Where we learn that following the direction
of the arrow is probably a good idea.
A potpourri of morphisms was my first tattoo and it’s all because of their arrows. Arrows have been my doodle of choice for as long as I can remember and I was itching to get my first tattoo; I was already midway through my 30’s, quite overdue for this so-called rebellious act.
For many years I had wanted, though had not gotten, a tattoo. This was simply because I was taking the whole permanent inking on my body process very seriously, so research upon research upon folders of ideas – and tattoo artists – had piled up.
Then I was flipping through my trusty HarperCollins Math Dictionary (c. 1991) – as one does when they’re wanting to be visually inspired 🤓 – and the diagrams of different morphisms lifted up off the page. I realized I was looking at Commutative Diagrams and hullo, could there be a more perfect thing for me to become enamored with?
Commutative Diagrams are visual equations. What a delightful alternative way to be introduced to the beauty of mathematics.
A diagram in a category is a collection of objects and arrows from the category, [though] possibly not all of them. A diagram is said to commute if any two composable strings with the same endpoints produce the same composite.“The Joy of Abstraction: An Exploration of Math, Category Theory, and Life”
by Eugenia Cheng (pg. 107)
When employing even the most standard and basic symbols and shapes, like arrows & squares, to me it looked like they were visual representations suggesting movement and action in a defined area, maybe even a contained space.
The icing on the morphism cake was finding out that in particular conditions, those objects within a category may be known as branes – and when that is the case that means that those objects/branes are then setting boundaries on that category and the actions of the relations that are exemplified by the morphisms. They are defining the constraints (cuz there’s always constraints) put upon that particular relationship.
This diagram is a mathematical structure. It represents a category, could be a group of some sort, where the letters represent objects and my dear sweet childhood doodles aka arrows are the morphisms that are not only connecting those objects – but they are the pathways that bind this structure-preserving map.
The morphism spaces between objects should be thought of as some kind of relational data. Morphisms themselves interact with one another, as they can be composed when the end object of one morphism is the start object of another.“The Princeton Companion to Mathematics“
The composition is associative,
so whether you compute abc as (ab)c or a(bc) does not matter.
Ok, seriously? The morphisms are a representation of relationships? Yup. So when there are arrows lining up with other arrows, that relationship they’re conveying is like a trail of breadcrumbs with the first one leading to the next, moving from the source to the target and so on, keeping in order as they carry to & leave some of their influence on the next one in succession. They sorta kinda morph along, following the movement of one object to the next.
No surprise that there are multiple types of morphisms out there – just as there are multiple types of relationships out there. The two most basic & common types are homomorphisms and isomorphisms. That square one above that is rep’n category theory would be considered a homomorphism, where it’s a simple structure-preserving map between two structures or groups, just mapping that overall common operation of the group within the category. It’s the relationship with the person you work with, you may even work on some projects together, though you probably aren’t meeting up outside of business-related stuff.
An isomorphism, though, doesn’t simply make and define the connection from one (source) object to the next (target) object. An isomorphism signifies that the connection is a 2-way street, emphasizing that there is more than just some commonality going on between the groups. An isomorphism is a mapping function that is calling out that the elements in one group are structurally the same as the elements in this other group. This relationship would be with a close friend or with a peer company; there’s a significant identity that is shared across this relationship, including at least a few brane-defining constraints. In mathematics, isomorphisms are commonly expressed with a ϕ.
Isomorphisms are ‘tighter’: they require exact sameness. Homomorphisms are ‘looser’: they require two groups to have common structure but not (necessarily) to be structurally identical.“How to Think About Abstract Algebra” by Lara Alcock (pg. 184)
The one other type of morphism that has my attention is a diffeomorphism, which (visually) fascinated me because of its squiggle, which I’ve come to think of as a swagger of sorts. It sure seems like there would be a swagger in the midst of a catastrophe, right? Cuz that diffeomorphism below represents Catastrophe Theory and it’s fabulous, what with its ‘abrupt change’ in direction.
It’s a reminder to me that when a relationship takes an unexpected turn – like, completely goes off course and takes an entirely not-even-considered-before different direction – it’s a sudden action, one that likely came to bear too much weight (literally or figuratively) and it just had to, well, swagger. For its own survival.
Much like my fascination with chaos, though, that unexpected turn of the relational path, the abrupt turn that that arrow takes, is not unexpected. Granted – sometimes you don’t fully see what might be happening when you’re a part of a relationship that’s getting heavy and doesn’t have the strength to persevere so you may only see that point of catastrophic action after it has occurred. But it’s clear that something got built up and, probably rather impressively, that morphism – that relationship – must have shown signs of the increasing strain it was bearing. Sometimes we’re intooclose to see it. Can’t see the forest for the trees, ya know?
There’s morphic relationships all over the place IRL. Once again, it aaaalll comes back to the relationships, right?
There is no backdrop against which reality happens. An object doesn’t sit anywhere absolute in space; its position is entirely a matter of its relation to every other object out there.“Team Human” by Douglas Rushkoff (pg. 183)
Like a dance where the only space that exists is defined by and between the dancers themselves, everything is happening in relationship to everything else. It’s never over, it’s never irrelevant, it’s never somewhere else.
These shifts, these changes – these morphings – are always happening whether you are actively choosing to be a part of the interaction or not. Change is constant and you get to decide if you want to be a part of affecting that relationship and where it may be headed, especially if there happens to be some boundary conditions put upon the structural-preserving map you may find yourself a part of.
That’s why I chose to have my collection-of-morphisms tattoo emanate upward from my favorite word: GO. The word GO is all about movement, sure, but the power of it lies in how it’s used; yell it, whisper it, put it before the words ‘away’ or ‘now’, or after the words ‘just’ or ‘don’t.
Looking at these Commutative Diagrams, these Structure-Preserving Maps, my strongly held belief is that if you are wanting to actively participate – if you want to GO – just maintain awareness of the direction of the arrows. That’ll be your clue of what kind of relationship you’re engaging in and the group you’re identifying yourself with.
…sometimes you’re already doing category theory even if you don’t realize it, just like you’re sometimes doing math even if you don’t realize it.“The Joy of Abstraction: An Exploration of Math, Category Theory, and Life”
by Eugenia Cheng (pg. 107)
Thanks math, you’re the best.