# Does this “leaky isomorphism” concept have a name?

I've been thinking about catalysis in chemical reaction networks while learning category theory at the same time, and it's given me a weird idea, which I'm asking about out of curiosity.

Suppose I have a symmetric monoidal category $$\mathscr{C}$$. Given two objects $$A,B \in \mathrm{Ob}(C)$$ can be the case that $$A\not\simeq B$$, but that there exists some $$Z\in\mathrm{Ob}(C)$$ such that $$A\otimes Z \simeq B \otimes Z$$. For want of a better name, let's say that $$A$$ and $$B$$ are leakily isomorphic if such a $$Z$$ exists.

If $$A$$ and $$B$$ are leakily isomorphic then they not isomorphic, but in some sense they 'almost' are, since they can be transformed into one another with the aid of $$Z$$, without transforming $$Z$$ in the process. This is closely analagous to a catayltic reaction $$A + Z \leftrightharpoons B + Z$$ in a chemical reaction network, where the catalyst $$Z$$ allows $$A$$ and $$B$$ to be interconverted without itself being altered by the reaction.

As an example, consider a category where the objects are solid shapes in $$n$$-dimensional spaces, the morphisms are rotations and translations, and the monoidal product is a Cartesian product type of operation that makes the dimensions of its components orthogonal to each other. Then let $$L$$ and $$J$$ be the L-shaped and J-shaped Tetris pieces in 2 dimensions, and let $$I$$ be the unit interval in 1 dimension. Then $$J$$ and $$L$$ are not isomorphic, but by composing $$J\otimes I$$ one gets a 3-dimensional object that can be rotated to obtain $$L\otimes I$$, and we have $$J\otimes I \simeq L\otimes I$$. As a diagram:

This shows that while the "J" and "L" shapes are not isomorphic they are nevertheless closely related in this category, since they can be interconverted by using the unit interval as a 'catalyst'.

In chemistry one can never interconvert $$A$$ and $$B$$, even with the aid of a catalyst, unless they are made of the same stuff. So I would expect that two objects related by this kind of "leaky" isomorphism will in general be closely related in some way, though not isomorphic. I would like to understand what properties this notion of relatedness will have in general.

As an obvious generalisation, one could consider morphisms in general rather than isomorphisms. One could also consider cases like $$A\otimes R \to B$$ (which could be interpreted as "$$A$$ can be converted into $$B$$ at the expense of using up a resource $$R$$") or $$A \to B \otimes W$$ ("$$A$$ can be converted into $$B$$ at the cost of creating a 'waste product' $$W$$"). These remind me of how irreversible computations can be embedded inside reversible ones, at the cost of needing to dispose of waste entropy.

One could also have other interesting variants, such as $$A\otimes A\to B\otimes B$$ ($$A$$ can be transformed into $$B$$ as long as you have two instances of it) or even $$A \otimes B \to B \otimes B$$ (you can convert as many $$A$$'s into $$B$$'s as you want, as long as you have at least one $$B$$ to start the process, this being analogous to chain reactions and biological growth).

My question is just whether this set of ideas have been developed and given names in the context of monoidal categories. It might be that I can learn something about chemistry, as well as about category theory, by following up on this idea. It would be interesting to know if these ideas have any use outside the context of modelling chemistry-like physical processes.

To expand on the question a bit: NDewolf points out in an answer that a special case of this concept has been called "stable isomorphism". This gives me a name, but somehow doesn't give me the kind of generality I was hoping for - I was hoping to get some kind of general category-theoretic insight from the idea. (The specific application to rings and modules is too far outside my expertise for me to easily appreciate it.) Above I give the example of reflections, which arise as a "leaky"/"stable" isomorphism in a category of rotations. There must be plenty of other cases where we can define one type of relationship between objects, together with a monoidal operation, and obtain another, closely related type of relationship via the "leaky isomprphism" concept and its relatives as discussed above. I'm hoping ideally for some more examples of this kind of thing, along with, hopefully, a better understanding of the implications of the concept.

• arxiv.org/abs/1409.5531 – Oscar Cunningham Feb 23 '20 at 20:21
• @OscarCunningham Thanks, that's very relevant, and does seem to define the same concept. – Nathaniel Feb 23 '20 at 20:59

I think the notion you are looking for is a "stable isomorphism". This notion is more prevalent in the context of direct sums (for modules, vector bundles, ...) but it can as well be defined for tensor products.

• +1, this is helpful. I had a quick look, and the way it's defined, e.g. here it seems to be a specific instance of this concept, with $A$, $B$ and $Z$ being specific types of object. Do you know of somewhere where it's developed as a general concept in the theory of (symmetric) monoidal categories? – Nathaniel Feb 21 '20 at 10:23
• I'm not sure if there is some general construction. I think it is more of a naming convention for this kind of operation (which possibly originated in the study of K-theory). – NDewolf Feb 21 '20 at 12:13