nLab over category

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Definition

The slice category or over category C/c\mathbf{C}/c of a category C\mathbf{C} over an object cCc \in \mathbf{C} has

  • objects that are all arrows fCf \in \mathbf{C} such that cod(f)=ccod(f) = c, and

  • morphisms g:XXCg: X \to X' \in \mathbf{C} from f:Xcf:X \to c to f:Xcf': X' \to c such that fg=ff' \circ g = f.

C/c={X g X f f c} 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 a comma category.

There is a forgetful functor U c:C/cCU_c: \mathbf{C}/c \to \mathbf{C} which maps an object f:Xcf:X \to c to its domain XX and a morphism g:XXC/cg: X \to X' \in \mathbf{C}/c (from f:Xcf:X \to c to f:Xcf': X' \to c such that fg=ff' \circ g = f) to the morphism g:XXg: X \to X'.

The dual notion is an under category.

Examples

Properties

Relation to codomain fibration

The assignment of overcategories C/cC/c to objects cCc \in C extends to a functor

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

Under the Grothendieck construction this functor corresponds to the codomain fibration

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

from the arrow category of CC. (Note that unless CC has pullbacks, this functor is not actually a fibration, though it is always an opfibration.)

Slicing of adjoint functors

Proposition

(sliced adjoints)
Let

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).

Then:

  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{\,,}

    where:

    • 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}

      of

      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{\,,}

    where:

    • 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}

      of

      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. 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 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)

(2a)

(2b)

(1b)

Here:

  • (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.

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 (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 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}

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=Δ 0K = \Delta^0.)

Proposition

(sliced adjoints)
Let

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

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

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

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

where:

  • L /cL_{/c} is the composite

    L /c:𝒞 /cL𝒟 /L(c)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 LL;

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

  • R /dR_{/d} is the composite

    R /d:𝒟 /dR𝒞 /R(d)(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 RR;

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

Presheaves on over-categories and over-categories of presheaves

Let CC be a category, cc an object of CC and let C/cC/c be the over category of CC over cc. Write PSh(C/c)=[(C/c) op,Set]PSh(C/c) = [(C/c)^{op}, Set] for the category of presheaves on C/cC/c and write PSh(C)/Y(c)PSh(C)/Y(c) for the over category of presheaves on CC over the presheaf Y(c)Y(c), where Y:CPSh(c)Y : C \to PSh(c) is the Yoneda embedding.

Proposition

There is an equivalence of categories

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

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

A weak inverse of ee is given by the functor

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

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

F:(f:dc)F(d)| c, F : (f : d \to c) \mapsto F'(d)|_c \,,

where F(d)| cF'(d)|_c is the pullback

F(d)| c F(d) η d pt f C(d,c). \array{ F'(d)|_c &\to& F'(d) \\ \downarrow && \downarrow^{\eta_d} \\ pt &\stackrel{f}{\to}& C(d,c) } \,.
Example

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

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

Then F(d)= fC(d,c)F(f)= fC(d,c)F(d)C(d,c)×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)×FF ' = Y(c) \times F with respect to the closed monoidal structure on presheaves.

See also functors and comma categories.

For the analogous statement in (∞,1)-category theory see at (∞,1)-category of (∞,1)-presheaves – Interaction with overcategories?.

Limits and colimits

Proposition

A colimit in an over category is computed as a colimit in the underlying category.

Precisely: let 𝒞\mathcal{C} be a category, t𝒞t \in \mathcal{C} an object, and 𝒞/t\mathcal{C}/t the corresponding overcategory, and p:𝒞/t𝒞p \colon \mathcal{C}/t \to \mathcal{C} the obvious projection.

Let F:D𝒞/tF \colon D \to \mathcal{C}/t be any functor. Then, if it exists, the colimit of pFp \circ F in 𝒞\mathcal{C} is the image under pp of the colimit over FF:

p(limF)lim(pF) p \big( \underset{\longrightarrow}{\lim} F \big) \;\simeq\; \underset{\longrightarrow}{\lim} (p \circ F)

and limF\underset{\longrightarrow}{\lim} F is uniquely characterized by lim(pF)\underset{\longrightarrow}{\lim} (p \circ F) this way.

This statement, and its proof, is the formal dual to the corresponding statement for undercategories, see there.

Proposition

For 𝒞\mathcal{C} a category, X:𝒟𝒞X \;\colon\; \mathcal{D} \longrightarrow \mathcal{C} a diagram, 𝒞 /X\mathcal{C}_{/X} the comma category (the over-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 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.

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, then 𝒞 /X\mathcal{C}_{/X} has an initial object, given by X\langle \emptyset \to X\rangle.

  • The terminal object of 𝒞 /X\mathcal{C}_{/X} is id X\mathrm{id}_X.

Last revised on September 22, 2022 at 07:34:52. See the history of this page for a list of all contributions to it.