nLab over category

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Definition
Examples
Properties

Comonadicity
Relation to codomain fibration
Slicing of adjoint functors
Presheaves on over-categories and over-categories of presheaves
Limits and colimits
Initial and terminal objects

Related concepts
References

Definition

Theslice category orover category $C / c \mathbf{C}/c$ of acategory $C \mathbf{C}$ over an object $c \in C c \in \mathbf{C}$ has

objects that are all arrows $f \in C f \in \mathbf{C}$ such that $cod (f) = c cod(f) = c$ , and
morphisms $g : X \to X' \in C g: X \to X' \in \mathbf{C}$ from $f : X \to c f:X \to c$ to $f' : X' \to c f': X' \to c$ such that $f' \circ g = f f' \circ g = f$ .

C / c = {\begin{matrix} X & \overset{g}{\to} & X' \\ _{f} ↘ & ↙_{f'} \\ c \end{matrix}} C/c  =  \left\lbrace    \array{       X &&\stackrel{g}{\to}&& X'       \\       & {}_f \searrow && \swarrow_{f'}       \\          && c    }  \right\rbrace

The slice category is a special case of acomma category.

There is aforgetful functor $U_{c} : C / c \to C U_c: \mathbf{C}/c \to \mathbf{C}$ which maps an object $f : X \to c f:X \to c$ to its domain $X X$ and a morphism $g : X \to X' \in C / c g: X \to X' \in \mathbf{C}/c$ (from $f : X \to c f:X \to c$ to $f' : X' \to c f': X' \to c$ such that $f' \circ g = f f' \circ g = f$ ) to the morphism $g : X \to X' g: X \to X'$ .

Thedual notion is anunder category.

Examples

If $C = P \mathbf{C} = \mathbf{P}$ is aposet and $p \in P p \in \mathbf{P}$ , then the slice category $P / p \mathbf{P}/p$ is thedown set $↓ (p) \downarrow (p)$ of elements $q \in P q \in \mathbf{P}$ with $q \leq p q \leq p$ .
If $11$ is aterminal object in $C \mathbf{C}$ , then $C / 1 \mathbf{C}/1$ is isomorphic to $C \mathbf{C}$ .
For $X X$ atopological space then thecategory of covering spaces over $X X$ is afull subcategory of the slice category ${Top}_{/ X} Top_{/X}$ of thecategory of topological spaces.
Thefundamental theorem of topos theory states that the slice category over any object in a topos is itself a topos.
For amonoidal category the slice category over anymonoid object is monoidal.
For instance, theslice topos of a giventopos over anymonoid object is canonically amonoidal topos (see the Examplethere).

Properties

Comonadicity

If $C C$ admits binary coproducts with the fixed object $c c$ , then the forgetful functor $C / c \to C C/c \to C$ iscomonadic. Seecoreader comonad for more details.

Relation to codomain fibration

The assignment of overcategories $C / c C/c$ to objects $c \in C c \in C$ extends to afunctor

C / (-) : C \to Cat C/(-) : C \to Cat

Under theGrothendieck construction this functor corresponds to thecodomain fibration

cod : [I, C] \to C cod : [I,C] \to C

from thearrow category of $C C$ . (Note that unless $C C$ haspullbacks, this functor is not actually afibration, though it is always an opfibration.)

Slicing of adjoint functors

Proposition

(sliced adjoints)
Let

𝒟 ⊥_{\underset{R}{⟶}}^{\overset{L}{⟵}} 𝒞 \mathcal{D}    \underoverset    {\underset{\;\;\;\;R\;\;\;\;}{\longrightarrow}}    {\overset{\;\;\;\;L\;\;\;\;}{\longleftarrow}}    {\bot}  \mathcal{C}

be a pair ofadjoint functors (adjoint ∞-functors), where thecategory (∞-category) $𝒞 \mathcal{C}$ has allpullbacks (homotopy pullbacks).

Then:

For everyobject $b \in 𝒞 b \in \mathcal{C}$ there is induced a pair ofadjoint functors between theslice categories (slice ∞-categories) of the form
(1) $𝒟_{/ L (b)} ⊥_{\underset{R_{/ b}}{⟶}}^{\overset{L_{/ b}}{⟵}} 𝒞_{/ b}, \mathcal{D}_{/L(b)} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/b} \mathrlap{\,,}$
where:
- $L_{/ b} L_{/b}$ is the evident induced functor (applying $L L$ to the entire trianglediagrams in $𝒞 \mathcal{C}$ which represent the morphisms in $𝒞_{/ b} \mathcal{C}_{/b}$ );
- $R_{/ b} R_{/b}$ is thecomposite
  $R_{/ b} : 𝒟_{/ L (b)} \overset{R}{⟶} 𝒞_{/ (R \circ L (b))} \overset{(η_{b})^{*}}{⟶} 𝒞_{/ b} R_{/b} \;\colon\; \mathcal{D}_{/{L(b)}} \overset{\;\;R\;\;}{\longrightarrow} \mathcal{C}_{/{(R \circ L(b))}} \overset{\;\;(\eta_{b})^*\;\;}{\longrightarrow} \mathcal{C}_{/b}$
  of
  1. the evident functor induced by $R R$ ;
  2. the (homotopy)pullback along the $(L ⊣ R) (L \dashv R)$ -unit at $b b$ (i.e. thebase change along $η_{b} \eta_b$ ).
For everyobject $b \in 𝒟 b \in \mathcal{D}$ there is induced a pair ofadjoint functors between theslice categories of the form
(2) $𝒟_{/ b} ⊥_{\underset{R_{/ b}}{⟶}}^{\overset{L_{/ b}}{⟵}} 𝒞_{/ R (b)}, \mathcal{D}_{/b} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/R(b)} \mathrlap{\,,}$
where:
- $R_{/ b} R_{/b}$ is the evident induced functor (applying $R R$ to the entire trianglediagrams in $𝒟 \mathcal{D}$ which represent the morphisms in $𝒟_{/ b} \mathcal{D}_{/b}$ );
- $L_{/ b} L_{/b}$ is thecomposite
  $L_{/ b} : 𝒟_{/ R (b)} \overset{L}{⟶} 𝒞_{/ (L \circ R (b))} \overset{(ϵ_{b})_{!}}{⟶} 𝒞_{/ b} L_{/b} \;\colon\; \mathcal{D}_{/{R(b)}} \overset{\;\;L\;\;}{\longrightarrow} \mathcal{C}_{/{(L \circ R(b))}} \overset{\;\;(\epsilon_{b})_!\;\;}{\longrightarrow} \mathcal{C}_{/b}$
  of
  1. the evident functor induced by $L L$ ;
  2. thecomposition with the $(L ⊣ R) (L \dashv R)$ -counit at $b b$ (i.e. the leftbase change along $ϵ_{b} \epsilon_b$ ).

The first statement appears, in the generality of(∞,1)-category theory, asHTT, prop. 5.2.5.1. For discussion inmodel category theory see atsliced Quillen adjunctions.

Proof

(in1-category theory)

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

theadjunction unit $η_{c} : c \overset{}{\to} R \circ L (c) \;\;\eta_c \colon c \xrightarrow{\;} R \circ L(c)$
theadjunction counit $ϵ_{d} : L \circ R (d) \overset{}{\to} d \;\;\epsilon_d \colon L \circ R(d) \xrightarrow{\;} d$

as follows:

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

(1a)

(2a)

(2b)

(1b)

Here:

(1a) and (1b) are equivalent expressions of the same morphism $f f$ in $𝒟_{/ L (b)} \mathcal{D}_{/L(b)}$ , by (at the top of the diagrams) the above expression ofadjuncts between $𝒞 \mathcal{C}$ and $𝒟 \mathcal{D}$ and (at the bottom) by thetriangle identity.
(2a) and (2b) are equivalent expression of the same morphism $\tilde{f} \tilde f$ in $𝒞_{/ b} \mathcal{C}_{/b}$ , by theuniversal property of thepullback.

Hence:

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) $\leftrightarrow \leftrightarrow$ (2) consitute ahom-isomorphism that witnesses an adjunction of the first claimed form(1).

