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Stream: theory: applied category theory

Topic: Categorification of storages, backups and shardings?

Ignat Insarov (Oct 22 2024 at 09:02):

Say I run an e-commerce web site that sells gadgets. It is implemented as an HTTP server program X that can write stuff to a PostgreSQL database Y or to an ElasticSearch installation Z. Then:

• When a manager adds a new gadget, X should write to Y and to Z.
• When a customer views a gadget, X may ask the details from either Y or Z.

Here, Y and Z are two backups of the same data. Here is another possible situation:

• When a gadget is written to Y, Y may write to either Y₁ or Y₂, both of which are SQL databases with identical schemata.
• When a gadget is asked of Y, X must ask the details from Y₁ and maybe also from Y₂ (or in the other order, or in parallel).

Whenever I see words and and or, I think there must be a category with sums and products in play. But in this case I cannot think of a way to formalize this intuition.

For one approach, we can say that objects are storages and arrows are connections, and we can ask that connections compose associatively. So, for example, X would have composite connection to Y₁ through Y. And maybe we can say that now there is a «virtual» storage (Y₁ + Y₂) × Z that X is connected to by virtue of being connected to Y and to Z.

One aspect this model does not account for is the difference between potentiality and actuality at a given time. Z can in principle hold any element of the set of gadgets, but Z is not the set of gadgets — at any given time it is some subset of that set. And X is either the empty set (when the HTTP server is idling) or a singleton set holding one gadget (when the HTTP server is working) at any given time, even though in potentiality it holds all possible gadgets.

Where can I go from here?

David Egolf (Oct 22 2024 at 17:23):

If I understand you correctly, you first note that we can form a category from "storages". We start as follows:

• the objects are storages
• we put an arrow from $A$ to $B$ if we have a "connection" from $A$ to $B$

It's not totally clear to me what exactly a "connection" is. I have at least two questions about that: (1) Can we have a connection from $A$ to $B$ without a connection from $B$ to $A$? (2) Can we have multiple distinct connections from $A$ to $B$?

At any rate, we end up with a (directed multi-)graph of storages and connections, and from that we can freely generate a category.

David Egolf (Oct 22 2024 at 17:23):

However, you then note that this category only describes (to some level of detail) how information can flow around between our storages. It doesn't keep track of the particular state of any storage at some moment in time.

David Egolf (Oct 22 2024 at 17:32):

I don't know enough about this kind of thing to give specific suggestions. (Maybe someone else can chime in though?)

Here are some resources that might possibly be helpful to you, that talk about ways to model things that change over time using categories:

• a series of blog posts on "Agent-Based Models" by John Baez: here
• "Categorical System Theory" by David Jaz Myers: here
• "Seven Sketches in Compositionality" by Brendan Fong and David Spivak: here

(I haven't read these yet - just bits and pieces of some of them. But I would like to read them properly someday!)

David Egolf (Oct 22 2024 at 17:38):

On a side note, I find it interesting that two kinds of information flow are present here: reading and writing. This reminds me of medical imaging, where a transducer can both transmit and receive. It seems like there ought to be a way to dream up a category where objects involve two-way interactions... but I'm not sure what morphisms should be! (What should a morphism be between two reading-writing systems? What should a morphism be between two transmitting-receiving systems?)

Ignat Insarov (Oct 22 2024 at 18:14):

Thank you David. These are great questions.

Say we have a logic. Then:

• There is a category of propositions and implications in this logic, which is a kind of a preorder category — either proposition B follows from proposition A or not. This is the way I think about mathematical problems where I do not usually try to do induction on proof terms or anything such.
• There is a category of propositions and proofs in this logic, which allows for multiple arrows — there is generally any number of ways to prove B given A. This is the way I think about code in Agda or another proof relevant dependently typed programming language.

By analogy, we may say that:

• There is a category of computers and ways they are allowed to connect. This is how I should think about virtual machines and other abstract resources when I design a cloud based deployment. Either connection from A to B is allowed and will be happening with some regularity, or such connection is not allowed and will never be happening.
• There is a category of storages and «connection options», where an object is also a computer, and an arrow is a specific private SSH key — or an encryption key of another type, such as a PostgreSQL connection string. Now it is possible to have multiple arrows from one computer to another. This lets us talk concretely about how two specific computers will be connected.

Either way, connections are not generally symmetric. In some situations, like say a blockchain, we have multiple independent communicating agents that all have connection to one another. In other situations, like say an e-commerce installation, there is a hierarchy with few if any cycles.

One way we can think about this «duality» of reading and writing is by saying that there are two opposite categories — one where arrows are reading connections and another where arrows are writing connections. Maybe this should even be one single dagger category. But I am not ready to do any meaningful mathematical work in this direction yet.

None of this brings us closer to understanding transactions as they happen over time. The straightforward way would be to ask for a time-indexed sequence of categories defined as above — but we still need some secret sauce in order to be able to say that at time t₁ A knows about gadget g but B does not, and at time t₂ both A and B know about g. The things we may wish to prove here are like so:

• A can always obtain knowledge about any g at any time t > t₁ given some network of connections.
• There are some g such that removing any connection will result in A being unable to obtain knowledge about g at some time t > t₁.

Practically, in a non-trivial installation there will be many storages, some storages will store more than one type of information, and not all storages will be allowed to store information for time intervals longer than some value. For example:

• An HTTP server is not guaranteed to store anything — it may cache stuff at its own discretion, but it may be forcibly rebooted any time, losing all knowledge.
• An HTTP server may push its logs to a third party logging service that will retain these logs indefinitely, without possibility of erasure.
• The same HTTP server may read and write stuff to the SQL database… but also delete stuff from it!
• The information stored in ElasticSearch may be dropped and regenerated from the SQL database — we may think that there is a one way connection from the SQL database to the ElasticSearch service, even though it goes through some other program that connects to both, reads form SQL, performs the necessary conversions and writes to ElasticSearch. It is as if the ElasticSearch service was a sub-object of the SQL database…

I have no idea how to formalize any of this.

I read some of David Jaz Myers's book, and I read some of Brendan Fong and David Spivak's book. But their ideas do not live in me yet. I can draw composition of lenses if you wake me at 3 AM — I have been trying to apply this stuff for the last two years. But still it is not easy to explain how, for example, a non-trivial React web app is a composite lens. I have not seen John Baez's series of blog posts before — it looks interesting!