nLab hom-object in a quasi-category

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.

Definition

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

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

where

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

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

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

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

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.

Definition

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

$$Hom_C^{L R}(x,y) := \{x\} \times_C C^{\Delta[1]} \times_C \{y\}$$

for the pullback

$$\array{ \{x\} \times_C C^{\Delta[1]} \times_C \{y\} &\to& C^{\Delta[1]} \\ \downarrow && \downarrow^{\mathrlap{d_1 \times d_0}} \\ \{x\} \times \{y\} &\to& C \times C }$$

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

Write

$$Hom^R_C(x,y) \in sSet$$

for the simplicial set whose $n$-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

$$Hom^L_C(x,y) \in sSet$$

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 $\{1, \cdots, n+1\}$.

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

$$(C_{/p})_n := Hom(\Delta[n],C^{/p}) := Hom_p(\Delta[n] \star K, C) \,,$$

where on the right we have the set of morphisms in $sSet$ 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

$$(C^{/p})_n := Hom_{K/sSet}( \Delta[n] \diamondsuit K, C) \,.$$

Obervation

We have

$$Hom^R_C(x,y) = C_{/{y}} \times_C \{x\} \,,$$

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

$$\array{ C_{/{y}} \times_C \{x\} &\to& C_{/y} \\ \downarrow && \downarrow \\ \{x\} &\to& C }$$

and the equality sign denotes an isomorphism in sSet.

And we have

$$Hom_C^{L R}(x,y) = C^{/y} \times_C \{x\}$$

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

$$\array{ && \Delta[n] \times \Delta[0] &\to & \Delta[0] \\ && \downarrow &\downarrow& \\ \Delta[n] \times \Delta[0] &\to& C \times \Delta[0] \times \Delta[1] &\to& \Delta[n+1] \\ \downarrow && \downarrow && \downarrow \\ \Delta[n] &\to& C \times \Delta[1] &\to& \Delta[n+1] } \,,$$

so that

$$C^{/y} = C^{\Delta[1]} \times_C \{y\}$$

because

$$(C^{/y})_n = Hom_{\Delta[0]/sSet}( \Delta[n]\times \Delta[1] \coprod_{\Delta[1] \times \Delta[0]} \Delta[0], C) = Hom(\Delta[n] \times \Delta[1], C) \times_{Hom(\Delta[1],C)} \{y\} = (C^{\Delta[1]} \times \{y\})_n \,.$$

Remark One advantage of the representative $\tau(C)(c,y)$ of $Hom_C(x,y)$ is that, by the fact that $\tau(C)$ is an sSet-enriched category, there is a strict composition operation

$$\tau(C)(x,y) \times \tau(C)(y,z) \to \tau(C)(x,z) \,.$$

This is not available for the $Hom_C^R(x,y)$ and $Hom_C^L(x,y)$

Properties

Proposition

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

Proof

This is 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.

Proposition

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

$$Hom_C^R(x,y) \to Hom_C^{L R}(x,y) \leftarrow Hom_C^L(x,y)$$

are homotopy equivalences of Kan complexes.

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 $Hom_C^R(x,y)$ 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 HTT, prop. 3.3.1.5. (2).

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