nLab
hom-object in a quasi-category

Contents

Idea

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

Definition

Let CC be a quasi-category (or any simplicial set), and x,yC 0x,y \in C_0 any two objects. Then write

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

where

This defines Hom C(x,y)Hom_C(x,y) as an equivalence class of \infty-groupoids, but at the same time defines a particular representative: if CC is a quasi-category then τ(C)(x,y)\tau(C)(x,y) is a Kan complex that represents this class.

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

Definition

For CC and x,yx,y as above, write

Hom C LR(x,y):={x}× CC Δ[1]× C{y} Hom_C^{L R}(x,y) := \{x\} \times_C C^{\Delta[1]} \times_C \{y\}

for the pullback

{x}× CC Δ[1]× C{y} C Δ[1] d 1×d 0 {x}×{y} C×C \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 Δ[1]C^{\Delta[1]} (the cartesian internal hom in sSet with the 1-simplex Δ[1]\Delta[1]) .

Write

Hom C R(x,y)sSet Hom^R_C(x,y) \in sSet

for the simplicial set whose nn-simplices are defined to be those morphisms σ:Δ[n+1]C\sigma : \Delta[n+1] \to C such that the restriction to Δ{0,,n}\Delta\{0, \cdots, n\} is the constant map to xx and the restriction to Δ{n+1}\Delta\{n+1\} is the map to yy.

Analogously, write

Hom C L(x,y)sSet Hom^L_C(x,y) \in sSet

for the simplicial set whose nn-simplices are morphisms Δ[n+1]X\Delta[n+1] \to X which restrict to xx on {0}\{0\} and are constant on yy when restricted to {1,,n+1}\{1, \cdots, n+1\}.

Remark The 1-cells in Hom C R(x,y)Hom_C^R(x,y), Hom C L(x,y)Hom_C^L(x,y) and Hom C LR(x,y)Hom_C^{L R}(x,y) are 2-globes in CC. 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:KCp : K \to C a morphism, we have the over quasi-category C /pC_{/p} defined by

(C /p) n:=Hom(Δ[n],C /p):=Hom p(Δ[n]K,C), (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 sSetsSet out of the join of simplicial sets that restrict on KK to pp.

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

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

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

Obervation

We have

Hom C R(x,y)=C /y× C{x}, Hom^R_C(x,y) = C_{/{y}} \times_C \{x\} \,,

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

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

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

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

Δ[n]×Δ[0] Δ[0] Δ[n]×Δ[0] C×Δ[0]×Δ[1] Δ[n+1] Δ[n] C×Δ[1] Δ[n+1], \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 Δ[1]× C{y} C^{/y} = C^{\Delta[1]} \times_C \{y\}

because

(C /y) n=Hom Δ[0]/sSet(Δ[n]×Δ[1] Δ[1]×Δ[0]Δ[0],C)=Hom(Δ[n]×Δ[1],C)× Hom(Δ[1],C){y}=(C Δ[1]×{y}) n. (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 τ(C)(c,y)\tau(C)(c,y) of Hom C(x,y)Hom_C(x,y) is that, by the fact that τ(C)\tau(C) is an sSet-enriched category, there is a strict composition operation

τ(C)(x,y)×τ(C)(y,z)τ(C)(x,z). \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)Hom_C^R(x,y) and Hom C L(x,y)Hom_C^L(x,y)

Properties

Proposition

If the simplicial set CC is a quasi-category, then Hom C R(x,y)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)Hom_C^R(x,y) has fillers for all inner and right outer horns Λ[n] 1in\Lambda[n]_{1 \leq i \leq n}, because these yield inner horns in Δ[n+1]=Δ[n]Δ[0]\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 CC is a quasi-category then the canonical inclusions

Hom C R(x,y)Hom C LR(x,y)Hom C L(x,y) 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 /yC /yC_{/y} \to C^{/y} is an equivalence of quasi-categories. So for the statement for Hom C R(x,y)Hom_C^R(x,y) it suffices to show that this induces an equivalence of fibers over CC. This follows from the fact that both C /yCC_{/y} \to C and C /yCC^{/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)Hom_C^L(x,y) follows dually.

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