nLab
category of presheaves

Definition

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

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

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

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

• A category of presheaves is a topos.

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

  $$[-,\mathrm{Set}]:\mathrm{Cat}\to \mathrm{Topos}$$[-,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).

Characterization

Theorem

A category $E$ 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

• closed monoidal structure on presheaves.

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 $\mathrm{PSh}(C/c)=[(C/c{)}^{\mathrm{op}},\mathrm{Set}]$ for the category of presheaves on $C/c$ and write $\mathrm{PSh}(C)/Y(y)$ for the over category of presheaves on $C$ over the presheaf $Y(c)$, where $Y:C\to \mathrm{PSh}(c)$ is the Yoneda embedding.

Proposition

There is an equivalence of categories

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

Proof

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

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

$$\overline{e}:\mathrm{PSh}(C)/Y(c)\to \mathrm{PSh}(C/c)$$\bar e : PSh(C)/Y(c) \to PSh(C/c)

which sends $\eta :F\prime \to Y(C))$ to $F\in \mathrm{PSh}(C/c)$ given by

$$F:(f:d\to c)\mapsto F\prime (d){\mid }_c,$$F : (f : d \to c) \mapsto F'(d)|_c \,,

where $F\prime (d){\mid }_c$ is the pullback

$$\begin{array}{ccc}F\prime (d){\mid }_c& \to & F\prime (d)\\ \downarrow & & {\downarrow }^{{\eta }_d}\\ \mathrm{pt}& \stackrel{f}{\to }& C(d,c)\end{array}.$$\array{ F'(d)|_c &\to& F'(d) \\ \downarrow && \downarrow^{\eta_d} \\ pt &\stackrel{f}{\to}& C(d,c) } \,.

Example

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

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

Then $F\prime (d)={\bigsqcup }_{f\in C(d,c)}F(f)={\bigsqcup }_{f\in C(d,c)}F(d)\simeq C(d,c)\times F(d)$ and hence $F\prime =Y(c)\times F$ with respect to the closed monoidal structure on presheaves.

See also functors and comma categories.

For the analog statement in (∞,1)-category theory see

Related concepts

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

Locally presentable categories: Large categories whose objects arise from small generators under small relations.

(n,r)-categories…	satisfying Giraud's axioms	inclusion of left exaxt localizations	generated under colimits from small objects		localization of free cocompletion		generated under filtered colimits from small objects
(0,1)-category theory	(0,1)-toposes	$\hookrightarrow$	algebraic lattices	$\simeq$ Porst’s theorem	subobject lattices in accessible reflective subcategories of presheaf categories
category theory	toposes	$\hookrightarrow$	locally presentable categories	$\simeq$ Adámek-Rosický’s theorem	accessible reflective subcategories of presheaf categories	$\hookrightarrow$	accessible categories
model category theory	model toposes	$\hookrightarrow$	combinatorial model categories	$\simeq$ Dugger’s theorem	left Bousfield localization of global model structures on simplicial presheaves
(∞,1)-topos theory	(∞,1)-toposes	$\hookrightarrow$	locally presentable (∞,1)-categories	$\simeq$ Simpson’s theorem	accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories	$\hookrightarrow$	accessible (∞,1)-categories