nLab over-(infinity,1)-category

Idea

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

$$\array{ Y_1 &&\stackrel{f}{\to}&& Y_2 \\ & {}_{\mathllap{p_1}}\searrow &\swArrow_{\simeq}^{\eta}& \swarrow_{\mathrlap{p_2}} \\ && X } \,,$$

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

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

in $C$ of the shape of the cocone under the $n$-simplex that take the tip of the cocone to the object $X$.

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-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.

Definition/Proposition

Let $C$ be a quasi-category. let $K$ be any simplicial set and let

$$F : K \to C$$

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

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

   $$Hom_{sSet}(S, C_{F/}) \cong (Hom_{(K\downarrow SSet)})(i_{K,S} , F) \,,$$

   where $i_{K,S} : K \to K \star S$ is the canonical inclusion of $K$ into its join of simplicial sets with $S$.

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

   $$Hom_{sSet}(S, C_{/F}) \simeq Hom_{(K\downarrow SSet)}( j_{K,S} , F )$$

   naturally in $S$, where $j_{K,S}$ is the canonical inclusion map $K\to S\star K$.

2. The functor

   $$sSet \to sSet_{K/}$$
   $$S \mapsto K \diamondsuit S$$

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

   has a right adjoint

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

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

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 $C_{/F}$ and $C_{F/}$ 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

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

is an equivalence of quasi-categories.

This is HTT, prop. 4.2.1.5

From the formula

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

we see that

• an object in the over quasi-category $C_{/F}$ is a cone over $F$;.

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

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

  in $C$.

• a morphism in $C_{/F}$ is a morphism of cones,

• etc:.

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

• the terminal object in $C_{/F}$ is (if it exists) the limit in $C$ over $F$;

• the initial object in $C_{F/}$ is (if it exists) the colimit of $F$ in $C$.

Properties

Relation to over-1-categories

Proposition

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

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

This appears as HTT, remark 1.2.9.6.

Functoriality of the slicing

Proposition

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

This appears as HTT, prop 1.2.9.3.

Proposition

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

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

and

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

are both left Kan fibrations.

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

Proposition

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

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

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

This is (Lurie, prop. 4.1.1.8).

Hom-spaces in a slice

Proposition

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

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

This is HTT, prop. 5.5.5.12.

Limits and Colimits in a slice

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

Proposition

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

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

For a proof see at (∞,1)-limit here.

Slicing of adjoint functors

Proposition

(sliced adjoints)
Let

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

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

Then:

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

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

   where:

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

   • $R_{/b}$ is the composite

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

     of

     1. the evident functor induced by $R$;

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

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

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

   where:

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

   • $L_{/b}$ is the composite

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

     of

     1. the evident functor induced by $L$;

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

The first statement appears, in the generality of (∞,1)-category theory, as HTT, prop. 5.2.5.1. For discussion in model category theory see at sliced Quillen adjunctions.

Proof

(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 $\;\;\eta_c \colon c \xrightarrow{\;} R \circ L(c)$

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

as follows:

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

(1a)

(2a)

(2b)

(1b)

Here:

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

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

Hence:

• starting with a morphism as in (1a) and transforming it to $(2)$ 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:

(1a)

(2)

(1b)

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

Remark

(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 $\tau$ to their adjuncts $\widetilde{\tau}$, in that (again by this Prop.):

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

The two adjunctions in admit the following joint generalisation, which is proven HTT, lem. 5.2.5.2. (Note that the statement there is even more general and here we only use the case where $K = \Delta^0$.)

Proposition

(sliced adjoints)
Let

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

be a pair of adjoint ∞-functors, where the ∞-category $\mathcal{C}$ has all homotopy pullbacks. Suppose further we are given objects $c \in \mathcal{C}$ and $d \in \mathcal{D}$ together with a morphism $\alpha: c \to R(d)$ and its adjunct $\beta:L(c) \to d$.

Then there is an induced a pair of adjoint ∞-functors between the slice ∞-categories of the form

(3)$$\mathcal{C}_{/c} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longleftarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longrightarrow}} {\bot} \mathcal{D}_{/d} \mathrlap{\,,}$$

where:

• $L_{/c}$ is the composite

  $$L_{/c} \;\colon\; \mathcal{C}_{/{c}} \overset{\;\;L\;\;}{\longrightarrow} \mathcal{D}_{/{L(c)}} \overset{\;\;\beta_!\;\;}{\longrightarrow} \mathcal{D}_{/d}$$

  of

  1. the evident functor induced by $L$;

  2. the composition with $\beta:L(c) \to d$ (i.e. the left base change along $\beta$).

• $R_{/d}$ is the composite

  $$R_{/d} \;\colon\; \mathcal{D}_{/{d}} \overset{\;\;R\;\;}{\longrightarrow} \mathcal{C}_{/{R(d)}} \overset{\;\;(\alpha^*\;\;}{\longrightarrow} \mathcal{C}_{/c}$$

  of

  1. the evident functor induced by $R$;

  2. the homotopy along $\alpha:c \to R(d)$ (i.e. the base change along $\alpha$).

Examples

• pointed homotopy types

Related concepts

• arrow category

• over-category

  • slice category

  • under category

  • over topos

• over-(∞,1)-category

  • model structure on an over category

  • over-(∞,1)-topos, etale geometric morphism

• arrow (∞,1)-category, arrow (∞,1)-topos

• slice 2-category

References

Section 1.2.9 of

• Jacob Lurie, Higher Topos Theory