equivalences in/of $(\infty,1)$-categories
For $C$ an (∞,1)-category and $X \in C$ an object, the over-$(\infty,1)$-category or slice $(\infty,1)$-category $C_{/X}$ is the $(\infty,1)$-category whose objects are morphism $p : Y \to X$ in $C$, whose morphisms $\eta : p_1 \to p_2$ are 2-morphisms in $C$ of the form
hence 1-morphisms $f$ as indicated together with a homotopy $\eta \colon p_2 \circ f \stackrel{\simeq}{\to} p_1$; and generally whose n-morphisms are diagrams
in $C$ of the shape of the cocone under the $n$-simplex that take the tip of the cocone to the object $X$.
This is the generalization of the notion of over-category in ordinary category theory.
We give the definition in terms of the model of (∞,1)-categories in terms of quasi-categories.
Recall from join of quasi-categories that there are two different but quasi-catgorically equivalent definitions of join, denoted $\star$ and $\diamondsuit$. Accordingly we have the following two different but quasi-categoricaly equivalent definitions of over/under quasi-categories.
Let $C$ be a quasi-category. let $K$ be any simplicial set and let
be an (∞,1)-functor – a morphism of simplicial sets.
The under-quasi-category $C_{F/}$ is the simplicial set characterized by the property that for any other simplicial set $S$ there is a natural bijection of hom-sets
where $i_{K,S} : K \to K \star S$ is the canonical inclusion of $K$ into its join of simplicial sets with $S$.
Similarly, the over quasi-category over $F$ is the simplicial set characterized by the property that
naturally in $S$, where $j_{K,S}$ is the canonical inclusion map $K\to S\star K$.
The functor
with $\diamondsuit$ denoting the other definition of join of quasi-categories (as described there)
has a right adjoint
and its image $C^{F/}$ is another definition of the quasi-category under $F$.
The first definition in terms of the the mapping property is due to Andre Joyal. Together with the discussion of the concrete realization it appears as HTT, prop 1.2.9.2. The second definition appears in HTT above prop. 4.2.1.5.
The simplicial sets $C_{/F}$ and $C_{F/}$ are indeed themselves again quasi-categories.
This appears as HTT, prop. 1.2.9.3
The two definitions yield equivalent results in that the canonical morphism
is an equivalence of quasi-categories.
This is HTT, prop. 4.2.1.5
From the formula
we see that
an object in the over quasi-category $C_{/F}$ is a cone over $F$;.
For instance if $K = \Delta[1]$ then an object in $C_{/F}$ is a 2-cell
in $C$.
a morphism in $C_{/F}$ is a morphism of cones,
etc:.
So we may think of the overcategory $C_{/F}$ as the quasi-category of cones over $F$. Accordingly we have that
the terminal object in $C_{/F}$ is (if it exists) the limit in $C$ over $F$;
the initial object in $C_{F/}$ is (if it exists) the colimit of $F$ in $C$.
For $C = N(\mathcal{C})$ (the nerve of) an ordinary category $\mathcal{C}$ and $K = *$, this construction coincides with the ordinary notion of overcategory $\mathcal{C}/F$ in that there is a canonical isomorphism of simplicial sets
This appears as HTT, remark 1.2.9.6.
If $q : C \to D$ is a categorical equivalence then so is the induced morphism $C_{/F} \to C_{/q F}$.
This appears as HTT, prop 1.2.9.3.
For $C$ a quasi-category and $p : X \to C$ any morphism of simplicial sets, the canonical morphisms
and
are both left Kan fibrations.
This is a special case of HTT, prop 2.1.2.1 and prop. 4.2.1.6.
For $v \colon K \to \tilde K$ an map of small (∞,1)-categories and $\mathcal{C}$ any $(\infty,1)$-category, the induced (∞,1)-functor between slice $(\infty,1)$-categories
is an equivalence of (∞,1)-categories precisely if $v$ if an op-final (∞,1)-functor (hence if $v^{op}$ is final).
This is (Lurie, prop. 4.1.1.8).
For $C$ an (∞,1)-category and $X \in C$ an object in $C$ and $f : A \to X$ and $g : B \to X$ two objects in $C/X$, the hom-∞-groupoid $C/X(f,g)$ is equivalent to the homotopy fiber of $C(A,B) \stackrel{g_*}{\to} C(A,X)$ over the morphism $f$: we have an (∞,1)-pullback diagram
This is HTT, prop. 5.5.5.12.
The forgetful functor $\mathcal{C}_{/X} \to \mathcal{C}$ out of a slice (dependent sum) reflects (∞,1)-colimits:
Let $f \colon K \to \mathcal{C}_{/X}$ be a diagram in the slice of an (∞,1)-category $\mathcal{C}$ over an object $X \in \mathcal{C}$. Then if the composite $K \stackrel{f}{\to} \mathcal{C}_{/X} \to \mathcal{C}$ has an (∞,1)-colimit, then so does $f$ itself and the projection $\mathcal{C}_{/q} \to \mathcal{C}$ takes the latter to the former. Conversely, a cocone $K \star \Delta^0 \to \mathcal{C}_{/X}$ under $f$ is an (∞,1)-colimit of $f$ precisely if the composite $K \star \Delta^0 \to \mathcal{C}_{/X} \to \mathcal{C}$ is an $(\infty,1)$-colimit of the projection of $f$.
This appears as (Lurie, prop. 1.2.13.8).
On the other hand (∞,1)-limits in the slice are computed as limits over the diagram with the slice-cocone adjoined:
For $\mathcal{C}$ an (∞,1)-category, $X \;\colon\; \mathcal{D} \longrightarrow \mathcal{C}$ a diagram, $\mathcal{C}_{/X}$ the comma category (the over-$\infty$-category if $\mathcal{D}$ is the point) and $F \;\colon\; K \to \mathcal{C}_{/X}$ a diagram in the comma category, then the (∞,1)-limit $\underset{\leftarrow}{\lim} F$ in $\mathcal{C}_{/X}$ coincides with the limit $\underset{\leftarrow}{\lim} F/X$ in $\mathcal{C}$.
For a proof see at (∞,1)-limit here.
over-(∞,1)-category
Section 1.2.9 of