More generally, given any category , an -valued presheaf on is a functor
Historically, the initial applications of presheaves and sheaves involved cases like (the category of commutative rings), Ab, -, etc. Later, especially with the development of topos theory, the primary importance of the category of set-valued (pre)sheaves as topos was recognized; these other cases could be considered algebraic objects which live in the topos. This article and the one on sheaf topos recognize these later developments by making the set-valued case the default (in other words, presheaf or sheaf without further qualification is understood to refer to the set-valued case).
The category of presheaves on , usually denoted or , but often abbreviated as , has:
functors as objects;
natural transformations between such functors as morphisms.
As such, it is an example of a functor category.
Speaking of functors as presheaves indicates operations that one wants to do apply to these functors, or certain properties that one wants to check.
One generally useful way to think of presheaves is in the sense of space and quantity.
In the case where and is small, an important general principle is that the presheaf category is the free cocompletion of ; see Yoneda extension. Intuitively, it is formed by taking and ‘freely throwing in small colimits’. The category is contained in via the Yoneda embedding
The Yoneda embedding sends each object to the presheaf
Presheaves of this form, or isomorphic to those of this form, are called representable; among their properties, representable presheaves always turn colimits into limits, in the sense that a representable functor from to turns colimits in (i.e., limits in ) into limits in (i.e., colimits in ). In general, such continuity is a necessary but not sufficient criterion for representability; however, nicely enough, it is sufficient when itself is a presheaf category. To see this, suppose is such a presheaf on , and let , a presheaf on . By the Yoneda lemma, we have a natural isomorphism between and . But by the free cocompletion property of the Yoneda embedding, a colimit-preserving functor on presheaves is entirely determined by its precomposition with ; accordingly, our isomorphism must extend to an identification of with , thus establishing the representability of .
Any category of presheaves is complete and cocomplete, with both limits and colimits being computed pointwise. That is, to compute the limit or colimit of a diagram , we think of it as a functor and take the limit or colimit in the variable.
An elegant way to express this colimit for a presheaf is in terms of the coend identity
and let the canonical forgetful functor. Then the colimit over representables expression is
This is often written with some convenient abuse of notation as
Notice that these formulas can also be understood as those for the left Kan extension (see there) of along the identity functor.
Notice that for every and using the property of the hom-functor we have
by the Yoneda lemma.
By the definition of limit we have that
so for each natural transformation and each object , is a map , that is, it is an element of . However, by Yoneda, we know that each object specifies a unique element . Then rephrasing this, specifies a function . The naturality of this assignment is guaranteed by the naturality of the map . Then induces a natural transformation . It’s easy to check that defines an isomorphism:
Since this holds for all , the claim follows, again using the Yoneda lemma.
Examples for presheaves are abundant. Here is a non-representative selection of some examples.
More generally, for a subcategory of a locally small category , every object gives rise to the presheaf
Let’s spell this out in more detail: given a mophism in , we can take any morphism in and turn it into a morphism in . This determines a map of set
So we have a functorial assignment of the form
Traditional standard examples include: the presheaf of smooth functions on , that assigns to each the set of smooth functions and to each inclusion the corresponding restriction operation of functions. This is further a sheaf.
Traditional standard example which is a presheaf but not a sheaf: the presheaf of exact forms on , that assigns to the set of exact forms on and to each inclusion the corresponding restriction operation of functions. Here, and like above, the site is made up by open sets in with inclusions as morphisms.
… etc. pp.