In category theory and general topology, a topological functor is one which has similar properties to the forgetful functor from the category of topological spaces. The domain of a topological functor admits construction similar to initial topology (and equivalently the final topology) of a family of functions. The notion of topological functors generalizes (and strengthens) that of fibered categories, for which one considers a single morphism instead of a family.
A topological functor is considered in the field of categorical topology.
Definition Source and sink A source in a category consists of the following data: an object , a (possibly proper) class of objects and a class of morphisms .
Dually, a sink in consists of an object , a class of objects and a class of morphisms .
In particular, a source is an object if is empty, a morphism if is a set of a single element. Similarly for a sink.
Initial source and final sink Let be a source in a category and let be a functor. The source is said to be a -initial source if it satisfies the following universal property. For every object , a morphism and a family of morphisms such that for each , there exists a unique -morphism such that and .
Similarly one defines the dual notion of -final sink.
When is a set of a single element, the initial source is called a Cartesian morphism.
Lift Let , be two categories. Let be a functor. A source in is a -structured source if for each we have for some . One similarly defines a -structured sink.
A lift of a -structured source is a source in such that and for each
A lift of a -structured sink is similarly defined. Since initial and final lifts are defined via universal properties, they are unique up to a unique isomorphism, if they exist.
If a -structured source has an initial lift , we say that is an initial -structure on with respect to . Similarly for a final -structure with respect to a -structured sink.
Topological functor Let be a functor. Then the following two conditions are equivalent. Every -structured source has an initial lift. That is, an initial structure always exists. Every -structured sink has a final lift. That is, a final structure always exists.
A functor satisfying this condition is called a topological functor.
One can define topological functors in a different way, using the theory of enriched categories.
A concrete category is called a topological (concrete) category if the forgetful functor is topological. (A topological category can also mean an enriched category enriced over the category of topological spaces.) Some require a topological category to satisfy two additional conditions. Constant functions in lift to -morphisms. Fibers () are small (they are sets and not proper classes).
Properties Every topological functor is faithful.
Let be one of the following four properties of categories: complete category cocomplete category well-powered category co-well-powered category If is topological and has property , then also has property .
Let be a category. Then the topological functors are unique up to natural isomorphism.
Examples An example of a topological category is the category of all topological spaces with continuous maps, where one uses the standard forgetful functor.
References