nLab
over-(infinity,1)-category

Contents

Idea

For CC an (∞,1)-category and XCX \in C an object, the over-(,1)(\infty,1)-category or slice (,1)(\infty,1)-category C /XC_{/X} is the (,1)(\infty,1)-category whose objects are morphism p:YXp : Y \to X in CC, whose morphisms η:p 1p 2\eta : p_1 \to p_2 are 2-morphisms in CC of the form

Y 1 f Y 2 p 1 η p 2 X, \array{ Y_1 &&\stackrel{f}{\to}&& Y_2 \\ & {}_{\mathllap{p_1}}\searrow &\swArrow_{\simeq}^{\eta}& \swarrow_{\mathrlap{p_2}} \\ && X } \,,

hence 1-morphisms ff as indicated together with a homotopy η:p 2fp 1\eta \colon p_2 \circ f \stackrel{\simeq}{\to} p_1; and generally whose n-morphisms are diagrams

Δ[n+1]=Δ[n]Δ[0]C \Delta[n+1] = \Delta[n] \star \Delta[0]\to C

in CC of the shape of the cocone under the nn-simplex that take the tip of the cocone to the object XX.

This is the generalization of the notion of over-category in ordinary category theory.

Definition

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.

Defnition/Proposition

Let CC be a quasi-category. let KK be any simplicial set and let

F:KC F : K \to C

be an (∞,1)-functor – a morphism of simplicial sets.

  1. The under-quasi-category C F/C_{F/} is the simplicial set characterized by the property that for any other simplicial set SS there is a natural bijection of hom-sets

    Hom sSet(S,C F/)(Hom (KSSet))(i K,S,F), Hom_{sSet}(S, C_{F/}) \cong (Hom_{(K\downarrow SSet)})(i_{K,S} , F) \,,

    where i K,S:KKSi_{K,S} : K \to K \star S is the canonical inclusion of KK into its join of simplicial sets with SS.

    Similarly, the over quasi-category over FF is the simplicial set characterized by the property that

    Hom sSet(S,C /F)Hom (KSSet)(j K,S,F) Hom_{sSet}(S, C_{/F}) \simeq Hom_{(K\downarrow SSet)}( j_{K,S} , F )

    naturally in SS, where j K,Sj_{K,S} is the canonical inclusion map KSKK\to S\star K.

  2. The functor

    sSetsSet K/ sSet \to sSet_{K/}
    SKS S \mapsto K \diamondsuit S

    with \diamondsuit denoting the other definition of join of quasi-categories (as described there)

    has a right adjoint

    sSet K/sSet sSet_{K/} \to sSet
    (F:KC)C F/ (F : K \to C) \mapsto C^{F/}

    and its image C F/C^{F/} is another definition of the quasi-category under FF.

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.

Proposition

The simplicial sets C /FC_{/F} and C F/C_{F/} are indeed themselves again quasi-categories.

This appears as HTT, prop. 1.2.9.3

Proposition

The two definitions yield equivalent results in that the canonical morphism

C /FC /F. C_{/F} \to C^{/F} \,.

is an equivalence of quasi-categories.

This is HTT, prop. 4.2.1.5

From the formula

(C /F) n=(Hom sSet) F(Δ nK,C) (C_{/F})_n = (Hom_{sSet})_F(\Delta^n \star K , C)

we see that

  • an object in the over quasi-category C /FC_{/F} is a cone over FF;.

    For instance if K=Δ[1]K = \Delta[1] then an object in C /FC_{/F} is a 2-cell

    v F(0) F(1) \array{ && v \\ & \swarrow &\swArrow& \searrow \\ F(0) &&\to&& F(1) }

    in CC.

  • a morphism in C /FC_{/F} is a morphism of cones,

  • etc:.

So we may think of the overcategory C /FC_{/F} as the quasi-category of cones over FF. Accordingly we have that

Properties

Relation to over-1-categories

Proposition

For C=N(𝒞)C = N(\mathcal{C}) (the nerve of) an ordinary category 𝒞\mathcal{C} and K=*K = *, this construction coincides with the ordinary notion of overcategory 𝒞/F\mathcal{C}/F in that there is a canonical isomorphism of simplicial sets

N(𝒞/F)N(𝒞)/F. N(\mathcal{C}/F) \simeq N(\mathcal{C})/F \,.

