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Stream: learning: questions

Topic: Reactivity categories

fosco (Dec 21 2020 at 09:43):

Has anyone ever considered a structure that is like a category, but where one can measure the tendency of objects to connect with each other via morphisms? So there will be "noble" objects, unwilling to bond, and "alkaline" objects, violently reacting when they are put in proximity with other objects.

Depending on one's will, there are many different possibilities to axiomatise such a thing; one can try to copy the definition of a quantum graph, attaching to each edge $A\to B$ of the underlying graph of a category $\mathcal C$ a real number $\ell$ , confused with the interval $[0,\ell]$ , and a repulsive or attractive force $\bf F$ , pointing inwards or outwards the object $A$ , depending on its will to "bond" with $B$ . This is a nice perspective, because one can then put differential equations on the underlying graph of $\mathcal C$ (and this is something I was thinking about a few years ago...).

Alternatively, one can consider an enrichment of the representable functor $\hom(A,-) : \mathcal C \to \sf Set$ associated with $A$ , into $\underline{\hom}(A,-) : \mathcal C \to {\sf Set}\times [0,\infty) : X\mapsto (\hom(A,X),\lambda_X)$ , with requests like

if $\hom(A,X)=\varnothing$ , then $\lambda_X=0$ ;
$\underline{\hom}(A,X\cup Y)$ has second component $\lambda_X + \lambda_Y$
$\underline{\hom}(A,X\times Y)$ has second component $\lambda_X \cdot \lambda_Y$
etc., etc..

I haven't put much thought into this, but I'd like to gather some suggestions. Hope you like the idea :-)

Chad Nester (Dec 21 2020 at 10:31):

Is chemical bonding in this sense compositional?

Chad Nester (Dec 21 2020 at 10:31):

(Should this be a “mere” graph instead?)

fosco (Dec 21 2020 at 10:40):

Chad Nester said:

Is chemical bonding in this sense compositional?

It is not clear whether given elements A,B the product $A+B$ should be more or less stable than A,B alone; I guess it depends on A,B?

Morgan Rogers (he/him) (Dec 21 2020 at 10:41):

Chad Nester said:

(Should this be a “mere” graph instead?)

Not necessarily: imagine a string of particles. The particles on either end might have some direct interactions, but most of the interactions between them will be transferred along the string. It seems a useful abstraction to be able to describe the total interaction along some pathway in terms of a composite morphism connecting them

Fabrizio Genovese (Dec 21 2020 at 11:56):

Traditionally, in categorical quantum mechanics, "interaction" is represented via monoidal products. Composition is mainly used to represent the idea that "process A causally depends on process B"

Fabrizio Genovese (Dec 21 2020 at 11:57):

This is clearly justified by the fact that Categorical Quantum Mechanics, in its essence, is a way to wrap the traditional linear algebra with which we do quantum mechanics into a nice categorical formalism. In this setting, interaction (e.g. in the sense of entangling things) is given by the tensor product of vector spaces

Fabrizio Genovese (Dec 21 2020 at 11:58):

In general, every time you want to describe the idea of morphisms/objects/whatever as interacting, and your intuition is at least partially based in physics, giving a look at categorical quantum mechanics is a wise idea. Researchers in the field have had roughly 16+ years to think about questions that are kinda similar :smile: