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• CommentRowNumber1.
• CommentAuthorKeithEPeterson
• CommentTimeDec 13th 2016
• (edited Dec 14th 2016)

Made the chemistry page slightly less stubby.

Slightly off topic but… I’ve been working on the category of recipes as side project as a way to bring the layman to category theory, the general idea also appears in Bob Coecke’s Quantum Pictoralism. In some exchanges I pointed out to Bob that recipes are just the edible full subcategory of chemistry. Btw, here’s the first ever (buggy) visual recipe made using monoidal category theory and string diagrams.

Would anyone be offended if I added in that recipes are a full subcategory of chemistry as a sidenote?

May I also start some pages on recipes… in some way? I think it really is a great way to teach people category theory. After all, everyone eats, and if you can follow a recipe, you can do monoidal category theory.

• CommentRowNumber2.
• CommentAuthorDavidRoberts
• CommentTimeDec 13th 2016

Are you aware of the work of John Baez and students on network theory? It’s much more comprehensive and general than recipes for food.

They make precise the notion of a diagram such as you’re produced and category theory, and it’s not so simple as just a monoidal category.

• CommentRowNumber3.
• CommentAuthorKeithEPeterson
• CommentTimeDec 13th 2016

Yes I am. I’ve never seen so much talk about cospans…

Baez and friends are partly the reason I decided to make a diagramming language.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeDec 13th 2016
• (edited Dec 13th 2016)

the edible full subcategory of chemistry

Is that a precise statement?

Edit: The “full subcategory” part is certainly a precise-sounding statement. But unless that has mathematically verifiable content (and not just rhetorical content), it sounds like something which wouldn’t easily fit well with the nLab.

• CommentRowNumber5.
• CommentAuthorKeithEPeterson
• CommentTimeDec 13th 2016
• (edited Dec 13th 2016)

Is there a a usual format the nlab sticks to? The chemistry page I updated looks off. What would chemists consider to be 2-morphisms? Equivalences of reactions (under different physical conditions) with reaction rates relating them? Is there any literature on chemistry from a categorical perspective?

The “full subcategory” part is certainly a precise-sounding statement. But unless that has mathematically verifiable content (and not just rhetorical content), it sounds like something which wouldn’t easily fit well with the nLab.

Okay. Fair enough. What would be a more proper way of saying that cooking is chemistry in the kitchen?

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeDec 14th 2016

Your edit to the chemistry page doesn’t seem so bad, except that the pronouncement in the “Definition” section warrants some discussion first. Maybe I wouldn’t say “definition”, for one thing. But maybe some interesting supporting material can be adduced from the Baez school, as a start.

What would be a more proper way of saying that cooking is chemistry in the kitchen?

I’m not in opposition to the sentiment of cooking being a kind of applied chemistry (and I agree some knowledge of chemical science is indeed helpful for cooking), but that doesn’t mean I’d shoot for a category-theoretic pronouncement or description right away.

The link you provided for recipes gives a kind of flowchart, so I gather you’d want to think more generally of flowcharts as fitting within string diagrams and such. Maybe so (I’m not sure), but let’s see if there isn’t some literature there to support that. However, that doesn’t seem in any way specific to the kitchen.

• CommentRowNumber7.
• CommentAuthorKeithEPeterson
• CommentTimeDec 14th 2016

I’m not in opposition to the sentiment of cooking being a kind of applied chemistry (and I agree some knowledge of chemical science is indeed helpful for cooking), but that doesn’t mean I’d shoot for a category-theoretic pronouncement or description right away.

Very well. This is why I asked before doing anything.

The link you provided for recipes gives a kind of flowchart, so I gather you’d want to think more generally of flowcharts as fitting within string diagrams and such. Maybe so (I’m not sure), but let’s see if there isn’t some literature there to support that. However, that doesn’t seem in any way specific to the kitchen.

In so far there is an equivalence between string diagrams and flowcharts. I wasn’t actually trying to create a flowchart, but a pictorial representation of a process algebra a’la Samson Abramsky and Bob Coecke. They claimed that cookery was also given by a process algebra and had a pictorial representation, and I took the idea and ran with it.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeDec 14th 2016

I have edited the entry chemistry. Mentioned chemical reaction diagrams, and that one might be inspired by them to identify a category of substances and reactions. Removed the “Definition” statement. Added as one working example of a mathematical formalization of fusion and decay processes the concept of triangulated categories with Bridgeland stability conditions.

If you have another actual formalization, say using network theory, it should be added. But beware that not every arrow one sees in real life is a morphism in a category.

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeDec 14th 2016

Is that relation between triangulated categories with stability conditions and chemistry more than an analogy?

I added a pointer to John Baez’s chemical reaction networks.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeDec 14th 2016

As I explain in the entry, it formalizes the decay and fusion reaction processes between species of D-branes for the topological string.

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeDec 14th 2016

I know, but what have species of D-branes for the topological string to do with chemistry?

