nLab over-(infinity,1)-category




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.


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-categorically 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.


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 The second definition appears in HTT above prop.


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

This appears as HTT, prop.


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.

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


Relation to over-1-categories


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

Functoriality of the slicing


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

This appears as HTT, prop


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


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

are both left Kan fibrations.

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


Let v:KK˜v \colon K \to \tilde K be a map of small (∞,1)-categories, 𝒞\mathcal{C} an (,1)(\infty,1)-category, p˜:K˜𝒞\tilde{p}: \tilde{K} \to \mathcal{C} and p=p˜vp = \tilde{p}v. The induced (∞,1)-functor between slice (,1)(\infty,1)-categories

𝒞 /p˜𝒞 /p \mathcal{C}_{/ \tilde{p}} \to \mathcal{C}_{/p}

is an equivalence of (∞,1)-categories for each diagram p˜\tilde{p} precisely if vv is an op-final (∞,1)-functor (hence if v opv^{op} is final).

This is (Lurie, prop.

Hom-spaces in a slice


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) & \longrightarrow & C(A,B) \\ \downarrow && \downarrow^{\mathrlap{g_*}} \\ {*} & \stackrel{\vdash f}{\longrightarrow} & C(A,X) } \,.

This is HTT, prop.

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:


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.

On the other hand (∞,1)-limits in the slice are computed as limits over the diagram with the slice-cocone adjoined:


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.

Slicing of adjoint functors


(sliced adjoints)

𝒟RL𝒞 \mathcal{D} \underoverset {\underset{\;\;\;\;R\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}

be a pair of adjoint functors (adjoint ∞-functors), where the category (∞-category) 𝒞\mathcal{C} has all pullbacks (homotopy pullbacks).


  1. For every object b𝒞b \in \mathcal{C} there is induced a pair of adjoint functors between the slice categories (slice ∞-categories) of the form

    (1)𝒟 /L(b)R /bL /b𝒞 /b, \mathcal{D}_{/L(b)} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/b} \mathrlap{\,,}


    • L /bL_{/b} is the evident induced functor (applying LL to the entire triangle diagrams in 𝒞\mathcal{C} which represent the morphisms in 𝒞 /b\mathcal{C}_{/b});

    • R /bR_{/b} is the composite

      R /b:𝒟 /L(b)R𝒞 /(RL(b))(η b) *𝒞 /b R_{/b} \;\colon\; \mathcal{D}_{/{L(b)}} \overset{\;\;R\;\;}{\longrightarrow} \mathcal{C}_{/{(R \circ L(b))}} \overset{\;\;(\eta_{b})^*\;\;}{\longrightarrow} \mathcal{C}_{/b}


      1. the evident functor induced by RR;

      2. the (homotopy) pullback along the (LR)(L \dashv R)-unit at bb (i.e. the base change along η b\eta_b).

  2. For every object b𝒟b \in \mathcal{D} there is induced a pair of adjoint functors between the slice categories of the form

    (2)𝒟 /bR /bL /b𝒞 /R(b), \mathcal{D}_{/b} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/R(b)} \mathrlap{\,,}


    • R /bR_{/b} is the evident induced functor (applying RR to the entire triangle diagrams in 𝒟\mathcal{D} which represent the morphisms in 𝒟 /b\mathcal{D}_{/b});

    • L /bL_{/b} is the composite

      L /b:𝒟 /R(b)L𝒞 /(LR(b))(ϵ b) !𝒞 /b L_{/b} \;\colon\; \mathcal{D}_{/{R(b)}} \overset{\;\;L\;\;}{\longrightarrow} \mathcal{C}_{/{(L \circ R(b))}} \overset{\;\;(\epsilon_{b})_!\;\;}{\longrightarrow} \mathcal{C}_{/b}


      1. the evident functor induced by LL;

      2. the composition with the (LR)(L \dashv R)-counit at bb (i.e. the left base change along ϵ b\epsilon_b).

The first statement appears, in the generality of (∞,1)-category theory, as HTT, prop. For discussion in model category theory see at sliced Quillen adjunctions.

(in 1-category theory)

Recall that (this Prop.) the hom-isomorphism that defines an adjunction of functors (this Def.) is equivalently given in terms of composition with

  • the adjunction unit η c:cRL(c)\;\;\eta_c \colon c \xrightarrow{\;} R \circ L(c)

  • the adjunction counit ϵ d:LR(d)d\;\;\epsilon_d \colon L \circ R(d) \xrightarrow{\;} d

as follows:

Using this, consider the following transformations of morphisms in slice categories, for the first case:






  • (1a) and (1b) are equivalent expressions of the same morphism ff in 𝒟 /L(b)\mathcal{D}_{/L(b)}, by (at the top of the diagrams) the above expression of adjuncts between 𝒞\mathcal{C} and 𝒟\mathcal{D} and (at the bottom) by the triangle identity.

  • (2a) and (2b) are equivalent expression of the same morphism f˜\tilde f in 𝒞 /b\mathcal{C}_{/b}, by the universal property of the pullback.


  • starting with a morphism as in (1a) and transforming it to (2)(2) and then to (1b) is the identity operation;

  • starting with a morphism as in (2b) and transforming it to (1) and then to (2a) is the identity operation.

In conclusion, the transformations (1) \leftrightarrow (2) consitute a hom-isomorphism that witnesses an adjunction of the first claimed form (1).

The second case follows analogously, but a little more directly since no pullback is involved:




In conclusion, the transformations (1) \leftrightarrow (2) consitute a hom-isomorphism that witnesses an adjunction of the second claimed form (2).


(left adjoint of sliced adjunction forms adjuncts)
The sliced adjunction (Prop. ) in the second form (2) is such that the sliced left adjoint sends slicing morphism τ\tau to their adjuncts τ˜\widetilde{\tau}, in that (again by this Prop.):

L /d(c τ R(b))=(L(c) τ˜ b)𝒟 /b L_{/d} \, \left( \array{ c \\ \big\downarrow {}^{\mathrlap{\tau}} \\ R(b) } \right) \;\; = \;\; \left( \array{ L(c) \\ \big\downarrow {}^{\mathrlap{\widetilde{\tau}}} \\ b } \right) \;\;\; \in \; \mathcal{D}_{/b}



Section 1.2.9 of

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