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

(1a)

(2)

(1b)

In conclusion, the transformations (1) $\leftrightarrow \leftrightarrow$ (2) consitute ahom-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 slicedleft adjoint sends slicing morphism $τ \tau$ to theiradjuncts $\tilde{τ} \widetilde{\tau}$ , in that (again bythis Prop.):

L_{/ d} (\begin{matrix} c \\ ↓^{τ} \\ R (b) \end{matrix}) = (\begin{matrix} L (c) \\ ↓^{\tilde{τ}} \\ b \end{matrix}) \in 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}

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

Proposition

(sliced adjoints)
Let

𝒞 ⊥_{\underset{R}{⟵}}^{\overset{L}{⟶}} 𝒟 \mathcal{C}    \underoverset    {\underset{\;\;\;\;R\;\;\;\;}{\longleftarrow}}    {\overset{\;\;\;\;L\;\;\;\;}{\longrightarrow}}    {\bot}  \mathcal{D}

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

Then there is an induced a pair ofadjoint ∞-functors between theslice ∞-categories of the form

(3)

𝒞_{/ c} ⊥_{\underset{R_{/ b}}{⟵}}^{\overset{L_{/ b}}{⟶}} 𝒟_{/ d}, \mathcal{C}_{/c}     \underoverset       {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longleftarrow}}       {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longrightarrow}}       {\bot}     \mathcal{D}_{/d}     \mathrlap{\,,}

where:

$L_{/ c} L_{/c}$ is thecomposite
$L_{/ c} : 𝒞_{/ c} \overset{L}{⟶} 𝒟_{/ L (c)} \overset{β_{!}}{⟶} 𝒟_{/ d} L_{/c} \;\colon\; \mathcal{C}_{/{c}} \overset{\;\;L\;\;}{\longrightarrow} \mathcal{D}_{/{L(c)}} \overset{\;\;\beta_!\;\;}{\longrightarrow} \mathcal{D}_{/d}$
of
1. the evident functor induced by $L L$ ;
2. thecomposition with $β : L (c) \to d \beta:L(c) \to d$ (i.e. the leftbase change along $β \beta$ ).
$R_{/ d} R_{/d}$ is thecomposite
$R_{/ d} : 𝒟_{/ d} \overset{R}{⟶} 𝒞_{/ R (d)} \overset{(α^{*}}{⟶} 𝒞_{/ c} R_{/d} \;\colon\; \mathcal{D}_{/{d}} \overset{\;\;R\;\;}{\longrightarrow} \mathcal{C}_{/{R(d)}} \overset{\;\;(\alpha^*\;\;}{\longrightarrow} \mathcal{C}_{/c}$
of
1. the evident functor induced by $R R$ ;
2. thehomotopy along $α : c \to R (d) \alpha:c \to R(d)$ (i.e. thebase change along $α \alpha$ ).

Presheaves on over-categories and over-categories of presheaves

Seeslice of presheaves is presheaves on slice.

Let $C C$ be acategory, $c c$ anobject of $C C$ and let $C / c C/c$ be theover category of $C C$ over $c c$ . Write $PSh (C / c) = [(C / c)^{op}, Set] PSh(C/c) = [(C/c)^{op}, Set]$ for thecategory of presheaves on $C / c C/c$ and write $PSh (C) / Y (c) PSh(C)/Y(c)$ for theover category ofpresheaves on $C C$ over the presheaf $Y (c) Y(c)$ , where $Y : C \to PSh (c) Y : C \to PSh(c)$ is theYoneda embedding.

Proposition

There is anequivalence of categories

e : PSh (C / c) \overset{≃}{\to} PSh (C) / Y (c) . e : PSh(C/c) \stackrel{\simeq}{\to} PSh(C)/Y(c)  \,.

Proof

The functor $e e$ takes $F \in PSh (C / c) F \in PSh(C/c)$ to the presheaf $F' : d \mapsto ⊔_{f \in C (d, c)} F (f) F' : d \mapsto \sqcup_{f \in C(d,c)} F(f)$ which is equipped with the natural transformation $η : F' \to Y (c) \eta : F' \to Y(c)$ with component map $η_{d} : ⊔_{f \in C (d, c)} F (f) \to C (d, c) \eta_d: \sqcup_{f \in C(d,c)} F(f) \to C(d,c)$ .

A weak inverse of $e e$ is given by the functor

\bar{e} : PSh (C) / Y (c) \to PSh (C / c) \bar e : PSh(C)/Y(c) \to PSh(C/c)

which sends $η : F' \to Y (C)) \eta : F' \to Y(C))$ to $F \in PSh (C / c) F \in PSh(C/c)$ given by

F : (f : d \to c) \mapsto F' (d) |_{c}, F : (f : d \to c) \mapsto F'(d)|_c  \,,

where $F' (d) |_{c} F'(d)|_c$ is thepullback

\begin{matrix} F' (d) |_{c} & \to & F' (d) \\ ↓ & ↓^{η_{d}} \\ pt & \overset{f}{\to} & C (d, c) \end{matrix} . \array{     F'(d)|_c &\to& F'(d)     \\     \downarrow && \downarrow^{\eta_d}     \\     pt &\stackrel{f}{\to}& C(d,c)  }  \,.

