For $C$ a small category, its category of presheaves is the functor category
from the opposite category of $C$ to Set.
An object in this category is a presheaf. See there for more details.
For $\mathcal{C}$ any category, consider $PSh(\mathcal{C})$ its category of Set-valued presheaves.
$PSh(\mathcal{C})$ has all limits and colimits (over small diagrams), and these are computed objectwise: For $\mathcal{I}$ any diagram (a small category) we have:
The defining universal property readily follows from that of the objectwise (co-)limits.
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.
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.
$PSh(\mathcal{C})$ is a cartesian closed category.
This is spelled out at closed monoidal structure on presheaves.
$PSh(\mathcal{C})$ is a topos.
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
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).
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.
See functoriality of categories of presheaves.
The following Giraud like theorem stems from Marta Bunge's dissertation (1966)
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):
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.
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.
There is an equivalence of categories
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
which sends $\eta : F' \to Y(C))$ to $F \in PSh(C/c)$ given by
where $F'(d)|_c$ is the pullback
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 $C$:
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
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:
Let $P:C^{op}\to Set$ be a presheaf. Then there is an equivalence of categories
On objects this takes $F : (\int_C P)^{op} \to Set$ to
with obvious projection to $P$. The inverse takes $f : Q \to P$ to
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.
The statements of the preceding subsection may be generalized further:
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.
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
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
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
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 $C$.
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
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))
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.
Original discussion:
https://ncatlab.org/nlab/files/Bunge-SetValuedFunctors.pdf
Basic exposition:
Introduction:
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,
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.