# nLab hom-object in a quasi-category

(∞,1)-category theory

# Contents

## 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 $\left(\infty ,1\right)$-category is incarnated as a quasi-category, there are several explicit ways to extract representatives of this hom-object.

## Definition

Let $C$ be a quasi-category (or any simplicial set), and $x,y\in {C}_{0}$ any two objects. Then write

${\mathrm{Hom}}_{C}\left(x,y\right):=\left[\tau \left(C\right)\left(x,y\right)\right]\in \mathrm{Ho}\left({\mathrm{sSet}}_{\mathrm{Quillen}}\right)\simeq \mathrm{Ho}\left(\infty \mathrm{Grpd}\right)\phantom{\rule{thinmathspace}{0ex}},$Hom_C(x,y) := [\tau(C)(x,y)] \in Ho(sSet_{Quillen}) \simeq Ho(\infty Grpd) \,,

where

• $\tau$ is the left adjoint to the homotopy coherent nerve;

• $\tau \left(C\right)$ is accordingly the simplicially enriched category incarnation of $C$,

• $\tau \left(C\right)\left(x,y\right)$ is the sSet-hom-object of that $\mathrm{sSet}$-enriched category;

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

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

This is useful for many purposes, but $\tau \left(C\right)$ is usually hard to compute explicitly. The following three other definitions of representatives of ${\mathrm{Hom}}_{C}\left(x,y\right)$ are often useful.

Definition

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

${\mathrm{Hom}}_{C}^{LR}\left(x,y\right):=\left\{x\right\}{×}_{C}{C}^{\Delta \left[1\right]}{×}_{C}\left\{y\right\}$Hom_C^{L R}(x,y) := \{x\} \times_C C^{\Delta[1]} \times_C \{y\}

for the pullback

$\begin{array}{ccc}\left\{x\right\}{×}_{C}{C}^{\Delta \left[1\right]}{×}_{C}\left\{y\right\}& \to & {C}^{\Delta \left[1\right]}\\ ↓& & {↓}^{{d}_{1}×{d}_{0}}\\ \left\{x\right\}×\left\{y\right\}& \to & C×C\end{array}$\array{ \{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 }

in sSet of the path object ${C}^{\Delta \left[1\right]}$ (the cartesian internal hom in sSet with the 1-simplex $\Delta \left[1\right]$) .

Write

${\mathrm{Hom}}_{C}^{R}\left(x,y\right)\in \mathrm{sSet}$Hom^R_C(x,y) \in sSet

for the simplicial set whose $n$-simplices are defined to be those morphisms $\sigma :\Delta \left[n+1\right]\to C$ such that the restriction to $\Delta \left\{0,\cdots ,n\right\}$ is the constant map to $x$ and the restriction to $\Delta \left\{n+1\right\}$ is the map to $y$.

Analogously, write

${\mathrm{Hom}}_{C}^{L}\left(x,y\right)\in \mathrm{sSet}$Hom^L_C(x,y) \in sSet

for the simplicial set whose $n$-simplices are morphisms $\Delta \left[n+1\right]\to X$ which restrict to $x$ on $\left\{0\right\}$ and are constant on $y$ when restricted to $\left\{1,\cdots ,n+1\right\}$.

Remark The 1-cells in ${\mathrm{Hom}}_{C}^{R}\left(x,y\right)$, ${\mathrm{Hom}}_{C}^{L}\left(x,y\right)$ and ${\mathrm{Hom}}_{C}^{LR}\left(x,y\right)$ 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

$\left({C}_{/p}{\right)}_{n}:=\mathrm{Hom}\left(\Delta \left[n\right],{C}^{/p}\right):={\mathrm{Hom}}_{p}\left(\Delta \left[n\right]\star K,C\right)\phantom{\rule{thinmathspace}{0ex}},$(C_{/p})_n := Hom(\Delta[n],C^{/p}) := Hom_p(\Delta[n] \star K, C) \,,

where on the right we have the set of morphisms in $\mathrm{sSet}$ 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 $\left(\Delta \left[n\right]\star K\to C\right)$ to the restriction $\left(\Delta \left[n\right]\to \Delta \left[n\right]\star K\to C\right)$.

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

$\left({C}^{/p}{\right)}_{n}:={\mathrm{Hom}}_{K/\mathrm{sSet}}\left(\Delta \left[n\right]♢K,C\right)\phantom{\rule{thinmathspace}{0ex}}.$(C^{/p})_n := Hom_{K/sSet}( \Delta[n] \diamondsuit K, C) \,.

