You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
Tomáš Gonda (Jun 01 2020 at 18:13):Simon Burton said:
Tomáš Gonda said:
A bit more specifically, I think about resource theories, as they offer a nice way to understand physics from a more agent-centric and less reductionist perspective, in contrast to the classic narratives.
Have you thought about a resource theory being a functor ? It would have codomain the positive reals (with as morphisms) and domain a category of quantum density matrices and CP maps (or something like that)... I'm wondering if anyone has thought about resource theories with codomain some other POSET's.
The functor you mention is commonly referred to as a resource monotone. These, in isolation, constitute some of the most coarse-grained information you can extract from a resource theory. Of course, using other ordered structures as codomains (such as, but not limited to, considering families of monotones) allows one to make more (but also less) fine-grained inferences.
Tomáš Gonda (Jun 01 2020 at 18:21):Generally, in the resource theory literature, category theory language is not wide-spread. However, thinking of resource monotones as certain kinds of functors does help when you want to generalize the ideas or get an overview.
Tomáš Gonda (Jun 01 2020 at 18:28):Simon Burton said:
Also, these functors don't seem to be monoidal, but they do obey a "sub-monoidal" equation: I wonder if the cateogory theorists have a name for such a thing. Here I'm thinking of the codomain of , as a monoidal category.
Indeed, in all instances of resource theories being studied, I'd say that one would require monotones (i.e., resource quantifiers) to be lax monoidal. This rests on the interpretation of the action of forming tensor products to be a "free operation", which may not always be satisfied, but of course most of the time this background assumption is fairly uncontroversial.
Note that the special case of monoidal monotones turns out to be quite important as far as the characterization of asymptotic and catalytic resource conversions is concerned. This has been noticed in specific instances of resource theories over time and proven more generally by @Tobias Fritz recently.
Tomáš Gonda (Jun 01 2020 at 18:50):As far as the syntax of resource theories goes, I tend towards the view that different applications of the idea may require different models with different levels of abstraction.
However, broadly speaking, there is usually some descriptive theory (like a category of CP maps that you mention, other notable example is of stochastic maps) which tells you not only what one can do with the resource, but also how. This may come in various flavours with varying detail of description:
The second layer then consists of stripping the descriptive theory of its how content and only leaving the information about the possible conversions. This is commonly some thin category (i.e., an ordered structure), which again could contain some extra compositional/logical structure. At this point, the ordering tells you which resources can be converted into each other, but generally lacks the information about how to achieve such a conversion.
The third layer would then consist of studying the resource ordering by coarse-grainings, such as real-valued monotones like you mentioned @Simon Burton, or indeed by using other simple ordered structures as codomains. I don't think of this layer as being an essential part of the resource theory as far as the descriptive content is concerned, for me it is more about being able to compute stuff and about extracting partial information :slight_smile: