Categorical Representation of Set-Theoretic Topology

I am an undergraduate with huge interest to the topology and foundations of mathematics. I have been studying Spanier's Algebraic Topology and Engelking's General Topology, and I noticed that while algebraic topology heavily depends on the category theory, set-theoretic topology does not.

Is there a reason why category theory is not used in set-theoretic topology? Is there no advantage of trying to introduce the categories to the set-theoretic foundations? At least from books in general topology, I did not see single use of categories except for spaces relevant to algebraic topology like quotient space.

• Category theory is used in "set theoretic" topology: for example, products, quotients, pullbacks and pushouts (of spaces) are all categorical notions. One can consider colimits and limits of spaces, and many other constructions (such as initial or final topologies) can be nicely stated in terms of universal properties. Perhaps you have the impression many different functors and functorial constructions are heavily exploited in algebraic topology, and this is true. – Pedro Tamaroff Aug 26 '16 at 3:45
• There certainly does exist a field of categorical general topology-it's just been incorporated less into traditional general topology. – Kevin Carlson Aug 26 '16 at 3:48
• @KevinCarlson Could you inform me some sources (books, articles, etc.) that deals with categorical general topology? – MathWanderer Aug 26 '16 at 3:53
• Set-theoretic topologists of my generation are much more likely to have backgrounds in set theory and model theory and to be comfortable with the tools and language of those areas. In general I find that category-theoretic language does more to obfuscate than to clarify the things that interest me as a topologist: I simply don't think in those terms and have seen nothing that persuades me that it would be advantageous to do so. Universal properties are nice and certainly have their uses, but I generally want to see the set-theoretic nuts and bolts. – Brian M. Scott Aug 26 '16 at 4:10
• Do you really mean set-theoretic topology or is it used in your post just as an incorrect name for general topology, which is also called point-set topology? (There is a difference between these two, I think they should be not confused. Set-theoretic topology can be considered as one branch of general topology.) The reason that you mention Engelking's book is the reason why my guess would be that you really mean general topology when you say set-theoretic topology. – Martin Sleziak Sep 3 '16 at 5:57

There are advantages in the use of category theory in the presentation of general topology, or any general theory.

One aim of category theory is to look at an object or construction in the light of its relation to all other objects through the notion of morphism - for topology the morphisms are the continuous functions. In this view the definition of a topological space (by open sets, closed sets, neighbourhoods, ...) is less important than the morphisms, and the global properties of the category of topological spaces and continuous maps.

This consideration led to the notion of convenient category of spaces, see also the Introduction to this 1963 paper.

Further defining constructions on, or properties of, spaces by their categorical properties allows for analogy and comparison with other categories. As one simple example, in my book Topology and Groupoids (T&G) I define a space $X$ to be connected if the only maps from $X$ to the discrete space with two elements are constant. This is sometimes useful for proofs of connectivity.

Here is another example: the Tychonoff topology on the product set $X= \Pi_{i \in I} X_i$ of a family of topological spaces $X_i$. This topology may be defined purely in terms of the open sets of the spaces $X_i$, and this is useful to know. However there are projections $p_i: X \to X_i$ and a key property of these which we want is that a function $f: Y \to X$ from a space $Y$ is continuous if and only if all the projections $p_i f: Y \to X_i$ are continuous. This property characterises the Tychonoff topology, and gives an analogy between this topological construction and properties of products in other categories.

Such more categorical presentation, and so further possibilities for analogy and comparison, gives prospects for analysis and evaluation, whose necessity for progress in science is discussed in this article by Einstein.

The two books you mention were published quite a while ago, when the wide value of category theory was not so clear as today. Also, some people are intrinsically interested in problems, others interested in the development of structures and viewpoints. Each should be encouraged, as a part of the methodology of research.