Obervation

We have

${\mathrm{Hom}}_{C}^{R}\left(x,y\right)={C}_{/y}{×}_{C}\left\{x\right\}\phantom{\rule{thinmathspace}{0ex}},$Hom^R_C(x,y) = C_{/{y}} \times_C \{x\} \,,

where on the right we have the pullback in sSet in the diagram

$\begin{array}{ccc}{C}_{/y}{×}_{C}\left\{x\right\}& \to & {C}_{/y}\\ ↓& & ↓\\ \left\{x\right\}& \to & C\end{array}$\array{ C_{/{y}} \times_C \{x\} &\to& C_{/y} \\ \downarrow && \downarrow \\ \{x\} &\to& C }

and the equality sign denotes an isomorphism in sSet.

And we have

${\mathrm{Hom}}_{C}^{LR}\left(x,y\right)={C}^{/y}{×}_{C}\left\{x\right\}$Hom_C^{L R}(x,y) = C^{/y} \times_C \{x\}

Proof For the first statement use the identification of $\Delta \left[n+1\right]$ with the join of simplicial sets $\Delta \left[n\right]\star \Delta \left[0\right]$, as described there.

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

$\begin{array}{ccccc}& & \Delta \left[n\right]×\Delta \left[0\right]& \to & \Delta \left[0\right]\\ & & ↓& ↓& \\ \Delta \left[n\right]×\Delta \left[0\right]& \to & C×\Delta \left[0\right]×\Delta \left[1\right]& \to & \Delta \left[n+1\right]\\ ↓& & ↓& & ↓\\ \Delta \left[n\right]& \to & C×\Delta \left[1\right]& \to & \Delta \left[n+1\right]\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ && \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] } \,,

so that

${C}^{/y}={C}^{\Delta \left[1\right]}{×}_{C}\left\{y\right\}$C^{/y} = C^{\Delta[1]} \times_C \{y\}

because

$\left({C}^{/y}{\right)}_{n}={\mathrm{Hom}}_{\Delta \left[0\right]/\mathrm{sSet}}\left(\Delta \left[n\right]×\Delta \left[1\right]\coprod _{\Delta \left[1\right]×\Delta \left[0\right]}\Delta \left[0\right],C\right)=\mathrm{Hom}\left(\Delta \left[n\right]×\Delta \left[1\right],C\right){×}_{\mathrm{Hom}\left(\Delta \left[1\right],C\right)}\left\{y\right\}=\left({C}^{\Delta \left[1\right]}×\left\{y\right\}{\right)}_{n}\phantom{\rule{thinmathspace}{0ex}}.$(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 \,.

Remark One advantage of the representative $\tau \left(C\right)\left(c,y\right)$ of ${\mathrm{Hom}}_{C}\left(x,y\right)$ is that, by the fact that $\tau \left(C\right)$ is an sSet-enriched category, there is a strict composition operation

$\tau \left(C\right)\left(x,y\right)×\tau \left(C\right)\left(y,z\right)\to \tau \left(C\right)\left(x,z\right)\phantom{\rule{thinmathspace}{0ex}}.$\tau(C)(x,y) \times \tau(C)(y,z) \to \tau(C)(x,z) \,.

This is not available for the ${\mathrm{Hom}}_{C}^{R}\left(x,y\right)$ and ${\mathrm{Hom}}_{C}^{L}\left(x,y\right)$

## Properties

Proposition

If the simplicial set $C$ is a quasi-category, then ${\mathrm{Hom}}_{C}^{R}\left(x,y\right)$ is a Kan complex.

Proof

This is HTT, prop 1.2.2.3.

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

Proposition

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

${\mathrm{Hom}}_{C}^{R}\left(x,y\right)\to {\mathrm{Hom}}_{C}^{LR}\left(x,y\right)←{\mathrm{Hom}}_{C}^{L}\left(x,y\right)$Hom_C^R(x,y) \to Hom_C^{L R}(x,y) \leftarrow Hom_C^L(x,y)

Proof

This is 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 ${\mathrm{Hom}}_{C}^{R}\left(x,y\right)$ 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 HTT, prop. 3.3.1.5. (2).

The statement for ${\mathrm{Hom}}_{C}^{L}\left(x,y\right)$ follows dually.

Revised on April 7, 2010 07:30:48 by Urs Schreiber (87.212.203.135)