by Brendan Fong
Hugo Nava-Kopp and I have a new paper on resource theories:
• Brendan Fong and Hugo Nava-Kopp, Additive monotones for resource theories of parallel-combinable processes with discarding.
A mathematical theory of resources is Tobias Fritz’s current big project. He’s already explained how ordered monoids can be viewed as theories of resource convertibility in a three part series on this blog.
Ordered monoids are great, and quite familiar in network theory: for example, a Petri net can be viewed as a presentation for an ordered commutative monoid. But this work started in symmetric monoidal categories, together with my (Oxford) supervisor Bob Coecke and Rob Spekkens.
The main idea is this: think of the objects of your symmetric monoidal category as resources, and your morphisms as ways to convert one resource into another. The monoidal product or ‘tensor product’ in your category allows you to talk about collections of your resources. So, for example, in the resource theory of chemical reactions, our objects are molecules like oxygen O2, hydrogen H2, and water H2O, and morphisms things like the electrolysis of water:
This is a categorification of the ordered commutative monoid of resource convertibility: we now keep track of how we convert resources into one another, instead of just whether we can convert them.
Categorically, I find the other direction easier to state: being a category, the resource theory is enriched over 
But the research programme didn’t start at resource theories either. The starting point was ‘partitioned process theories’.
Here’s an example that guided the definitions. Suppose we have a bunch of labs with interacting quantum systems, separated in space. With enough cooperation and funding, they can do big joint operations on their systems, like create entangled pairs between two locations. For ‘free’, however, they’re limited to classical communication between the locations, although they can do the full range of quantum operations on their local system. So you’ve got a symmetric monoidal category with objects quantum systems and morphisms quantum operations, together with a wide (all-object-including) symmetric monoidal subcategory that contains the morphisms you can do with local quantum operations and classical communication (known as LOCC operations).
This general structure: a symmetric monoidal category (or SMC for short) with a wide symmetric monoidal subcategory, is called a partitioned process theory. We call the morphisms in the SMC processes, and those in the subSMC free processes.
There are a number of methods for building a resource theory (i.e. an SMC) from a partitioned process theory. The unifying idea though, is that your new SMC has the processes 
But we don’t have to go to fancy sounding quantum situations to find examples of partitioned process theories. Instead, just look at any SMC in which each object is equipped with an algebraic structure. Then the morphisms defining this structure can be taken as our ‘free’ processes.
For example, in a multigraph category every object has the structure of a ‘special commutative Frobenius algebra’. That’s a bit of a mouthful, but John defined it a while back, and examples include categories where morphisms are electrical circuits, and categories where morphisms are signal flow diagrams.
So these categories give partitioned process theories! This idea of partitioning the morphisms into ‘free’ ones and ‘costly’ ones is reminiscent of what I was saying earlier about the operad of wiring diagrams about it being useful to separate behavioural structure from interconnection structure.
This suggests that we can also view the free processes as generating some sort of operad, that describes the ways we allow ourselves to use free processes to turn processes into other processes. If we really want to roll a big machine out to play with this stuff, framed bicategories may also be interesting; Spivak is already using them to get at questions about operads. But that’s all conjecture, and a bit of a digression.
To get back to the point, this was all just to say that if you find yourself with a bunch of resistors, and you ask ‘what can I build?’, then you’re after the resource theory apparatus.
You can read more about this stuff here:
• Bob Coecke, Tobias Fritz and Rob W. Spekkens, A mathematical theory of resources.
• Tobias Fritz, The mathematical structure of theories of resource convertibility I.
Brendan wrote:
I haven’t read your paper yet, and I should, but I’ll just ask the obvious question here.
It’s a bit of a ‘level shift’ to start treating morphisms as objects and look at morphisms between those. If you just take your morphisms, treat them as objects in a new category, and start defining morphisms between those, you’ve thrown out a lot of interesting stuff: namely, your ability to compose your original morphisms.
The most famous way around this is to build a bicategory where you now have 2-morphisms going between your original morphisms. Of course, this only works if all your new morphisms-between-morphisms go between morphisms having the same source and target: if we have
we can now talk about 
I bet you need more flexibility here—right? If a morphism from
to 
is ‘a way to use free processes to build 
from 
‘, there’s no reason 
and 
should have the same source and target.
So, one option is to consider a double category. For example, you could make one where the vertical arrows are your original morphisms and the horizontal ones are your ‘free processes’: morphisms in some subcategory. There’s an easy obvious way to build such a double category.
