a commuting triangle
in $E$, such that $p(\hat f ) = f$.
$n=3$ – …
Consider first the case of degenrate 3-simplices on $N(B)$, on 2-simplices as above.
Suppose in the above situation two lifts $(\hat f)_1$ and $(\hat f)_2$ are found. Together these yield a $\Lambda[3]_3$-horn in $N(E)$. The filler condition says this can be filled, which implies that $(\hat f)_1 = (\hat f)_2$.
So the $n=3$-condition implies that the lift whose existence is guaranteed by the $n=2$-condition is unique.
By similar reasoning one sees that this is all the $n=3$-condition yields.
In total, these three lifting conditions are precisely those for a Grothendieck fibration in groupoids.
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
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.
Over a Kan complex $T$, left fibrations $S \to T$ are automatically Kan fibrations.
This appears as HTT, prop. 2.1.3.3.
As an important special case of this we have
For $C \to *$ a right (left) fibration over the point, $C$ is a Kan complex, i.e. an ∞-groupoid.
This is originally 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 $LAn_0$:
$LAn_0 =$
$\{ \Lambda[n]_{i} \to \Delta[n] | 0 \leq i \lt n \}$;
the collection of all inclusions of the form
the collection of all inclusions of the form
for all inclusions of simplicial sets $S \hookrightarrow S'$.
This is due to Andre Joyal, recalled as HTT, prop 2.1.2.6.
…
For $i : A \to A'$ left-anodyne and $j : B \to B'$ a cofibration in the model structure for quasi-categories, the canonical morphism
is left-anodyne.
This appears as HTT, cor. 2.1.2.7.
For $p : X \to S$ a left fibration and $i : A \to B$ a cofibration of simplicial sets, the canonical morphism
is a left fibration. If $i$ is furthermore left anodyne, then it is an acyclic Kan fibration.
This appears as HTT, cor. 2.1.2.9.
For $f : A_0 \to A$ and $g : B_0 \to B$ two inclusions of simplicial sets with $f$ left anodyne, we have that the canonical morphism
into the join of simplicial sets is left anodyne.
This is due to Andre Joyal. It appears as HTT, lemma 2.1.4.2.
(restriction of over-quasi-categories along left anodyne inclusions)
Let $p : B \to S$ be a morphism of simplicial sets and $i : A \to B$ a left anodyne morphism, then the restriction morphism of under quasi-categories
is an acyclic Kan fibration.
This is a special case of what appears as HTT, prop. 2.1.2.5, which is originally due to Andre Joyal.
Let $p : X \to S$ be a morphism of simplicial sets with section $s : S \to X$. If there is a fiberwise simplicial homotopy $X \times \Delta[1] \to S$ from $s \circ p$ to $Id_X$ then $s$ is left anodyne.
This appears as HTT, prop. 2.1.2.11.
right/left Kan fibration, right/left anodyne map
David Ayala, John Francis, Fibrations of $\infty$-Categories (arXiv:1702.02681)