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\begin{document}

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\section*{hom-object in a quasi-category}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Between any two [[objects]] $x,y$ in an [[(∞,1)-category]] $C$ there is an [[∞-groupoid]] of morphisms. It is well-defined up to equivalence. When the $(\infty,1)$-category is incarnated as a [[quasi-category]], there are several explicit ways to extract representatives of this [[hom-object]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $C$ be a [[quasi-category]] (or any [[simplicial set]]), and $x,y \in C_0$ any two objects. Then write

\begin{displaymath}
Hom_C(x,y) := [\tau(C)(x,y)] \in Ho(sSet_{Quillen}) \simeq Ho(\infty Grpd)
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $\tau$ is the [[left adjoint]] to the [[homotopy coherent nerve]];


\item $\tau(C)$ is accordingly the [[simplicially enriched category]] incarnation of $C$,


\item $\tau(C)(x,y)$ is the [[sSet]]-[[hom-object]] of that $sSet$-[[enriched category]];


\item $[\tau(C)(x,y)]$ is the equivalence class of this object.



\end{itemize}
This defines $Hom_C(x,y)$ as an equivalence class of $\infty$-groupoids, but at the same time defines a particular representative: if $C$ is a [[quasi-category]] then $\tau(C)(x,y)$ is a [[Kan complex]] that represents this class.

This is useful for many purposes, but $\tau(C)$ is usually hard to compute explicitly. The following three other definitions of representatives of $Hom_C(x,y)$ are often useful.

\textbf{Definition}

For $C$ and $x,y$ as above, write

\begin{displaymath}
Hom_C^{L R}(x,y) := \{x\} \times_C C^{\Delta[1]} \times_C \{y\}
\end{displaymath}
for the [[pullback]]

\begin{displaymath}
\itexarray{
    \{x\} \times_C C^{\Delta[1]} \times_C \{y\} &\to& C^{\Delta[1]}
    \\
    \downarrow && \downarrow^{\mathrlap{d_1 \times d_0}}
    \\
    \{x\} \times \{y\}
    &\to&
    C \times C
  }
\end{displaymath}
in [[sSet]] of the [[path object]] $C^{\Delta[1]}$ (the cartesian [[internal hom]] in [[sSet]] with the 1-[[simplex]] $\Delta[1]$) .

Write

\begin{displaymath}
Hom^R_C(x,y) \in sSet
\end{displaymath}
for the simplicial set whose $n$-[[simplex|simplices]] are defined to be those morphisms $\sigma : \Delta[n+1] \to C$ such that the restriction to $\Delta\{0, \cdots, n\}$ is the constant map to $x$ and the restriction to $\Delta\{n+1\}$ is the map to $y$.

Analogously, write

\begin{displaymath}
Hom^L_C(x,y) \in sSet
\end{displaymath}
for the simplicial set whose $n$-simplices are morphisms $\Delta[n+1] \to X$ which restrict to $x$ on $\{0\}$ and are constant on $y$ when restricted to $\{1, \cdots, n+1\}$.

\textbf{Remark} The 1-cells in $Hom_C^R(x,y)$, $Hom_C^L(x,y)$ and $Hom_C^{L R}(x,y)$ are 2-[[globes]] in $C$. The 2-cells are commuting squares of vertical composites of 2-globes forming a 3-globe.

Equivalently this may be understood in terms of fibers of [[over quasi-categories]].

Recall that for $p : K \to C$ a morphism, we have the [[over quasi-category]] $C_{/p}$ defined by

\begin{displaymath}
(C_{/p})_n := Hom(\Delta[n],C^{/p}) :=
  Hom_p(\Delta[n] \star K, C)
  \,,
\end{displaymath}
where on the right we have the set of morphisms in $sSet$ out of the [[join of simplicial sets]] that restrict on $K$ to $p$.

This comes with the canonical projection $C^{p/} \to C$, which sends $(\Delta[n] \star K \to C)$ to the restriction $(\Delta[n] \to \Delta[n] \star K \to C)$.

