But when I started working on Category Theory, the connections became more obvious.
As a general introduction to this topic let me mention several areas of particular interest .
From Emanation we map from different spheres to lower level spheres. But we lose content. Only the spheres of Emanation are Divine as stated openly in the introduction to the Tikunai HaZohar.(באצילות איהו וחיוהי חד ולא בעלמין תתאין) and the Ari himself brings this Tikunai Zohar. [and the Remak, Moshe from Cordoba, also in his magnum opus the Pardes]
In Category Theory there is a well defined system to map from mathematical structures like rings or topological spaces like spheres to lower or higher levels. To me the connection seems obvious. Though I can imagine that this connection is ignored in academia for the sad reason that most of 20th century LA/P Philosophy [Linguistic Analytic] has been an attempt to deny the validity of metaphysics. Only recently has philosophy stated to climb out of their hole by the books of people like Jerold Katz that have exposed the intellectual bankruptcy of 20th century philosophy.
In the map from a sphere to the union of two spheres [(S^n) Psi to S^n/S^n-1 seems to be related to the idea of connection or joining (זיווג) between spheres in kabalah] The difference is that spheres in kabalah have numinous content while mathematical structures do not. But outside of that the connections seems clear.
I noticed some connections also between Logic and Kabalah a few years ago but this is mentioned in the "Book of Ideas."
A general observation is that mathematics is usually horizontal. It is only with the advent of Category Theory has it become vertical. And this seems also related to the difference between the philosophy of Plato [vertical ideas or forms] and Aristotle [horizontal] and both are implicit in the general structure of the world of Isaac Luria. In particular you can see this in the Reshash (רש''ש)--Shalom Sharabi who holds that the entire system of the Ari [which starts out vertical and then morphs slowly into horizontal vessels] will change at some future time--the revival of the dead--when all the vertical world will become horizontal.
There is also some interesting parallels between Isaac Luria and Kant. Kant like Luria divides understanding into two parts (אמא עלאה ואמא תתאה). For Kant one is empirical, and the other for a priori. The most striking parallel between Kant and Luria is concerning Kant's idea of a function of the mind that accomplishes synthesis between objects of the understanding and empirical objects --the exact idea of Daat דעת
There is also knot theory which explicitly uses a concept from the Ari [ in the title of the book itself].
The Ari himself is positing worlds that exist and were created on higher spiritual planes before this physical universe. Ar first glance this seems obvious. This world seems to have aspects to it that are not physical and seem to be embedded in it. For example the number "2." It does not seem to be something you stumble on as you walk down the street. But it would be hard to argue that it does not exist. The Ari mainly deals with these in between levels between God and this physical universe. And they are not considered outside the limits of experience. In particular the moral plane, the world of moral law, though not physical is considered to be well within the range of experience.
In other words, the Ari is a highly Neo Platonic system that seems to reflect deep understanding of the metaphysical nature of the world
There is another connection between Math and kabalah that astounded me when I encountered it.
It is the breaking of the vessels called מיתת המלכים. This refers to the kings of Edom that ruled before king ruled in Israel. [Genesis and in Chronicles.]. This small seemingly insignificant paragraph in the Torah occupies a major extremely complicated and involved subject in the kabalah of Isaac Luria.
There are seven kings of Edom. And this is considered to be the source of the breaking of the vessels and the root of all problems that come into the world. The root of all catastrophes. And in Catastrophe Theory there are seven kinds of catastrophe in lower dimensions. [Seven major types.]
I noticed this Physics also. But that was a long time ago when I had ideas that I could defend on this subject.
Further look at the subdomains after integral domains. That is all the subdomains after you have the domain that can't be a clock. That is in clock 3^4 =0. The order of spaces under a domain which is not a clock comes out to be the order of the ten sepherot. So to me it looks like Isaac Luria did have some important insights. [And besides that I should mention that Yaakov Abuchatzeira calls Isaac Luria "Rabainu" (our teacher) with no other appellation. See for example the Pituchei Chotam of Yaakov Abuchatzaira]