The n-Category Café

Pseudo-Dionysius, a Christian Neoplatonist who wrote in the late fifth or early sixth century CE, described the 9-fold hierarchy of angels, 3 hierarchies of 3 levels in De coelesti hierarchia.

Aquinas explains this organisation in the Summa Theologica, First Part, Qustion 108, Article 6.

Let us then first examine the reason for the ordering of Dionysius, in which we see, that, as said above (Article [1]), the highest hierarchy contemplates the ideas of things in God Himself; the second in the universal causes; and third in their application to particular effects. And because God is the end not only of the angelic ministrations, but also of the whole creation, it belongs to the first hierarchy to consider the end; to the middle one belongs the universal disposition of what is to be done; and to the last belongs the application of this disposition to the effect, which is the carrying out of the work; for it is clear that these three things exist in every kind of operation. So Dionysius, considering the properties of the orders as derived from their names, places in the first hierarchy those orders the names of which are taken from their relation to God, the “Seraphim,” “Cherubim,” and “Thrones”; and he places in the middle hierarchy those orders whose names denote a certain kind of common government or disposition—the “Dominations,” “Virtues,” and “Powers”; and he places in the third hierarchy the orders whose names denote the execution of the work, the “Principalities,” “Angels,” and “Archangels.”

As regards the end, three things may be considered. For firstly we consider the end; then we acquire perfect knowledge of the end; thirdly, we fix our intention on the end; of which the second is an addition to the first, and the third an addition to both. And because God is the end of creatures, as the leader is the end of an army, as the Philosopher says (Metaph. xii, Did. xi, 10); so a somewhat similar order may be seen in human affairs. For there are some who enjoy the dignity of being able with familiarity to approach the king or leader; others in addition are privileged to know his secrets; and others above these ever abide with him, in a close union. According to this similitude, we can understand the disposition in the orders of the first hierarchy; for the “Thrones” are raised up so as to be the familiar recipients of God in themselves, in the sense of knowing immediately the types of things in Himself; and this is proper to the whole of the first hierarchy. The “Cherubim” know the Divine secrets supereminently; and the “Seraphim” excel in what is the supreme excellence of all, in being united to God Himself; and all this in such a manner that the whole of this hierarchy can be called the “Thrones”; as, from what is common to all the heavenly spirits together, they are all called “Angels.”

As regards government, three things are comprised therein, the first of which is to appoint those things which are to be done, and this belongs to the “Dominations”; the second is to give the power of carrying out what is to be done, which belongs to the “Virtues”; the third is to order how what has been commanded or decided to be done can be carried out by others, which belongs to the “Powers.”

The execution of the angelic ministrations consists in announcing Divine things. Now in the execution of any action there are beginners and leaders; as in singing, the precentors; and in war, generals and officers; this belongs to the “Principalities.” There are others who simply execute what is to be done; and these are the “Angels.” Others hold a middle place; and these are the “Archangels,” as above explained.

This explanation of the orders is quite a reasonable one. For the highest in an inferior order always has affinity to the lowest in the higher order; as the lowest animals are near to the plants. Now the first order is that of the Divine Persons, which terminates in the Holy Ghost, Who is Love proceeding, with Whom the highest order of the first hierarchy has affinity, denominated as it is from the fire of love. The lowest order of the first hierarchy is that of the “Thrones,” who in their own order are akin to the “Dominations”; for the “Thrones,” according to Gregory (Hom. xxiv in Ev.), are so called “because through them God accomplishes His judgments,” since they are enlightened by Him in a manner adapted to the immediate enlightening of the second hierarchy, to which belongs the disposition of the Divine ministrations. The order of the “Powers” is akin to the order of the “Principalities”; for as it belongs to the “Powers” to impose order on those subject to them, this ordering is plainly shown at once in the name of “Principalities,” who, as presiding over the government of peoples and kingdoms (which occupies the first and principal place in the Divine ministrations), are the first in the execution thereof; “for the good of a nation is more divine than the good of one man” (Ethic. i, 2); and hence it is written, “The prince of the kingdom of the Persians resisted me” (Dan. 10:13).

Aquinas then proceeds to discuss Gregory’s slightly different hierarchy. The material for a Galoisian angelology?

