Given a category of “test objects”, one can consider “things” which can detect. If one already has a class of objects in mind, one can consider -structures on those objects by looking at morphisms to and from objects of . Then one considers questions only in so far as they can be distinguished after mapping into . A simple example is that of smooth manifolds where many questions are solved by transporting the problem to Euclidean spaces via charts. Another example is the use of weak topologies on locally convex topological spaces. Without a class of objects in mind, the natural definition of an object probeable by -objects involves presheaves and copresheaves and is as follows.
Let be an essentially small category. Then the Isbell envelope of , written , is defined as follows. An object is a triple where
(We conventionally write .) A morphism is a pair of natural transformations with which satisfy the relation .
The requirement that be essentially small implies that the collections of natural transformations form sets, and thus that this is a locally small category.
Certain elementary properties are easy to prove.
There is a (“double/twosided Yoneda”) embedding of as a full subcategory of via
Identifying with its image, there are natural isomorphisms
Other elementary properties to follow
The Isbell envelope of a category can be viewed as a category of profunctors. In short, the Isbell envelope of consists of the lax factorisations of through .
Let us spell this out. Recall that a profunctor is a functor . Both covariant and contravariant functors to are special examples of profunctors: a covariant functor is a profunctor , where is the terminal category, and a contravariant functor is a profunctor . The composition of these as profunctors produces the obvious profunctor . Thus extracting the functor part of the definition of an object in the Isbell envelope of produces a profunctor which factors through .
There is an obvious profunctor given by the -bifunctor. This is the identity for profunctor composition. The natural transformation from the definition of an object in the Isbell envelope of defines a morphism of profunctors from the corresponding profunctor to . Thus an object in the Isbell envelope of corresponds to an object in the subcategory of the slice category of profunctors over of those objects which factor through .
In other words, a lax factorisation of through .
A variant of the above involves a background category, say . The test objects should be viewable also as objects of , usually via a faithful functor. Any object of defines a profunctor (which factors through ) via
Then one can consider those objects of the Isbell envelope of that are sub-profunctors of one of this type. In addition one should restrict the morphisms to those that are induced by morphism of objects of . Translating this back to the language of functors yields the following definition.
Let and be categories with a faithful functor which we shall write as . The -concrete Isbell envelope of , which we shall write , is the category whose objects are triples where
such that the image of the natural transformation
which comes from composition in , lies in the image of the natural transformation
A morphism in is a -morphism on the underlying -objects that defines natural transformations and .
Having a background category ensures that the size issues with natural transformations do not occur and so we can drop the requirement that be essentially small.
Within the Isbell envelope of one can consider various subcategories where the objects satisfy extra conditions. An obvious condition is that the presheaf is actually a sheaf. Another useful condition is that of Isbell duality.
An object of is said to be -saturated if the obvious natural transformations
are isomorphisms. It is said to be -saturated if the obvious natural transformations
are isomorphisms. It is said to satisfy Isbell duality if it is both - and -saturated.
Within one can consider the full subcategories of -saturated objects, of -saturated objects, and those satisfying Isbell duality. Clearly, the last is the intersection of the first two. There are idempotent functors onto the first two categories given by replacing one of or by the natural transformations of the other. An interesting question is to ask whether or not the obvious iteration stabilises after a finite number of steps (which would result in an object satisfying Isbell duality).
If the test category has, or “morally has”, a representable functor to then there is a strong relationship between saturation and concreteness.
A constant separator in a category is an object, say , with the property that if then there is a constant morphism such that .
As the constant morphisms form a two-sided ideal, any object in a category defines a covariant functor which on objects sends to the set of constant morphisms from to . A morphism defines a natural transformation (in the opposite direction) between the corresponding functors. From the properties of constant morphisms one can easily deduce that two different morphisms between the same objects induce the same natural transformations between the functors. Thus if there are morphisms between two objects in both directions the two functors are naturally isomorphic.
The property of being a constant separator is clearly equivalent to the condition that this constant functor be faithful. Moreover, it is easy to show that in a non-trivial category, any two constant separators have morphisms between them in both directions and so induce naturally isomorphic functors. Indeed, not just naturally isomorphic but naturally naturally isomorphic in that there is a canonical choice of natural isomorphism.
Now we transfer this to the Isbell envelope of .
Let be an object of . An element , for an object of , is said to be constant if is constant for all objects of and .
Let us write for the set of constant elements in .
Let be an object in . The assignment is functorial and extends the assignment . A –morphism defines a natural transformation of functors.
Let , , be two objects in . Let be a morphism from the first to the second. Then is a natural transformation so in particular defines a morphism of sets . Let be a constant element. Let be an object in and . We need to show that is a constant morphism. By the definition of a morphism in , and this latter is a constant morphism since is a constant element.
For the extension, we observe that for then if is a constant morphism, is constant for all ; whilst if is constant for all then in particular is constant. That the morphisms correspond is obvious.
A –morphism defines a map as is a contravariant functor. Let be a constant element. Then for an object in and , and is a constant morphism in so is a constant morphism in . Hence is a constant element in .
The result that the natural transformations depend only on the existence of a morphism does not carry over to this extended setting.
Andrew: Useful to have an example here, of course.
Let us fix a –object . We have two bifunctors given by
Let us, to simplify notation, identify with its image in . Then is naturally isomorphic to so the fact that is a functor defines a natural transformation from the first to the second via
In a similar fashion, we obtain a natural transformation
We therefore obtain a functorial choice of underlying concrete object (with the underlying category being ). We can refine the notion of concreteness slightly.
Let be an essentially small category with a concrete separator, say . Let be an object of . We say that is –concrete if the map of sets
is injective for all –objects, .
Similarly, we say that is –concrete if the map of sets
is injective for all –objects, .
Clearly, if it is both -concrete and -concrete then it is concrete. Changing the concrete separator does not alter concreteness.
Let be an essentially small category admitting a constant separator. Then for an object of , the following hold:
Clearly the first and second statements imply the third.
Let us consider the first.
Let be an object of such that for all –objects . Let be a concrete separator in . We shall write for . Let be an object of . Let . As , their inequality means that they differ as natural transformations. Thus there is some for which and induce different morphisms . Thus there is some such that .
As is a constant separator, there is thus a constant morphism such that . Using composition notation, we rewrite this as . As is a constant morphism, and are constant elements of . Since , they must be different constant elements. Thus the maps are different and so is –concrete.
The second statement is very similar.
Let be an object of such that for all –objects . Let be a concrete separator in . Let be an object of . Let . As , their inequality means that they differ as natural transformations. Thus there is some for which and induce different morphisms . Thus there is some such that .
As is a constant separator, there is thus a constant morphism such that . Using the composition notation, we rewrite this as . As is a constant morphism, is a constant element. We therefore have an element of which distinguishes between the induced maps from and . Hence is –concrete.