nLab category of presheaves

Definition

For $C$ a small category, its category of presheaves is the functor category

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

from the opposite category of $C$ to Set.

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

Properties

General

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

Proposition

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

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

Proof

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

Proposition

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

Proof

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 $\mathcal{C}$ is a small category.

Proposition

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

Proof

This is spelled out at closed monoidal structure on presheaves.

Proposition

$PSh(\mathcal{C})$ is a topos.

Proof

This is the base case of sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes.

• The category of presheaves $PSh(\mathcal{C})$ is the free cocompletion of $\mathcal{C}$.

• the Yoneda lemma says that the Yoneda embedding $j \colon \mathcal{C} \to PSh(\mathcal{C})$ is – in particular – a full and faithful functor.

• The construction of forming (co)-presheaves extends to a 2-functor

  $$[-,Set] : Cat \to Topos$$

  from the 2-category Cat to the 2-category Topos. (See at geometric morphism the section Between presheaf toposes for details.)

• A reflective subcategory of a category of presheaves is a locally presentable category if it is closed under $\kappa$-directed colimits for some regular cardinal $\kappa$ (the embedding is an accessible functor).

• A sub-topos of a category of presheaves is a Grothendieck topos: a category of sheaves (see there for details).

(Co-)Monadicity over Families

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

Functoriality

See functoriality of categories of presheaves.

Characterization

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

Theorem

A category $E$ 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):

Theorem

A category $E$ 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 $C$ be a category, $c$ an object of $C$ and let $C/c$ be the over category of $C$ over $c$. Write $PSh(C/c) = [(C/c)^{op}, Set]$ for the category of presheaves on $C/c$ and write $PSh(C)/Y(c)$ for the over category of presheaves on $C$ over the presheaf $Y(c)$, where $Y : C \to PSh(C)$ is the Yoneda embedding.

Proposition

There is an equivalence of categories

$$e : PSh(C/c) \stackrel{\simeq}{\to} PSh(C)/Y(c) \,.$$

Proof

The functor $e$ takes $F \in PSh(C/c)$ to the presheaf $F' : d \mapsto \sqcup_{f \in C(d,c)} F(f)$ which is equipped with the natural transformation $\eta : F' \to Y(c)$ with component map $\eta_d \sqcup_{f \in C(d,c)} F(f) \to C(d,c)$.

A weak inverse of $e$ is given by the functor

$$\bar e : PSh(C)/Y(c) \to PSh(C/c)$$

which sends $\eta : F' \to Y(C))$ to $F \in PSh(C/c)$ given by

$$F : (f : d \to c) \mapsto F'(d)|_c \,,$$

where $F'(d)|_c$ is the pullback

$$\array{ F'(d)|_c &\to& F'(d) \\ \downarrow && \downarrow^{\eta_d} \\ pt &\stackrel{f}{\to}& C(d,c) } \,.$$

Example

Suppose the presheaf $F \in PSh(C/c)$ does not actually depend on the morphisms to $C$, i.e. suppose that it factors through the forgetful functor from the over category to $C$:

$$F : (C/c)^{op} \to C^{op} \to Set \,.$$

Then $F'(d) = \sqcup_{f \in C(d,c)} F(f) = \sqcup_{f \in C(d,c)} F(d) \simeq C(d,c) \times F(d)$ and hence $F ' = Y(c) \times F$ 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

Remark

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

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

Proposition

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

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

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

$$i(F)(A \in C) = \{ (p,a) | p \in P(A), a \in F(A,p) \} = \Sigma_{p \in P(A)} F(A,p)$$

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

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

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:

Theorem

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

Proof

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

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

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

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

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 $Fam(Set^{C^{op}})$ is itself a presheaf topos: there are equivalences

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

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

Finite presheaves

A finite presheaf on a category $C$ is a functor $C^{op}\to FinSet$ 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 $FinSet$ 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 $C$ i.e. one with only a finite set of morphisms. Hence, one has the following

Proposition

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

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

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

Models in presheaf toposes

See at models in presheaf toposes.

Related concepts

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.

$\phantom{A}$(n,r)-categories$\phantom{A}$	$\phantom{A}$toposes$\phantom{A}$	locally presentable	loc finitely pres	localization theorem	free cocompletion	accessible
(0,1)-category theory	locales	suplattice	algebraic lattices	Porst’s theorem	powerset	poset
category theory	toposes	locally presentable categories	locally finitely presentable categories	Gabriel–Ulmer’s theorem	presheaf category	accessible categories
model category theory	model toposes	combinatorial model categories		Dugger's theorem	global model structures on simplicial presheaves	n/a
(∞,1)-category theory	(∞,1)-toposes	locally presentable (∞,1)-categories		Simpson’s theorem	(∞,1)-presheaf (∞,1)-categories	accessible (∞,1)-categories

References

Original discussion:

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

  Reprints in Theory and Applications of Categories, 30 (2024) 1-84 [tac:tr30]

• M.Artin, A.Grothendieck, J. L. Verdier (eds.), Théorie des Topos et Cohomologie Etale des Schémas - SGA 4, LNM 269 Springer (1972)

Basic exposition:

• Michael Barr, Colin McLarty, Charles Wells, Variable set theory, prepared for Scientific American but unpublished (~1985) [pdf, pdf]

Introduction:

• Marie La Palme Reyes, Gonzalo E. Reyes, Houman Zolfaghari, Generic figures and their glueings. A constructive approach to functor categories, Polimetrica (2008) [pdf, ISBN:8876990046]

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

• Francis Borceux, Handbook of Categorical Algebra 3 : Categories of Sheaves, Cambridge UP 1994.

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves , Springer Heidelberg 2006.

• Saunders Mac Lane, Ieke Moerdijk, Sheaves in Geometry and Logic, Springer Heidelberg 1994.

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,

• Tom Leinster, Higher Categories, Higher Operads [arXiv:math/0305049]