category of presheaves


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For CC a small category, its category of presheaves is the functor category

PSh(C):=[C op,Set] PSh(C) := [C^{op}, Set]

from the opposite category of CC to Set.

An object in this category is a presheaf. See there for more details.




See functoriality of categories of presheaves.



A category EE is equivalent to a presheaf topos if and only if it is cocomplete, atomic, and regular.

This is due to Marta Bunge.

Cartesian closed monoidal structure

As every topos, a category of presheaves is a cartesian closed monoidal category.

For details on the closed structure see

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)\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) } \,.

Suppose the presheaf FPSh(C/c)F \in PSh(C/c) does not actually depend on the morphisms 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


Consider CY(c)\int_C Y(c) , the category of elements of Y(c):C opSetY(c):C^{op}\to Set. This has objects (d 1,p 1)(d_1,p_1) with p 1Y(c)(d 1)p_1\in Y(c)(d_1), hence p 1p_1 is just an arrow d 1cd_1\to c in CC. A map from (d 1,p 1)(d_1, p_1) to (d 2,p 2)(d_2, p_2) is just a map u:d 1d 2u:d_1\to d_2 such that p 2u=p 1p_2\circ u =p_1 but this is just a morphism from p 1p_1 to p 2p_2 in C/cC/c.

Hence, the above proposition 1 can be rephrased as PSh( CY(c))PSh(C)/Y(c)PSh(\int_C Y(c))\simeq PSh(C)/Y(c) which is an instance of the following formula:


Let P:C opSetP:C^{op}\to Set be a presheaf. Then there is an equivalence of categories

PSh( CP)PSh(C)/P. PSh(\int_C P) \simeq PSh(C)/P \,.

For a proof see Kashiwara-Schapira (2006, p.26). For a more general statement involving slices of Grothendieck toposes see Mac Lane-Moerdijk (1994, p.157).

In particular, this equivalence shows that slices of presheaf toposes are presheaf toposes.

Finite presheaves

A finite presheaf on a category CC is a functor C opFinSetC^{op}\to FinSet valued in the category of finite sets. Categories of finite presheaves will hardly be Grothendieck toposes for want of infinite limits but they still can turn out to be elementary toposes as e.g. in the case of FinSetFinSet itself.

By going through the proof that ordinary categories of presheaves are toposes, one observes that the constructions stay within finite presheaves when applied to a finite category CC i.e. one with only a finite set of morphisms. Hence, one has the following


Let CC a finite category. Then the category of finite presheaves [C op,FinSet][C^{op},FinSet] is a topos. \qed

Note, that the category [G,FinSet][G,FinSet] of finite GG-sets is topos even when the group GG is infinite! In this case it is crucial that Ω={,G}\Omega =\{\emptyset , G\} in [G,Set][G,Set] is a finite set.

(Cf. Borceux (1994, p.299))

Models in presheaf toposes

See at models in presheaf toposes.

For (∞,1)-category theory see (∞,1)-category of (∞,1)-presheaves.

Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.

(n,r)-categoriestoposeslocally presentableloc finitely preslocalization theoremfree cocompletionaccessible
(0,1)-category theorylocalessuplatticealgebraic latticesPorst’s theorempowersetposet
category theorytoposeslocally presentable categorieslocally finitely presentable categoriesAdámek-Rosický’s theorempresheaf categoryaccessible categories
model category theorymodel toposescombinatorial model categoriesDugger’s theoremglobal model structures on simplicial presheavesn/a
(∞,1)-topos theory(∞,1)-toposeslocally presentable (∞,1)-categoriesSimpson’s theorem(∞,1)-presheaf (∞,1)-categoriesaccessible (∞,1)-categories


A classical (advanced) reference is exposé 1 of

An elementary introduction to presheaf toposes emphasizing finite underlying categories CC is

  • M. La Palme Reyes, G. E. Reyes, H. Zolfaghari, Generic Figures and their Glueings , Polimetrica Milano 2004.

Standard references are

Revised on April 8, 2016 06:10:25 by Thomas Holder (