A pair of mathematical structures are isomorphic to each other if they are “essentially the same”, even if they aren’t necessarily equal.
An isomorphism is a morphism between isomorphic structures which translates one to the other in a way that preserves all the relevant structure. An important property of an isomorphism is that it can be ‘undone’ by its inverse isomorphism.
An isomorphism from an object to itself is called an automorphism. They can be thought of as symmetries: different ways in which an object can be mapped onto itself without changing it.
Equality and Identity
The simplest isomorphism is equality: if two things are equal then they are actually the same thing (and so not actually two things at all). Anything is obviously indistinguishable from itself under whatever measure you might use (it has any property in common with itself) and so regardless of the theory or language, anything is isomorphic to itself. This is represented by the identity (iso)morphism.
Group Isomorphisms
For a more technical example, the theory of groups only talks about the way that elements are combined via group operation. The theory does not care in what order elements are put, or what they are labelled or even what they are. Hence, if you are using the language and theory of groups, you want to say two groups are essentially indistinguishable if you can pair up the elements such that their group operations act the same way.
Isomorphisms in Category Theory
In category theory, an isomorphism is a morphism which has a two-sided inverse function. That is to say, is an isomorphism if there is a morphism where and cancel each other out.
Formally, this means that both composites and are equal to identity morphisms (morphisms which ‘do nothing’ or declare an object equal to itself). That is, and .
Eric Rogstad But… but… poset office was a pun, not a typo.
I support the creation of a poset-office, but it’s gotta be about posets!
Oh, I thought it might be a pun. But nothing about the surrounding description sounded like a poset (weights are totally ordered, right?), so I figured it was a typo :P
Eric Rogstad Elmo comes to visit. Does that seem fine you think?
Why not just count explicitly?
I think the answer is, “because we want to teach what a bijection is,” but readers might be confused why we’re doing this. Maybe some of the flavor text about the Count should say that he’s not actually good at counting? :P (Though if we did that, I’d be worried about starting to be too long-winded.)
Or maybe there should just be a parenthetical saying there’s a reason for not counting explicitly, which we’ll come back later. And then we’d need to come back to that when we introduce “bijection” further down the page.
Patrick Stevens I agree completely. Along with some other pictures. However, due tomy current circumstances I can’t make any pictures at thr moment.
If someone else is willing to, I would be very grateful. Otherwise I could probably do it in about a month? Month and a half?
I think this probably wants a diagram of the two graphs, being differently laid out in the plane but isomorphic.
Eric Rogstad The post has been updated with an isomorphic version of what you suggested. Thanks!
Joke stolen shamelessly from the latest post on slatestarcodex.com
I might try to introduce these terms one at a time, and a bit more slowly—the paragraph up to this point reads like Simple English (good!), and then in the last two sentences I’ve got two terms thrown at me.
I think if a reader doesn’t yet know what an isomorphism is, it would be helpful to spend more time building the intuition that there’s something the same about both boxes, maybe like this:
What do you think?
Note that two sets have to have the same number of elements to be bijective, but that’s not enough — you also need some way to say which item in one should be paired with which item in the other. In the case above, we paired the items up using the order in which they were removed from their boxes.
I’d like to add some pictures to this page at some point, but due to current circumstances I can’t for now. If anyone wants to add pics (say different station maps with the same connections, two ‘boxes’ with random items) please feel welcome.
I also think I’ll change the names of the stations from a, b etc. to funny made up station names.
The majority of this page will probably end up in the least technical lens.
Eric Bruylant Thank you very much! just to be clear, are you talking about the ‘clickbait’, the intro paragraph in the text itself, or both?
Feel free to suggest / make your own changes if you have anything specific in mind by the way.
The intro paragraph, the clickbait seems fine.
This is a great page! I think the intro/summary could be made a little more accessible though? The use case I’m thinking of is a person who wants a brief overview in relatively non-technical language, which is valuable for the popups from links to here.
Patrick Stevens Yeah I’ve been wondering about the convention of things like this. I’ve been calling my pages things like category_mathematics.
“identity” is probably not a sufficiently specific link; I’d go for math_identity, probably.