Category Of Topological Spaces

Alright, let's dissect this. You want me to take something that's supposed to be informative, something sterile and factual like a Wikipedia article, and inject it with… me. You want the cold, hard facts, but delivered with a certain… disdain. Fine. Let’s see what we can do with this "category of topological spaces." Frankly, the very idea of a "category" in this context feels… overly organized. Like someone trying to impose order on chaos. But I suppose that’s the point, isn’t it? To find patterns, to define relationships, even when the underlying elements are, well, spaces. And maps. How quaint.

Category whose objects are topological spaces and whose morphisms are continuous maps

In the realm of mathematics, specifically within the abstract framework of category theory, there exists a construct known as the category of topological spaces. It’s usually abbreviated as Top. Think of it as a meticulously curated, yet fundamentally bleak, collection. The objects within this category are, rather predictably, topological spaces. These are sets, of course, but not just any sets. They are sets endowed with a specific kind of structure, a topology, that dictates notions of closeness and continuity. The morphisms, the connections between these spaces, are what we call continuous maps. These aren’t just arbitrary functions; they are functions that preserve this delicate topological structure. They don’t tear the space apart, so to speak.

This arrangement forms a proper category for a few rather obvious reasons. Firstly, if you have two continuous maps, say f from space X to space Y, and g from space Y to space Z, their composition, g o f, which maps X to Z, is also guaranteed to be continuous. It’s like a chain reaction of unbroken structure. Secondly, for any topological space X, the identity function on X is, by definition, a continuous map. It does nothing, and therefore preserves everything. The study of this category, Top, and the properties of topological spaces through the lens of category theory, is what some rather dedicated individuals refer to as categorical topology. It’s an attempt to see the forest, not just the individual, rather depressing, trees.

Now, a minor point, though I doubt it will trouble most of you: some mathematicians, in their infinite capacity for nuance or perhaps just a penchant for pedantry, might use the designation "Top" to refer to slightly different categories. They might be talking about categories where the objects are topological manifolds, or perhaps compactly generated spaces with continuous maps as morphisms, or even the more specific category of compactly generated weak Hausdorff spaces. It’s all a matter of how much structure you want to burden yourself with, I suppose.

As a concrete category

The category Top is what we call a concrete category. This means its objects aren't just abstract notions; they are, at their core, sets that have been dressed up with some extra structure – in this case, topologies. And the morphisms? They are simply functions that have the good sense to respect that added structure. There's a rather natural forgetful functor that bridges Top and the much simpler category of sets. We can call this functor U. It takes any topological space and strips away its topology, leaving just the underlying set. Similarly, it takes a continuous map and reduces it to its basic function without regard for continuity. It’s a rather brutal process, but efficient.

This forgetful functor U, bless its heart, has companions. It’s accompanied by both a left adjoint and a right adjoint. The left adjoint, let’s call it D, acts on a plain old set and bestows upon it the discrete topology. Every point becomes an open set. It’s an explosion of openness, really. The right adjoint, which we can label I, does the opposite: it equips a set with the indiscrete topology. Only the empty set and the whole space are open. It’s a claustrophobic existence. Both D and I are, in a sense, inverses to U. Applying U after D, or U after I, brings you back to the original set. They are right inverses to U. Furthermore, because any function between discrete spaces is continuous, and any function between indiscrete spaces is also continuous, these functors D and I provide what are called full embeddings of the category of sets into Top. They essentially embed the simpler world of sets into the more complex world of topological spaces, fully and faithfully.

Top also exhibits a property called fiber-completeness. This means that if you fix a set, say X, the collection of all possible topologies you can put on X – the fiber of U above X – forms a complete lattice when ordered by set inclusion. The most open topology, the discrete one, sits at the very top, the greatest element. The most closed, the indiscrete one, is at the bottom, the least element. It’s a hierarchy of openness, or perhaps, a spectrum of control.