Example

Suppose the presheaf $F \in PSh (C / c) F \in PSh(C/c)$ does not actually depend on the morphisms to $C C$ , i.e. suppose that it factors through the forgetful functor from theover category to $C C$ :

F : (C / c)^{op} \to C^{op} \to Set . F : (C/c)^{op} \to C^{op} \to Set  \,.

Then $F' (d) = ⊔_{f \in C (d, c)} F (f) = ⊔_{f \in C (d, c)} F (d) ≃ C (d, c) \times F (d) F'(d) = \sqcup_{f \in C(d,c)} F(f) = \sqcup_{f \in C(d,c)} F(d) \simeq C(d,c) \times F(d)$ and hence $F' = Y (c) \times F F ' = Y(c) \times F$ with respect to theclosed monoidal structure on presheaves.

Limits and colimits

Proposition

Acolimit in anover category is computed as a colimit in the underlying category.

Precisely: let $𝒞 \mathcal{C}$ be acategory, $t \in 𝒞 t \in \mathcal{C}$ anobject, and $𝒞 / t \mathcal{C}/t$ the correspondingovercategory, and $p : 𝒞 / t \to 𝒞 p \colon \mathcal{C}/t \to \mathcal{C}$ the obvious projection.

Let $F : D \to 𝒞 / t F \colon D \to \mathcal{C}/t$ be anyfunctor. Then, if it exists, thecolimit of $p \circ F p \circ F$ in $𝒞 \mathcal{C}$ is the image under $p p$ of the colimit over $F F$ :

p (\lim_{⟶} F) ≃ \lim_{⟶} (p \circ F) p  \big(    \underset{\longrightarrow}{\lim}    F  \big)    \;\simeq\;   \underset{\longrightarrow}{\lim} (p \circ F)

and $\lim_{⟶} F \underset{\longrightarrow}{\lim} F$ is uniquely characterized by $\lim_{⟶} (p \circ F) \underset{\longrightarrow}{\lim} (p \circ F)$ this way.

This statement, and its proof, is theformal dual to the corresponding statement forundercategories, seethere.

Proposition

For $𝒞 \mathcal{C}$ acategory, $X : 𝒟 ⟶ 𝒞 X \;\colon\; \mathcal{D} \longrightarrow \mathcal{C}$ adiagram, $𝒞_{/ X} \mathcal{C}_{/X}$ thecomma category (the over-category if $𝒟 \mathcal{D}$ is the point) and $F : K \to 𝒞_{/ X} F \;\colon\; K \to \mathcal{C}_{/X}$ adiagram in thecomma category, then thelimit $\lim_{\leftarrow} F \underset{\leftarrow}{\lim} F$ in $𝒞_{/ X} \mathcal{C}_{/X}$ coincides with the limit $\lim_{\leftarrow} F / X \underset{\leftarrow}{\lim} F/X$ in $𝒞 \mathcal{C}$ .

For a proof see at(∞,1)-limithere.

Initial and terminal objects

As a special case of the above discussion of limits and colimits in a slice $𝒞_{/ X} \mathcal{C}_{/X}$ we obtain the following statement, which of course is also immediately checked explicitly.

Corollary

If $𝒞 \mathcal{C}$ has an initial object $\emptyset \emptyset$ , then $𝒞_{/ X} \mathcal{C}_{/X}$ has aninitial object, given by $⟨ \emptyset \to X ⟩ \langle \emptyset \to X\rangle$ .
Theterminal object of $𝒞_{/ X} \mathcal{C}_{/X}$ is ${id}_{X} \mathrm{id}_X$ .

Related concepts

over-category
slice 2-category
over (∞,1)-category,
- model structure on an over category
- over-(∞,1)-topos

References

Formalization incubical Agda:

1lab:Slice categories

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nLab over category

Context

Category theory

Concepts

Universal constructions

Theorems

Extensions

Applications

Contents

Definition

Examples

Properties

Comonadicity

Relation to codomain fibration

Slicing of adjoint functors

Proposition

Proof

Remark

Proposition

Presheaves on over-categories and over-categories of presheaves

Proposition

Proof

Example

Limits and colimits

Proposition

Proposition

Initial and terminal objects

Corollary

Related concepts

References