There is also the other, equivalent, definition $C^{/p}$ of [[over quasi-category]], defined using the other, equivalent, definition $\diamondsuit$ of [[join of quasi-categories]] by

\begin{displaymath}
(C^{/p})_n := Hom_{K/sSet}( \Delta[n] \diamondsuit K, C)
  \,.
\end{displaymath}
\textbf{Obervation}

We have

\begin{displaymath}
Hom^R_C(x,y) = C_{/{y}} \times_C \{x\}
  \,,
\end{displaymath}
where on the right we have the [[pullback]] in [[sSet]] in the diagram

\begin{displaymath}
\itexarray{
    C_{/{y}} \times_C \{x\} &\to& C_{/y}
    \\
    \downarrow && \downarrow
    \\
    \{x\} &\to& C
  }
\end{displaymath}
and the equality sign denotes an [[isomorphism]] in [[sSet]].

And we have

\begin{displaymath}
Hom_C^{L R}(x,y) = C^{/y} \times_C \{x\}
\end{displaymath}
\textbf{Proof} For the first statement use the identification of $\Delta[n+1]$ with the [[join of simplicial sets]] $\Delta[n] \star \Delta[0]$, as described there.

For the second statement use that $\Delta[n] \diamondsuit \Delta[0]$ is the colimit $\Delta[n+1]$ in

\begin{displaymath}
\itexarray{
  && \Delta[n] \times \Delta[0] &\to & \Delta[0] 
  \\
  && \downarrow &\downarrow&
  \\
  \Delta[n] \times \Delta[0] &\to& C \times \Delta[0] \times \Delta[1] &\to& \Delta[n+1]
  \\
  \downarrow && \downarrow && \downarrow
  \\
  \Delta[n] &\to& C \times \Delta[1] &\to& \Delta[n+1]
  }
  \,,
\end{displaymath}
so that

\begin{displaymath}
C^{/y} = C^{\Delta[1]} \times_C \{y\}
\end{displaymath}
because

\begin{displaymath}
(C^{/y})_n = Hom_{\Delta[0]/sSet}( 
    \Delta[n]\times \Delta[1] \coprod_{\Delta[1] \times \Delta[0]} \Delta[0], C)
    =
    Hom(\Delta[n] \times \Delta[1], C) \times_{Hom(\Delta[1],C)} \{y\}
    = (C^{\Delta[1]} \times \{y\})_n
    \,.
\end{displaymath}
\textbf{Remark} One advantage of the representative $\tau(C)(c,y)$ of $Hom_C(x,y)$ is that, by the fact that $\tau(C)$ is an [[sSet]]-[[enriched category]], there is a strict composition operation

\begin{displaymath}
\tau(C)(x,y) \times \tau(C)(y,z) \to \tau(C)(x,z)
  \,.
\end{displaymath}
This is not available for the $Hom_C^R(x,y)$ and $Hom_C^L(x,y)$

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\textbf{Proposition}

If the simplicial set $C$ is a [[quasi-category]], then $Hom_C^R(x,y)$ is a [[Kan complex]].

\textbf{Proof}

This is [[Higher Topos Theory|HTT, prop 1.2.2.3]].

From the definition it is clear that $Hom_C^R(x,y)$ has fillers for all inner and right outer horns $\Lambda[n]_{1 \leq i \leq n}$, because these yield inner horns in $\Delta[n+1] = \Delta[n] \star \Delta[0]$. The claim follows then with the fact that every [[right fibration]] over the point is a Kan complex, as described there.

\textbf{Proposition}

If $C$ is a [[quasi-category]] then the canonical inclusions

\begin{displaymath}
Hom_C^R(x,y) \to Hom_C^{L R}(x,y) \leftarrow Hom_C^L(x,y)
\end{displaymath}
are [[homotopy equivalences]] of [[Kan complexes]].

\textbf{Proof}

This is [[Higher Topos Theory|HTT, cor. 4.2.1.7]].

As described at [[join of quasi-categories]] the canonical morphism $C_{/y} \to C^{/y}$ is an equivalence of quasi-categories. So for the statement for $Hom_C^R(x,y)$ it suffices to show that this induces an equivalence of fibers over $C$. This follows from the fact that both $C_{/y} \to C$ and $C^{/y} \to C$ are [[Cartesian fibrations]].

See [[Cartesian fibrations]] for these statements. This is [[Higher Topos Theory|HTT, prop. 3.3.1.5. (2)]].

The statement for $Hom_C^L(x,y)$ follows dually.



\end{document}