The category Top is considered a prime example of what's termed a topological category. These categories share a common characteristic: for any given source of structured objects (represented as a diagram from an index set I to the underlying set functor UA_i), there exists a unique "initial lift." In Top, this initial lift is achieved by bestowing the initial topology upon the underlying set. Topological categories, as a class, share many of Top's features, like fiber-completeness, the presence of discrete and indiscrete functors, and the ability to uniquely "lift" limits and colimits. It’s a family resemblance, I suppose, bound by shared structural properties.

Limits and colimits

The category Top is, thankfully, both complete and cocomplete. This simply means that all small limits and colimits exist within it. It’s not a barren wasteland; it has all the necessary constructions. The forgetful functor U, which we’ve already met, plays a rather crucial role here. It not only preserves limits and colimits from Top to the category of sets, but it also uniquely lifts them. This implies that to find a limit or colimit in Top, you first find the corresponding limit or colimit in Set, and then you equip it with the appropriate topology.

Specifically, if you have a diagram in Top, let's call it F, and you’ve found its limit in the category of sets, denoted (L, φ : L → UF), the corresponding limit in Top is obtained by placing the initial topology on L, using the structure provided by φ. Dually, for colimits, you take the set-theoretic colimit and equip it with the final topology. It’s a process of construction, building complexity from simplicity.

Unlike many categories built around algebraic structures, the forgetful functor U from Top to Set doesn't quite create or reflect limits. This is because there can be universal cones in Set that don't quite translate into universal cones in Top. There are extra, non-universal cones that can exist in Top. It’s a subtle distinction, but it matters.

Let’s look at some concrete examples of these limits and colimits in Top:

The empty set, when considered as a topological space, is the initial object of Top. It's the starting point, the void from which everything else can potentially emerge. Any singleton topological space, a set with just one point, serves as a terminal object. It’s the endpoint, the place where all paths can converge. Because of this, Top doesn't have zero objects. There's no object that is simultaneously initial and terminal.
The product in Top is formed by taking the Cartesian product of the underlying sets and then equipping it with the product topology. It’s a way of combining spaces. The coproduct, on the other hand, is the disjoint union of topological spaces. It’s like placing spaces side-by-side without letting them touch.
For a pair of continuous maps, the equalizer in Top is found by taking the set-theoretic equalizer and equipping it with the subspace topology. It's the part where the maps agree. Dually, the coequalizer is constructed by taking the set-theoretic coequalizer and applying the quotient topology. It's about identifying parts of the space.
Direct limits and inverse limits are also present. They correspond to the set-theoretic limits, but are endowed with the final topology and initial topology, respectively. It's about how sequences or systems of spaces relate to each other.
Adjunction spaces are a specific, and rather elegant, example of pushouts within the category Top. They represent a way of gluing spaces together along a common boundary.

Other properties

Let's delve into some of the finer points of Top:

The monomorphisms in Top are precisely the injective continuous maps. If a map is one-to-one and continuous, it's a monomorphism. The epimorphisms, conversely, are the surjective continuous maps. If a map is onto and continuous, it's an epimorphism. And the isomorphisms? Those are the homeomorphisms – the maps that are continuous, bijective, and have a continuous inverse. They are the structure-preserving bijections, the true equivalences between topological spaces.
The extremal monomorphisms in Top are, up to isomorphism, the embeddings of subspaces. When you take a subset and give it the inherited topology, that embedding is an extremal monomorphism. In fact, all extremal monomorphisms in Top possess a stronger property: they are regular monomorphisms.
The extremal epimorphisms are, in essence, the quotient maps. These are the surjective maps where the topology on the codomain is determined by the preimages of open sets. Every extremal epimorphism in Top is also regular.
The split monomorphisms correspond to inclusions of retracts into their ambient spaces. A retract is a subspace that can be mapped back onto itself by a continuous map.
The split epimorphisms, up to isomorphism, are continuous surjective maps from a space onto one of its retracts.
You won't find any zero morphisms in Top. This means the category is not preadditive. There's no sensible way to add or subtract maps in a general sense.
Top is not cartesian closed. This means it doesn't have exponential objects for all pairs of spaces. This is a significant limitation if you want to work with function spaces in a very abstract way. Consequently, it's not a topos either. To address this, mathematicians often restrict their attention to the full subcategory of compactly generated Hausdorff spaces, denoted CGHaus, or the category of compactly generated weak Hausdorff spaces. These categories are Cartesian closed and still encompass most of the spaces people are actually interested in. However, Top itself is contained within the exponential category of pseudotopologies, which, in turn, is a subcategory of the (also exponential) category of convergence spaces. So, while Top might lack certain desirable properties, it exists within a broader ecosystem that provides them.