A similar sort of situation led Shulman to develop framed bicategories: for example, you have both homomorphisms between rings, and bimodules between rings, and the former is a special case of the latter. Is this what you were alluding to?
John,
Before I go and read the manual, could you clear up one thing for me?
Is the term “framed bicategory” a bit of a red herring? In the sense that a framed bicategory in your example (rings and bimodules) does not appear to be a bicategory. Rather it looks like a generalization of a double category, with the composition of one sort of 1-arrows not quite associative. Is this right?
Yes, I’d say the term “framed bicategory” is a bit of a red herring. However, the main point is not that composition of one sort of 1-arrows is ‘not quite associative’!
It’s true that composition of bimodules is associative only up to coherent isomorphism. And it’s true that we should generalize double categories to allow that kind of situation. But a double category where composition of (say) horizontal 1-arrows is associative only up to coherent isomorphism is called a pseudo double category or weak double category, or sometimes, by very modern people, just a double category. There’s a theorem saying that every pseudo double category is equivalent to an ordinary double category (it’s Theorem 7.5 in Grandis and Pare). So this issue winds up being no big deal, except at a technical level.
The idea of ‘framing’ is rather different, and more substantial. If you look at Mike Shulman’s paper he says:
The paper looks a bit scary if you read it from the beginning, but since you know what a double category is, I suggest you go to Theorem 4.1 and take condition (iii) as the definition of a framed bicategory. It’s a double category where each vertical arrow can be turned into a horizontal one in two ways, mimicking how a ring homomorphism
gives rise both to an 
-bimodule and a 
-bimodule!
Another good example is the double category with sets as objects, functions as vertical arrows and spans as horizontal arrows. A function
can be turned into a span from 
to 
in an obvious way, but also into a span from 
to 
There are lots of other examples, listed in Shulman’s paper. So while the name ‘framed bicategory’ annoys me a bit, the concept does not. More recently Shulman has been using the term ‘fibrant double category’ for the same concept.
Hi John, the loss of structure is on purpose. While the passage from algebraic (i.e. order-theoretic) logic to categorical logic is all about passing from
can we prove this' tohow can we prove this’, here it’s the opposite, we are really only interested in the `can’ part. This compression is in the same spirit as assigning a measure to something e.g. entropy, cost, energy. However, for many measures the main thing of interest is what is more than what, rather than the actual number, and in some cases an additional monoid structure like addition in the case of money is crucial. That is, the manner in which things compose. So that’s how the ordered monoid pops up. Also, some things simply don’t compare, and therefore unlike a measure which induces a total order, here one allows for a partial order.Hi! I like the ability to collapse a category down to a poset by saying
if there was a morphism from 
to 
it’s a nice halfway house between the category and taking the set of isomorphism classes of objects. However, I like to postpone this loss of structure until we need to do it. If you can prove something for categories (or monoidal categories, etc.) then you’ve automatically proved it for posets (or partially ordered monoids,
etc.).
There’s a fun and probably little-read section about such ideas in my paper with Mike Shulman, section 5.1—the section called “Enrichment and posets”. The paper argued that truth values are (-1)-categories, and that we can only understand how category theory connects to logic and topology if we realize this. But this section goes further, saying:
We then run with this and discuss a version of the periodic table of n-categories based on this philosophy, where the columns are truth values, posets, 2-posets (that is, categories where the homsets are partially ordered), and so on.
First, sorry for not being more active here — I have a bunch of deadlines coming up, such as a colloquium…
Personally I wholeheartedly agree with that. Ultimately, I’d like to be able to do many of the things that one can do with ordered commutative monoids also for symmetric monoidal categories! But, my paper shows that the former includes some hard functional analysis, such as the Hahn-Banach theorem.
So, in order to do completely without the loss of structure, one will have to categorify the Hahn-Banach theorem. Here’s a guess as to what this could look like: given
• a symmetric monoidal category
,
,
,
which is suitably compatible with the lax and oplax structures,
• a lax monoidal functor
• an oplax monoidal functor
• natural transformation
then there exists some strong monoidal functor
such that 
factors as 
.
Does anyone know whether something like this could actually be true?
Whoops, apologies for the horrible formatting… the third-to-last paragraph should have been an unordered list, and the second-to-last paragraph is missing a ‘latex’.
Fixed. By the way: isn’t ‘unordered list’ a contradicto in adjecto?
I didn’t reply because I don’t have any bright ideas about this proposed ‘categorified Hahn–Banach theorem’. It would be fun to examine it for some very small choices of
, say one with only four objects 
forming a little sup-semilattice. Or perhaps a ‘free’ example where 
with addition as tensor product.