functors and comma categories


Category theory


Universal constructions

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This entry is about special properties of functors on comma categories. See also category of presheaves.


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.


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) \,.

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):((fC(d,c),θF(f))f. \eta_d : \sqcup_{f \in C(d,c)} F(f) \to C(d,c) : ((f \in C(d,c), \theta \in F(f)) \mapsto f \,.

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) } \,.

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 over-topos for more.

Over-categories of presheaf categories and presheaves on categories of elements

Generalizing the above,


For every P:Set DP \colon \mathbf{Set}^D, there is an equivalence of categories

φ:Set el(P)Set D/P. \varphi : \mathbf{Set}^{el(P)} \stackrel{\simeq}{\to} \mathbf{Set}^D / P \,.

where el(P)=*/Pel(P) = \ast / P is the category of elements of PP.


The construction is completely analogous to the above; Given F:Set el(P)F \colon \mathbf{Set}^{el(P)}, φ(F)\varphi(F) is defined pointwise as a coproduct:

φ(F)(c)= (x,Pc)F(x,Pc) \varphi(F)(c) = \coprod_{(x,Pc)} F(x,Pc)

where (x,Pc)=(*xPc)(x,Pc) = (\ast \stackrel{x}{\to} Pc) is an object of el(P)el(P). The action on morphisms is defined analogously. This comes equipped with a natural transformation α:φ(F)P\alpha \colon \varphi(F) \to P, with component

α c:φ(F)(c)Pc α c(F(x,Pc))=xPc \array{ \alpha_c \colon \varphi(F)(c) \to Pc \\ \alpha_c(F(x,Pc)) = x \in Pc \\ }

Given an object (QαP)(Q \stackrel{\alpha}{\to} P) of Set D/P\mathbf{Set}^D / P the action of a weak inverse φ¯\bar \varphi can be specified as φ¯(α)(x,Pc)=α c 1(x) \bar{\varphi}(\alpha)(x,Pc) = \alpha_c^{-1}(x), that is, the wedge of the pullback:

φ¯(α)(x,Pc) Qc α c pt x Pc. \array{ \bar{\varphi}(\alpha)(x,Pc) &\to& Qc \\ \downarrow && \downarrow^{\alpha_c} \\ pt &\stackrel{x}{\to}& Pc } \,.

The action of φ¯(α)\bar{\varphi}(\alpha) on arrows of el(P)el(P), functoriality, etc is derived from its definition as a pullback and the def of morphisms in el(P)el(P).

Relationship with the over-categories statement

Putting D=C op,P=Y(c)D = C^{op}, P = Y(c) in the above yields:

Set el(Y(c))Set D/Y(c) \mathbf{Set}^{el(Y(c))} \simeq \mathbf{Set}^D / Y(c)

Now it is easy to see that el(Y(c))(C/c) opel(Y(c)) \simeq (C / c)^{op}; we get then:

Set (C/c) opSet C op/Y(c) \mathbf{Set}^{(C / c)^{op}} \simeq \mathbf{Set}^{C^{op}} / Y(c)

In higher category theory

For the analogous result in the context of (∞,1)-category theory see (∞,1)-Category of (∞,1)-presheaves – Interaction with overcategories

Last revised on June 18, 2013 at 19:04:07. See the history of this page for a list of all contributions to it.