My course on applied category theory is continuing! After a two-week break where the students did exercises, I’m back to lecturing about Fong and Spivak’s book Seven Sketches. Now we’re talking about “resource theories”. Resource theories help us answer questions like this:
- Given what I have, is it possible to get what I want?
- Given what I have, how much will it cost to get what I want?
- Given what I have, how long will it take to get what I want?
- Given what I have, what is the set of ways to get what I want?
Resource theories in their modern form were arguably born in these papers:
• Bob Coecke, Tobias Fritz and Robert W. Spekkens, A mathematical theory of resources.
• Tobias Fritz, Resource convertibility and ordered commutative monoids.
We are lucky to have Tobias in our course, helping the discussions along! He’s already posted some articles on resource theory here on this blog:
• Tobias Fritz, Resource convertibility (part 1), Azimuth, 7 April 2015.
• Tobias Fritz, Resource convertibility (part 2), Azimuth, 10 April 2015.
• Tobias Fritz, Resource convertibility (part 3), Azimuth, 13 April 2015.
We’re having fun bouncing between the relatively abstract world of monoidal preorders and their very concrete real-world applications to chemistry, scheduling, manufacturing and other topics. Here are the lectures so far:
• Lecture 18 – Chapter 2: Resource Theories
• Lecture 19 – Chapter 2: Chemistry and Scheduling
• Lecture 20 – Chapter 2: Manufacturing
• Lecture 21 – Chapter 2: Monoidal Preorders
• Lecture 22 – Chapter 2: Symmetric Monoidal Preorders
• Lecture 23 – Chapter 2: Commutative Monoidal Posets
• Lecture 24 – Chapter 2: Pricing Resources
• Lecture 25 – Chapter 2: Reaction Networks
• Lecture 26 – Chapter 2: Monoidal Monotones
• Lecture 27 – Chapter 2: Adjoints of Monoidal Monotones
• Lecture 28 – Chapter 2: Ignoring Externalities
• Lecture 29 – Chapter 2: Enriched Categories
• Lecture 30 – Chapter 2: Preorders as Enriched Categories
• Lecture 31 – Chapter 2: Lawvere Metric Spaces
• Lecture 32 – Chapter 2: Enriched Functors
• Lecture 33 – Chapter 2: Tying Up Loose Ends
Azimuth 
However, neither engineers, nor software developers, nor business leaders were consulted for or contributed to the development of the book.
Am I right?
David Spivak helps run a software company, and he’s gotten a large grant from NASA to help design certain aspects of the air traffic control system. Does that count?
I believe it’ll be fun to think about “ignoring externalities” and adjoints of monoidal monotones – I posted two puzzles about that here:
• Lecture 28 – Chapter 2: Ignoring Externalities
The idea is that sometimes our view of a process:
is a simplification where we ignore features of a more complex process:
Can we go backwards and recover the more complex view from the simpler one?
I’ll have too look at the puzzles but i’ll make a comment first. Since i claim to have studied some economics , economists have 2 answers–one theoretical and one pragmatic.
Theoretically you can’t ignore externalities—only problem is, in earlier and evolving forms of economics they sort of kept finding more externalities–things they neglected to account for (e.g. things like land, water, air, biodiversity, human capital or education, social capital (‘diversity and social in/exclusion) , etc. ).
Pragmatically, alot of economists decide to ignore externalities because it complicates the analyses, and complications incur costs–including computing time. You may have to truncate the series or algorithm (be a Turing oracle). So you may just put all the trash in the creek and it will be taken away to the ocean, rather than buy a trash can and make a dump or recycling center.
I rarely eat sweets anymore so its hard for me to process the example–but its the same as any such process.
In statistical physics Jarzynski in/equality or Crooks fluctuation theorem (i view them as equivalent but that’s an opinion) point out there are no externalities. My view is they formalized what had been folklore (i.e. well known knowledge, just as indigenous americans knew about potatoes before there were frito lay potato chips–they just didn’t patent them). That view points out if you ask when entropy increases in a subsystem where did the negentropy go—the answer is out in the universe. Its still there. Thats the difference between the canonical and microcanonical ensembles in statphys. In canonical there are externalities, in microcanonical there aren’t any.