nLab 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.



For 𝒞\mathcal{C} any category, consider PSh(𝒞)PSh(\mathcal{C}) its category of Set-valued presheaves.


PSh(𝒞)PSh(\mathcal{C}) has all limits and colimits (over small diagrams), and these are computed objectwise: For \mathcal{I} any diagram (a small category) we have:

X :PSh(𝒞)cObj(𝒞)(limiX i)(c)limi(X i(c))and(limiX i)(c)limi(X i(c)) X_{\bullet} \,\colon\, \mathcal{I} \longrightarrow PSh(\mathcal{C}) \;\;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\;\;\; \underset{c \in Obj(\mathcal{C})}{\forall} \;\;\;\; \big( \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{lim} X_i \big) (c) \;\simeq\; \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{lim} \big( X_i(c) \big) \;\;\;\;\text{and}\;\;\;\; \big( \underset{\underset{i \in \mathcal{I}}{\longleftarrow}}{lim} X_i \big) (c) \;\simeq\; \underset{\underset{i \in \mathcal{I}}{\longleftarrow}}{lim} \big( X_i(c) \big)


The defining universal property readily follows from that of the objectwise (co-)limits.


A morphism XfYX \xrightarrow{\;f\;} Y of presheaves X,YPSh(𝒞)X,Y \,\in\, PSh(\mathcal{C}) is an epimorphism or monomorphism precisely if it is so over each object of 𝒞\mathcal{C}, hence precisely if it is object-wise a surjection or injection, respectively.


Using for instance the characterization of epimorphisms by pushouts (this Prop.) and of monomorphism by pullbacks (this Prop.), the statement follows by Prop. .

Now assume that 𝒞\mathcal{C} is a small category.


PSh(𝒞)PSh(\mathcal{C}) is a cartesian closed category.


This is spelled out at closed monoidal structure on presheaves.


PSh(𝒞)PSh(\mathcal{C}) is a topos.

(Co-)Monadicity over Families

There is a functor from presheaves to families of sets U:Psh(C)Set Ob(C)Set/Ob(C)U : Psh(C) \to Set^{Ob(C)} \cong Set/Ob(C) given by “forgetting” the functorial action on morphisms of CC. This functor is both monadic and comonadic.


See functoriality of categories of presheaves.


The following Giraud like theorem stems from Marta Bunge's dissertation (1966)


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

This characterisation is proven for enriched presheaf categories in Theorem 4.16 of Bunge 1969 (Corollary 4.19 for the unenriched statement).

A second characterization using exact completions can be found in Carboni-Vitale (1998) or Centazzo-Vitale (2004):


A category EE is equivalent to a presheaf topos if and only if it is locally small, extensive, exact and has a small set of projective and indecomposable generators.

(1998) has also an interesting comparison to a classical characterization of categories monadic over Set.

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 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 \,.

On objects this takes F:( CP) opSetF : (\int_C P)^{op} \to Set to

i(F)(AC)={(p,a)|pP(A),aF(A,p)}=Σ pP(A)F(A,p)i(F)(A \in C) = \{ (p,a) | p \in P(A), a \in F(A,p) \} = \Sigma_{p \in P(A)} F(A,p)

with obvious projection to PP. The inverse takes f:QPf : Q \to P to

i 1(f)(A,pP(A))=f A 1(p).i^{-1}(f)(A, p \in P(A)) = f_A^{-1}(p)\;.

For a proof see Kashiwara-Schapira (2006, Lemma 1.4.12, 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.

Artin gluings

The statements of the preceding subsection may be generalized further:


Let T:Set C opSet D opT: Set^{C^{op}} \to Set^{D^{op}} be a functor that preserves wide pullbacks. Then the Artin gluing (Set D opT)(Set^{D^{op}} \downarrow T) is also a presheaf topos.


The functor TT is familially representable: there is a functor D opFam(Set C op)D^{op} \to Fam(Set^{C^{op}}) into the small coproduct cocompletion of Set C opSet^{C^{op}}, taking dOb(D)d \in Ob(D) to a formal coproduct xT(1)(d)W (d,x)\sum_{x \in T(1)(d)} W_{(d, x)} of presheaves W (d,x):C opSetW_{(d, x)}: C^{op} \to Set, such that T(F)T(F) is given by the formula

T(F)(d)= xT(1)(d)Set C op(W (d,x),F).T(F)(d) = \sum_{x \in T(1)(d)} Set^{C^{op}}(W_{(d, x)}, F).

The Artin gluing itself is then describable as the collage of the profunctor W:C(y DT(1))W: C \nrightarrow (y_D \downarrow T(1)) defined by the formula

W(c;(d,x:D(,d)T(1)))=W (d,x)(c).W(c; (d, x: D(-, d) \to T(1))) = W_{(d, x)}(c).

For details, see for example Appendix C.3 of Leinster. The result is due to Carboni and Johnstone.

In passing, we note that the small coproduct cocompletion Fam(Set C op)Fam(Set^{C^{op}}) is itself a presheaf topos: there are equivalences

Fam(Set C op)Set C opΔSet C + opFam(Set^{C^{op}}) \simeq Set^{C^{op}} \downarrow \Delta \simeq Set^{C_+^{op}}

where Δ:SetSet C op\Delta: Set \to Set^{C^{op}} is the diagonal functor, and C +C_+ is the result of freely adjoining an initial object to CC, i.e., the ordinal sum of categories 1+ σC1 +_\sigma C of categories (11 being terminal), aka the cone of CC.

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 a 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 reflective localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.

A\phantom{A}(n,r)-categoriesA\phantom{A}A\phantom{A}toposesA\phantom{A}locally 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)-category theory(∞,1)-toposeslocally presentable (∞,1)-categoriesSimpson’s theorem(∞,1)-presheaf (∞,1)-categoriesaccessible (∞,1)-categories


Original discussion:

  • Marta Bunge, Categories of Set-Valued Functors, PhD thesis, University of Pennsylvania (1966) [pdf]

Basic exposition:


See also most accounts of general topos theory, such as:

The characterizations of categories of presheaves are discussed in

  • Marta Bunge, Relative functor categories and categories of algebras, J. of Algebra 11, Issue 1 (1969), 64–101. web, MR236238 doi

  • A. Carboni, E. M. Vitale, Regular and exact completions, JPAA 125 (1998), 79-116.

  • C. Centazzo, E. M. Vitale, Sheaf theory, pp. 311-358 in Pedicchio, Tholen (eds.), Categorical Foundations, Cambridge UP 2004. (draft)

The result about Artin gluings of presheaf toposes is due to Carboni and Johnstone,

  • Aurelio Carboni and Peter Johnstone, Connected limits, familial representability and Artin glueing, Mathematical Structures in Computer Science, Volume 5, Issue 4 (December 1995), 441-459. (link)

and is explained in section C.3 of Tom Leinster’s book,

Last revised on January 30, 2024 at 21:46:09. See the history of this page for a list of all contributions to it.