I started periodic table of the chemical elements, and added reference to Feynman on the elements to quantum chemistry.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeDec 14th 2016
• (edited Dec 14th 2016)

I find it serves as an excellent example of a category theoretic formalization of reaction processes

$A + B \leftrightarrow C + D$

of just the same kind as one is faced with in chemistry.

It serves to show both that 1) the naive idea of how to model this in category theory may be too naive, and 2) that something of genuine mathematical interest may come out of a good formalization.

And I think the general idea of how to formalize “reaction processes” in category theory will hardly depend on the specific nature of the the chemical elements, will it? The formalization via networks that you point to is also more general and probably will also apply to other reactions processes, such as nuclear reactions or (I gather) biological processes or (possibly) topological D-brane reactions.

Incidentally, I would find it useful if anyone could actually write into the entry what that formalization via networks (Petri nets, I gather) actually is. I clicked around following the pointer that you gave, but I didn’t find a definition before my time was up.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeDec 14th 2016

I have edited chemistry in an attempt to clarify, as in the above discussion.

Also I touched chemical element.

• CommentRowNumber14.
• CommentAuthorDavid_Corfield
• CommentTimeDec 14th 2016

Yes, John’s networks also model predator-prey.

I’ve added something to the effect that this isn’t just about chemistry. (We wouldn’t want people thinking we thought D-brane modes correspond to kinds of chemical ;)

At some point, I’ll see whether there’s an easy summary of these chemical reaction networks.

• CommentRowNumber15.
• CommentAuthorDavid_Corfield
• CommentTimeDec 14th 2016

I put in some kind of definition at least.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeDec 14th 2016
• (edited Dec 14th 2016)

Thanks, now I see that we have the definition at Petri net.

Okay, but I’ll note this is not a category theoretic definition in the style that the OP had envisioned in revision 3, where substances are identified with objects of a category and reaction processes are supposed to be given by morphisms. The concept of triangulated category with stability condition comes much closer to this.

• CommentRowNumber17.
• CommentAuthorDavid_Corfield
• CommentTimeDec 14th 2016

I believe they take these nets as objects of a category. Not sure what constructions that allows.

There’s also a linear logic-Petri net link.

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeDec 14th 2016

I believe they take these nets as objects of a category.

Sure, but that’s more like a category “of chemistries” then, not one category that witnesses the substances and processes of “one chemistry”.

Anyway, it’s the latter that I think the OP was after. And so, for whatever it’s worth, I still think that the best available answer to fix the OP’s idea expressed in revision 3 is triangulated categories with stability condition.

It might be a fun exercise to see if one may build an example of such inside which some actual chemical reactions might sensibly be identified. For that one should probably drop the full power of “triangulated” and just keep the core idea of modeling fusion/decay by exact sequences.

• CommentRowNumber19.
• CommentAuthorTim_Porter
• CommentTimeDec 14th 2016
• (edited Dec 14th 2016)

Does remembering that a Petri net can be considered as a presentation of a certain type of symmetric monoidal category help at all? This was in part the viewpoint that Mesequer and Montanari adopted and used in concurrency. (I am not 100% sure that that is the best article on this. There is also: International Journal of Foundations of Computer Science, December 1991, Vol. 02, No. 04 : pp. 297-399

FROM PETRI NETS TO LINEAR LOGIC THROUGH CATEGORIES: A SURVEY NARCISO MARTÍ -OLIET and JOSÉ MESEGUER (doi: 10.1142/S0129054191000182)

but that is behind a paywall.)

If I remember correctly they did look at a category of Petri nets.

• CommentRowNumber20.
• CommentAuthorKeithEPeterson
• CommentTimeDec 14th 2016
• (edited Dec 14th 2016)

Oh neat, a chemical complex $C: S \rightarrow \mathbb{N}$ is just a multiset.

At least I got one thing right with my recipe formulation…

• CommentRowNumber21.
• CommentAuthorDavid_Corfield
• CommentTimeDec 15th 2016

the core idea of modeling fusion/decay by exact sequences

That reminds me that I never got quite clear on the appropriate setting for exact sequences. We should point out at exact sequence, the notion of homological category and semi-abelian category (both in need of some treatment). This MathStackExchange answer

for a semi-abelian category, simplicial objects are chain complexes with additional structure in the sense that the category if simplicial objects is monadic over the category of chain complexes.

Consequently, semi-abelian categories provide a setting in which one can try to understand non-abelian cohomology.

Well, anyway, I added to exact sequence

An exact sequence may be defined in a semi-abelian category, and more generally in a homological category. It is a sequence of objects and morphisms in which the cokernel of each morphism is equal to the kernel of the next morphism.

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeDec 15th 2016

Well, anyway, I added to exact sequence

An exact sequence may be defined in a semi-abelian category, and more generally in a homological category. It is a sequence of objects and morphisms in which the cokernel of each morphism is equal to the kernel of the next morphism.

Thanks, good point that a remark like this needed to be added. I made that paragraph of yours the previously missing Idea-section.