This appears as HTT, remark 1.2.9.6.

Functoriality of the slicing

Proposition

If q:CDq : C \to D is a categorical equivalence then so is the induced morphism C /FC /qFC_{/F} \to C_{/q F}.

This appears as HTT, prop 1.2.9.3.

Proposition

For CC a quasi-category and p:XCp : X \to C any morphism of simplicial sets, the canonical morphisms

C p/C C_{p/} \to C

and

C p/C C^{p/} \to C

are both left Kan fibrations.

This is a special case of HTT, prop 2.1.2.1 and prop. 4.2.1.6.

Proposition

For v:KK˜v \colon K \to \tilde K an map of small (∞,1)-categories and 𝒞\mathcal{C} any (,1)(\infty,1)-category, the induced (∞,1)-functor between slice (,1)(\infty,1)-categories

𝒞 /K𝒞 /K˜ \mathcal{C}_{/K} \to \mathcal{C}_{/\tilde K}

is an equivalence of (∞,1)-categories precisely if vv if an op-final (∞,1)-functor (hence if v opv^{op} is final).

This is (Lurie, prop. 4.1.1.8).

Hom-spaces in a slice

Proposition

For CC an (∞,1)-category and XCX \in C an object in CC and f:AXf : A \to X and g:BXg : B \to X two objects in C/XC/X, the hom-∞-groupoid C/X(f,g)C/X(f,g) is equivalent to the homotopy fiber of C(A,B)g *C(A,X)C(A,B) \stackrel{g_*}{\to} C(A,X) over the morphism ff: we have an (∞,1)-pullback diagram

C/X(f,g) C(A,B) g * * f C(A,X). \array{ C/X(f,g) &\to& C(A,B) \\ \downarrow && \downarrow^{\mathrlap{g_*}} \\ {*} & \stackrel{f}{\to} & C(A,X) } \,.

This is HTT, prop. 5.5.5.12.

Limits and Colimits in a slice

The forgetful functor 𝒞 /X𝒞\mathcal{C}_{/X} \to \mathcal{C} out of a slice (dependent sum) reflects (∞,1)-colimits:

Proposition

Let f:K𝒞 /Xf \colon K \to \mathcal{C}_{/X} be a diagram in the slice of an (∞,1)-category 𝒞\mathcal{C} over an object X𝒞X \in \mathcal{C}. Then if the composite Kf𝒞 /X𝒞K \stackrel{f}{\to} \mathcal{C}_{/X} \to \mathcal{C} has an (∞,1)-colimit, then so does ff itself and the projection 𝒞 /q𝒞\mathcal{C}_{/q} \to \mathcal{C} takes the latter to the former. Conversely, a cocone KΔ 0𝒞 /XK \star \Delta^0 \to \mathcal{C}_{/X} under ff is an (∞,1)-colimit of ff precisely if the composite KΔ 0𝒞 /X𝒞K \star \Delta^0 \to \mathcal{C}_{/X} \to \mathcal{C} is an (,1)(\infty,1)-colimit of the projection of ff.

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:

Proposition

For 𝒞\mathcal{C} an (∞,1)-category, X:𝒟𝒞X \;\colon\; \mathcal{D} \longrightarrow \mathcal{C} a diagram, 𝒞 /X\mathcal{C}_{/X} the comma category (the over-\infty-category if 𝒟\mathcal{D} is the point) and F:K𝒞 /XF \;\colon\; K \to \mathcal{C}_{/X} a diagram in the comma category, then the (∞,1)-limit limF\underset{\leftarrow}{\lim} F in 𝒞 /X\mathcal{C}_{/X} coincides with the limit limF/X\underset{\leftarrow}{\lim} F/X in 𝒞\mathcal{C}.

For a proof see at (∞,1)-limit here.

References

Section 1.2.9 of

Revised on December 29, 2013 08:17:09 by Urs Schreiber (82.113.106.34)