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.
Definition.
Source and sink.
A source formula_1 in a category formula_2 consists of the following data:
Dually, a sink formula_6 in formula_2 consists of
In particular, a source formula_5 is an object formula_12 if formula_13 is empty, a morphism formula_14 if formula_13 is a set of a single element. Similarly for a sink.
Initial source and final sink.
Let formula_5 be a source in a category formula_2 and let formula_18 be a functor. The source formula_19 is said to be a formula_20-initial source if it satisfies the following universal property.
Similarly one defines the dual notion of formula_20-final sink.
When formula_13 is a set of a single element, the initial source is called a Cartesian morphism.
Lift.
Let formula_2, formula_34 be two categories. Let formula_18 be a functor. A source formula_36 in formula_34 is a formula_20-structured source if for each formula_39 we have formula_40 for some formula_41. One similarly defines a formula_20-structured sink.
A lift of a formula_20-structured source formula_44 is a source formula_45 in formula_2 such that formula_47 and formula_48 for each formula_25
formula_50
A lift of a formula_20-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 formula_20-structured source formula_53 has an initial lift formula_54, we say that formula_12 is an initial formula_2-structure on formula_57 with respect to formula_53. Similarly for a final formula_2-structure with respect to a formula_20-structured sink.
Topological functor.
Let formula_18 be a functor. Then the following two conditions are equivalent.
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 formula_64 is called a topological (concrete) category if the forgetful functor formula_65 is topological. (A "topological category" can also mean an enriched category enriced over the category formula_66 of topological spaces.) Some require a topological category to satisfy two additional conditions.
Properties.
Every topological functor is faithful.
Let formula_71 be one of the following four properties of categories:
If formula_18 is topological and formula_34 has property formula_71, then formula_2 also has property formula_71.
Let formula_2 be a category. Then the topological functors formula_78 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.