Locke’s Essay on Human Understanding, Book XVI, section 12 for reasoning about more perfect beings:

…finding in all parts of the creation, that fall under human observation, that there is a gradual connexion of one with another, without any great or discernible gaps between, in all that great variety of things we see in the world, which are so closely linked together, that, in the several ranks of beings, it is not easy to discover the bounds betwixt them; we have reason to be persuaded that, by such gentle steps, things ascend upwards in degrees of perfection. It is a hard matter to say where sensible and rational begin, and where insensible and irrational end: and who is there quick-sighted enough to determine precisely which is the lowest species of living things, and which the first of those which have no life? Things, as far as we can observe, lessen and augment, as the quantity does in a regular cone; where, though there be a manifest odds betwixt the bigness of the diameter at a remote distance, yet the difference between the upper and under, where they touch one another, is hardly discernible. The difference is exceeding great between some men and some animals: but if we will compare the understanding and abilities of some men and some brutes, we shall find so little difference, that it will be hard to say, that that of the man is either clearer or larger. Observing, I say, such gradual and gentle descents downwards in those parts of the creation that are beneath man, the rule of analogy may make it probable, that it is so also in things above us and our observation; and that there are several ranks of intelligent beings, excelling us in several degrees of perfection, ascending upwards towards the infinite perfection of the Creator, by gentle steps and differences, that are every one at no great distance from the next to it.

Thanks to Brandon Watson for this reference.

A nifty additional example of an operation whose fourth power is the identity, closely related to Hodge duality, is the operation of taking ‘duals’ of morphisms in a fusion category (semisimple linear monoidal category where each object has a dual). If $C$ is our fusion category, then the operation of taking duals can be seen as a contravariant op-monoidal functor

‘Taking the dual twice’ should be something like the identity, and this is precisely what a pivotal structure is — a monoidal natural isomorphism

For instance, and roughly speaking, in the Hodge duality situation the isomorphism $\gamma$ is precisely that minus sign.

A pivotal structure is what enables you to make “closed circles” from the cups and caps and hence form “dimensions” of objects. Interestingly, whether a fusion category always admits a pivotal structure is an intricate question (see these lecture notes) and is one of the things I’m currently working on. (I am actually simultaneously trying to promote an equivalent but more streamlined and simplified paradigm of the concept of pivotal structure, which I call an even-handed structure, see here and here.)

Remarkably though, although it’s tough in general to find an isomorphism between the identity and the square of the duality functor, there is always a more-or-less canonical isomorphism between the identity and the fourth power of the duality functor:

Even more remarkably, if you draw this out in string diagrams (for instance, see page 16 of this paper by Hagge and Hong) it turns out to be a Dirac rope trick kind of situation: you have to do two full rotations (fourth power of the duality functor $*$) to get back to where you started!

“How many such cases of an operation whose fourth power is the identity have a common ‘cause’?”

In Quantum Computing, but not in classical computing, we have the operation: “square root of not.”

That, to the 4th power, is logically an identity operation, preserving truth value.

I have spent some effort generalizing that, but this is not the place for the tangent.

I love this fourth root of unity stuff. It’s odd that nobody has yet mentioned this example:

$$i^4 = 1$$

I didn’t mention it because it was the answer to my puzzle! I hope the rest of you aren’t mentioning it because it’s so obvious.

The fact that the Fourier transform satisfies

$$F^4 = 1$$

is a direct consequence of the fact that the Fourier transform is the second quantization of the action of $i$ on $\mathbb{C}$. Irving Segal was one of the first to really understand this.

I like the idea of “square root of not”. If this operation were available in everyday language, politicians would use it all the time as they sought to halfway reverse their positions.

Of course the very meaning of “$i$” is “turning 90 degrees”, which is halfway reversing yourself. We need the complex plane because we can’t turn halfway around if we live on a mere line.

I don’t see yet how Bruce Bartlett’s remarks are related to $i^4 = 1$, but I bet they are — perhaps with marvelous consequences!

I wrote:

I don’t see yet how Bruce Bartlett’s remarks are related to $i^4 = 1$, but I bet they are — perhaps with marvelous consequences!

I think I’m seeing how it goes.

For starters, $i$ is an element of $SO(2) = U(1)$ whose fourth power is 1. Here $SO(2)$ is the group of rotations of the plane, and $U(1)$ is the unit complex numbers.

Bruce, on the other hand, is noting that dualization in string diagrams corresponds to a ‘half twist’: the element $-1$ in $SO(3)$, whose square is 1. Then he’s lifting this to an element of $SU(2)$ whose fourth power is 1. Here $SO(3)$ is the group of rotations of 3d space, and its double cover $SU(2)$ is the unit quaternions.

So, Bruce’s observation seems to involve a ‘higher-dimensional’ version of the concept of $i$, which lives in $SU(2)$ instead of $U(1)$.

But in fact there’s a standard name for a unit quaternion whose fourth power is 1: namely, $i$! But this is $i$ viewed as a unit quaternion, not just a unit complex number.

If we think about this hard enough we may learn some surprising things. After all, Bruce’s observations apply even to categories that seemingly don’t involve the complex numbers or quaternions. I would like to see these number systems ‘emerge’ from string diagram considerations.