Relationships to other categories

The category Top doesn't exist in isolation. It has connections to many other mathematical structures:

The category of pointed topological spaces, denoted Top•, is a coslice category over Top. It's like a category built "above" Top, focusing on spaces with a distinguished base point.
The homotopy category of topological spaces, hTop, is another important construction. Its objects are topological spaces, but its morphisms are not continuous maps themselves, but rather homotopy equivalence classes of continuous maps. This means maps that can be continuously deformed into one another are considered the same. hTop is a quotient category of Top, meaning it identifies certain morphisms. Similarly, one can construct the pointed homotopy category, hTop•.
Top contains the category Haus of Hausdorff spaces as a full subcategory. Hausdorff spaces are those where distinct points can always be separated by open sets – a fundamental property for many mathematical investigations. The added structure of Haus allows for more epimorphisms. In Haus, epimorphisms are precisely those morphisms with dense images in their codomains. This means an epimorphism in Haus doesn't have to be surjective; its image just needs to be dense.
As mentioned before, Top contains the full subcategory CGHaus of compactly generated Hausdorff spaces. This subcategory is particularly favored because it is a Cartesian closed category, making it much more convenient for working with function spaces, while still retaining all the "typical" spaces of interest. For many purposes, CGHaus is preferred over the broader Top.
We've already discussed how the forgetful functor U to Set has both a left and a right adjoint. This is a significant relationship, linking the structured world of topology to the fundamental world of sets.
There's a functor that maps a topological space to its locale of open sets, leading to the category of locales, Loc. This functor has a right adjoint that maps each locale back to its topological space of points. This adjunction establishes an equivalence between the category of sober spaces and spatial locales. It's a deep connection between topological spaces and a more abstract notion of "open set structure."
The homotopy hypothesis posits a profound connection between Top and ∞Grpd, the category of ∞-groupoids. The conjecture suggests that ∞-groupoids are essentially equivalent to topological spaces, modulo the notion of weak homotopy equivalence. It’s a bridge between topology and higher algebraic structures.

Citations and References

Naturally, for those who feel the need to cite everything:

Dolecki, Szymon (2009). "An initiation into convergence theory" (PDF). In Mynard, Frédéric; Pearl, Elliott (eds.). Beyond Topology. Contemporary Mathematics. Vol. 486. pp. 115–162. doi:10.1090/conm/486/09509. Retrieved 14 January 2021. (This is where you'll find it, buried deep.)
Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917. (More on convergence, if you must.)
Dolecki, Szymon; Mynard, Frédéric (2014). "A unified theory of function spaces and hyperspaces: local properties" (PDF). Houston J. Math. 40 (1): 285–318. Retrieved 14 January 2021. (Function spaces. Thrilling.)
Herrlich, Horst: Topologische Reflexionen und Coreflexionen. Springer Lecture Notes in Mathematics 78 (1968). (An older, presumably foundational text.)
Herrlich, Horst: Categorical topology 1971–1981. In: General Topology and its Relations to Modern Analysis and Algebra 5, Heldermann Verlag 1983, pp. 279–383. (A historical overview, for those who enjoy looking back.)
Herrlich, Horst & Strecker, George E.: Categorical Topology – its origins, as exemplified by the unfolding of the theory of topological reflections and coreflections before 1971. In: Handbook of the History of General Topology (eds. C.E.Aull & R. Lowen), Kluwer Acad. Publ. vol 1 (1997) pp. 255–341. (Even more history. Fascinating.)

There. A breakdown of Top. It’s all there, meticulously laid out. Don't expect me to find this particularly exhilarating. It’s just… structure. Order. And a whole lot of spaces.