An object in this category is a presheaf. See there for more details.
The category of presheaves is the free cocompletion of .
A category of presheaves is a topos.
The construction of forming (co)-presheaves extends to a 2-functor
This is due to Marta Bunge.
For details on the closed structure see
Let be a category, an object of and let be the over category of over . Write for the category of presheaves on and write for the over category of presheaves on over the presheaf , where is the Yoneda embedding.
There is an equivalence of categories
The functor takes to the presheaf which is equipped with the natural transformation with component map .
A weak inverse of is given by the functor
which sends to given by
where is the pullback
Suppose the presheaf does not actually depend on the morphsims to , i.e. suppose that it factors through the forgetful functor from the over category to :
Then and hence 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.
|(n,r)-categories…||satisfying Giraud's axioms||inclusion of left exact 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||algebraic lattices||Porst’s theorem||subobject lattices in accessible reflective subcategories of presheaf categories|
|category theory||toposes||locally presentable categories||Adámek-Rosický’s theorem||accessible reflective subcategories of presheaf categories||accessible categories|
|model category theory||model toposes||combinatorial model categories||Dugger’s theorem||left Bousfield localization of global model structures on simplicial presheaves|
|(∞,1)-topos theory||(∞,1)-toposes||locally presentable (∞,1)-categories|| |
|accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories||accessible (∞,1)-categories|