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overcategory

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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

  • If C=P\mathbf{C} = \mathbf{P} is a poset and pPp \in \mathbf{P}, then the slice category P/p\mathbf{P}/p is the down set (p)\downarrow (p) of elements qPq \in \mathbf{P} with qpq \leq p.

  • If 11 is a terminal object in C\mathbf{C}, then C/1\mathbf{C}/1 is isomorphic to C\mathbf{C}.

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 the 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(y)PSh(C)/Y(y) 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 analog 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 limit in an under category is computed as a limit in the underlying category.

Precisely: let CC be a category, tCt \in C an object, and t/Ct/C the corresponding under category, and p:t/CCp : t/C \to C the obvious projection.

Let F:Dt/CF : D \to t/C be any functor. Then, if it exists, the limit over pFp \circ F in CC is the image under pp of the limit over FF:

p(limF)lim(pF) p(\lim F) \simeq \lim (p F)

and limF\lim F is uniquely characterized by lim(pF)\lim (p F).

Proof

Over a morphism γ:dd\gamma : d \to d' in DD the limiting cone over pFp F (which exists by assumption) looks like

limpF pF(d) pF(γ) pF(d) \array{ && \lim p F \\ & \swarrow && \searrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') }

By the universal property of the limit this has a unique lift to a cone in the under category t/Ct/C over FF:

t limpF pF(d) pF(γ) pF(d) \array{ && t \\ & \swarrow &\downarrow & \searrow \\ && \lim p F \\ \downarrow & \swarrow && \searrow & \downarrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') }

It therefore remains to show that this is indeed a limiting cone over FF. Again, this is immediate from the universal property of the limit in CC. For let tQt \to Q be another cone over FF in t/Ct/C, then QQ is another cone over pFp F in CC and we get in CC a universal morphism QlimpFQ \to \lim p F

t Q limpF pF(d) pF(γ) pF(d) \array{ && t \\ & \swarrow & \downarrow & \searrow \\ && Q \\ \downarrow & \swarrow &\downarrow & \searrow & \downarrow \\ && \lim p F \\ \downarrow & \swarrow && \searrow & \downarrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') }

A glance at the diagram above shows that the composite tQlimpFt \to Q \to \lim p F constitutes a morphism of cones in CC into the limiting cone over pFp F. Hence it must equal our morphism tlimpFt \to \lim p F, by the universal property of limpF\lim p F, and hence the above diagram does commute as indicated.

This shows that the morphism QlimpFQ \to \lim p F which was the unique one giving a cone morphism on CC does lift to a cone morphism in t/Ct/C, which is then necessarily unique, too. This demonstrates the required universal property of tlimpFt \to \lim p F and thus identifies it with limF\lim F.

Remark

One often says “pp reflects limits” to express the conclusion of this proposition. A conceptual way to consider this result is by appeal to a more general one: if U:ACU: A \to C is monadic (i.e., has a left adjoint FF such that the canonical comparison functor A(UF)AlgA \to (U F)-Alg is an equivalence), then UU both reflects and preserves limits. In the present case, the projection p:A=t/CCp: A = t/C \to C is monadic, is essentially the category of algebras for the monad T()=t+()T(-) = t + (-), at least if CC admits binary coproducts. (Added later: the proof is even simpler: if U:ACU: A \to C is the underlying functor for the category of algebras of an endofunctor on CC (as opposed to algebras of a monad), then UU reflects and preserves limits; then apply this to the endofunctor TT above.)

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 𝒞 /X\mathcal{C}_{/X} is id X\mathrm{id}_X.

Revised on October 17, 2013 11:46:34 by Urs Schreiber (188.138.1.73)