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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}

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 October 9, 2021 at 04:12:51. See the history of this page for a list of all contributions to it.