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 morphisms 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
Consider , the category of elements of . This has objects with , hence is just an arrow in . A map from to is just a map such that but this is just a morphism from to in .
Hence, the above proposition 1 can be rephrased as which is an instance of the following formula:
Let be a presheaf. Then there is an equivalence of categories
In particular, this equivalence shows that slices of presheaf toposes are presheaf toposes.
A finite presheaf on a category is a functor 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 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 i.e. one with only a finite set of morphisms. Hence, one has the following
Let a finite category. Then the category of finite presheaves is a topos.
Note, that the category of finite -sets is topos even when the group is infinite! In this case it is crucial that in is a finite set.
(Cf. Borceux (1994, p.299))
See at models in presheaf toposes.
Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.
|(n,r)-categories||toposes||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||Adámek-Rosický’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)-topos theory||(∞,1)-toposes||locally presentable (∞,1)-categories||Simpson’s theorem||(∞,1)-presheaf (∞,1)-categories||accessible (∞,1)-categories|
A classical (advanced) reference is exposé 1 of
An elementary introduction to presheaf toposes emphasizing finite underlying categories is
Standard references are
Francis Borceux, Handbook of Categorical Algebra 3 : Categories of Sheaves , Cambridge UP 1994.
Masaki Kashiwara, Pierre Schapira, Categories and Sheaves , Springer Heidelberg 2006.