Applied Category Theory
johncarlosbaez.wordpress.com
johncarlosbaez.wordpress.com
Brendan Fong has worked on applying category theory to things that act like interconnected networks, such as electrical circuits. This fits into a larger ambitious program that John Baez is part of to use category theory to understand physical systems that are still much too complicated to understand with current mathematical tools, such as ecosystems. Fong's thesis is on my reading list [2].
The two authors are doing great research, like this investigation of the algebra of backpropagation [3].
I really look forward to reading this too!
[1] https://arxiv.org/abs/0904.2012
Brendan Fong (2012) Causal Theories: A Categorical Perspective on Bayesian Networks https://arxiv.org/abs/1301.6201
On another pedagogical note and as someone who reads a lot of math texts, I wish math textbooks would spend way more time on introducing the intuition (geometrical if in any way possible) of the ideas before or simultaneous with going into the formulas, theorems, and details. This not only vastly improves comprehension but I find having a geometrical understanding ( even if rough and partially incorrect) greatly helps in me being able to later derive the same formulas and theorems.
Having written about various math topics over the years I know its much more difficult to follow this approach, so I don't blame anyone for not going out of their way to imbue intuition, but its still something I wish more mathematicians who teach would advocate for.
I am not affiliated, just love how he geometrically visualizes the concepts of math, and how he focuses on he intuition.
i agree about geometric intuition, as long as there's a wall of separation between the intuition+motivation and the main results.
it frustrates me when an important result is only proved in the context of a motivating example where i don't have the background. usually this is when a math book dips into physics or engineering for whatever reason.
It uses fairly simple language all the way through with lots of diagrams.
Is this the case with this text?
As an amateur mathematician, don't worry about attempts to apply mathematics to the real world; mathematics is the pattern which emerges from the real world's incidental complexity, and we have always been applying maths to real-world concepts.
Category theory actually provides ways to go from abstract domains to concrete domains. Read the text; it might loosen your view a bit.
If mathematics is applied poorly to reality, it's usually because an inadequate model is used, not because attempting this is a wrong thing to do.
I am saying that mathematical abstraction is not the right tool to get a better understanding for a lot of subjects. My questions was asking if this is an instance of someone with a categorical hammer and seeing categorical nails everywhere, or that it is a super natural fit.
Most mathematicians (I know) do mathematics for the mathematics. It is in itself a goal that does not need an external use. So yes, mathematics for a lot of mathematicians is just mental gymnastics that turned out to be very useful. Probably the reasons for some people's love for math and the usefulness are related.
[0] https://www.quora.com/Are-there-applications-of-group-theory...
[1] https://en.wikipedia.org/wiki/Alain_Badiou#Mathematics_as_on...
It's an extremely good fit and highly productive, to answer your question directly.
In this paper we describe a functorial data migration scenario about the manufacturing service capability of a distributed supply chain. The scenario is a category-theoretic analog of an OWL ontology-based semantic enrichment scenario developed at the National Institute of Standards and Technology (NIST). The scenario is presented using, and is included with, the open-source FQL tool, available for download at categoricaldata.net/fql.html.
This is part of a series of work on applying category theory to databases. The initial work was to cast database concepts into categorical concepts, this led to a clarification of various concepts such as many kinds of SQL query being instances of limits and colimits. The theory was then used to extrapolate via category theoretic concepts to develop new database manipulation concepts.
This is a general recipe for applying category theory, though there are other approaches.
Obviously all the work can be done without category theory but since the mid-2000's I gather that many insights have been gained by exporing various categories and their relations.
Edit: I think one of the main benefits is efficiency. Even if one doesn't start from categorical formalisms, you can later use them to pare things down to what's absolutely necessary.
You also get into a bit of a tangle if you believe mathematics is not some separable set of properties or things 'in the world' but an expression of some fundamental laws of human reasoning--i.e. it captures the way in which, whether we like it or not we simply must reason, because that just happens to be how the brain is constructed. If this truly is the case, and mathematics captures some fundmanetal kernel of truth about 'proper' reasoning, it should, in theory, be widely applicable in many human endeavors. For example, someone might argue, from this perspective, that something like function application doesn't only express some property or rule of reasoning with numbers, but rather indicates a form of processing or mental operation that underlies many if not all forms of conceptual reasoning. (Frege and Leibniz are two historical figures who play with some of these ideas in their pursuit of a universal language for concepts).
Programming is very much unlike that; machines strictly follow known abstract principles.
As a possible example, I found Categories for Software Engineering to be...unspectacular, both in terms of category theory and software engineering.
https://github.com/kummahiih/python-domain-equations
but seems like, I am the only one who likes simplicity