The term ‘topological category’ is traditional, and comes from the frequent examples in topology. It does not mean an internal category or enriched category in Top! (Fortunately the term topological groupoid is not taken by this tradition; indeed, the only groupoid that is a topological category over is trivial. On the other hand, they do seem to use the term ‘topological functor’, which here we avoid.)
Most generally, the definition relates to a functor (such as the forgetful functor from to Set), but one can think of this as giving as a bundle over . Sometimes, when is in fact Set, the category satisfying the properties described belows is called a topological construct (Preuss). Usually and will be large categories. Let a space be an object of , an algebra be an object of , a map be a morphism in , and a homomorphism be a morphism in . (The reason is that, typically, will be a category of spaces with some kind of topological structure while will be, if not , then some kind of algebraic category.)
Then is a topological category over if, given any algebra and any (possibly large) family of spaces and homomorphisms (that is, a ”-structured” source from ), there exists an initial lift, which is to say
Here are some illustrative commutative diagrams (if you can read them):
It follows by a clever argument that must be faithful; see Theorem 21.3 of ACC. That is also often included in the definition, in which case the uniqueness of can be left out. Thus we may think of objects of as objects of equipped with extra structure. The idea is then that is equipped with the initial structure or weak structure determined by the requirement that the homomorphisms be structure-preserving maps.
The dual concept could be called a cotopological category. However, this is not actually anything new; is topological if and only if is. This is a categorification of the theorem that any complete semilattice is a complete lattice. Thus, every topological category also has final (not usually called terminal) or strong structures, each determined by a family of homomorphisms (a -structured sink to ).
Both of these results (faithfulness and self-duality) depend on the fact that we have allowed the family to be potentially large. Counterexamples are easy to find. For instance, if is a large category with all (small) products, then the functor to the terminal category satisfies the above lifting property for small families . However, it need not satisfy the dual property (unless also has all small coproducts) nor need it be faithful.
It also follows that is a fibration and opfibration, in the weakened bicategorical sense of Street. One also often assumes in the definition and that is the identity morphism, which in particular makes into a fibration in the original sense of Grothendieck. This is a bit evil, but it is convenient and satisfied in almost all examples, and any example not satisfying it is equivalent to one which does (via fibrant replacement by an isofibration).
In contrast, the category of locales is not topological over , apparently not even the category of spatial locales (equivalent to the category of sober spaces), essentially because soberification of a topological space may not preserve the underlying set.
Outside of topology, the category of measurable spaces is topological over .
If is topological over , then so is any full retract of , as long as the functors involved live in .
If is mono-complete or epi-cocomplete, then so is .
If is well-powered or co-well-powered, then so is .
If has a factorization structure for sinks , then has one , where is the collection of morphisms in lying over -morphisms in , and the collection of final sinks in lying over -sinks in . This generalizes the lifting of orthogonal factorization systems along Grothendieck fibrations.
In particular, if is Set, then is a concrete category that is complete, cocomplete, well powered, and well copowered.
If is any algebra, then there is a discrete space over induced by the empty family of maps. Similarly, we have an indiscrete space with the final structure induced by no maps. This defines functors that are respectively left and right adjoints of .
More generally, suppose that has products or coproducts (indexed by whichever cardinalities you may wish to consider). Then also has (co)products, lying over the (co)products in , with structures induced by the product projections or coproduct inclusions.
More general limits and colimits are constructed in a similar way. However, it is not typically the case that creates (co)limits in because creation of a limit requires that every preimage of the limiting cone is limiting. This fails for since we can coarsen the topology on the limit vertex to obtain a counterexample.
If a single algebra has been given the structure of several spaces, then there are a supremum structure and an infimum structure on induced (as the initial and final structures) by the various incarnations of its identity homomorphism. Exploiting this shows how to construct final structures out of initial ones and conversely.
If is a regular subalgebra of some , then the inclusion homomorphism makes into a subspace of , which is also a subobject in . Every regular subobject of is of this form; note however that there may be nonregular subobjects in even if all subobjects in are regular.