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.
The category of presheaves $PSh(C)$ is the free cocompletion of $C$.
the Yoneda lemma says that the Yoneda embedding $j : C \to PSh(C)$ 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
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).
See functoriality of categories of presheaves.
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.
As every topos, a category of presheaves is a cartesian closed monoidal category.
For details on the closed structure see
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 $PSh(C)/Y(y)$ 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 morphsims 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
See at models in presheaf toposes.
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 |