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.)

Adjunctions on overcategories

Proposition

Let

(LR):DRLC (L \dashv R) : D \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} C

be a pair of adjoint functors, where the category CC has all pullbacks.

Then for every object XCX \in C there is induced a pair of adjoint functors between the slice categories

(L/XR/X):D/(LX)R/XL/XC/X (L/X \dashv R/X) : D/(L X) \stackrel{\overset{L/X}{\leftarrow}}{\underset{R/X}{\to}} C/X

where

  • L/XL/X is the evident induced functor;

  • R/XR/X is the composite

    R/X:D/LXRC/(RLX)i X *C/X R/X : D/{L X} \stackrel{R}{\to} C/{(R L X)} \stackrel{i_{X}^*}{\to} C/X

    of the evident functor induced by RR with the pullback along the (LR)(L \dashv R)-unit at XX.

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 (∞,1)-category of (∞,1)-presheaves – Interaction with overcategories

at (∞,1)-category of (∞,1)-presheaves.

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 August 25, 2020 at 00:51:43. See the history of this page for a list of all contributions to it.