There is an important concept in math called an isomorphism. This is typically introduced when you study a subject called abstract algebra, although it applies beyond abstract algebra too. While its definition generally requires some technical background, in this post we attempt to provide a non-technical introduction.
Let’s get started!
Starting With Bookshelves
Imagine you had two bookshelves. Let’s say both bookshelves are messy and disorganized, and you want to clean them up. So you go to the first bookshelf and start organizing it, say by genre. So you have separate sections for fantasy, mystery, nonfiction, and so on. You do this separately for both bookshelves, organizing each one the same way (defining the sections the same way.)
Now, both shelves would have different books. But let’s say you take a step back and realize that after organizing them, the bookshelves happen to look the exact same!
To be more specific, let’s say that for each section, whether fantasy or mystery or something else, there are an equal number of items in that section in each bookshelf.
Then, we would say that the bookshelves are isomorphic.
In general, let’s say you have two sets of data (here the books are the data) and some structure you’re placing or imposing on the data (here organization by section is the structure.) If after you impose the structure on both sets it happens to be the case that the sets “look the same under that structure,” then we would say that the two sets are isomorphic.
Let’s go back to the bookshelves, and let’s assume they are isomorphic. As we just discussed, this means that the number of books in each section is the same. Let’s concentrate on one section, say fantasy. We’ll go through each fantasy book in the first shelf and assign it to a corresponding fantasy book in the second shelf. In general, this kind of assignment, where we go through each object in a set and assign it to a corresponding object, is known in math as a function, and we say that a function sends an object to its correspondent. So with this terminology, we are making a function that sends fantasy books in the first shelf to fantasy books in the second.
Since in our case there are the same number of fantasy books, we can make this function in such a way that:
- No two books in the first shelf get sent to the same book in the second shelf. This wouldn’t be possible if there were fewer books in the second shelf. In that case, if we went through each book in the first shelf and tried to send it to a different book in the second, at some point we’d have to repeat.
- Every book in the second shelf is accounted for — in other words, every book in the second shelf has a book from the first shelf that is sent to it. This wouldn’t be possible if there were fewer books in the first shelf. In that case, if we went through each book in the first shelf and tried to cover all the books in the second shelf, there just wouldn’t be enough.
This kind of function is known in math as a bijection. So with this terminology, what we are saying is that we can form a bijection for the fantasy books. This bijection is only possible because there are an equal number of such books in each shelf. In fact, these two conditions are really the same:
- The number of books is the same.
- We can form a bijection for the books.
Building Up to the Isomorphism
Now, this function goes through books in the first shelf and assigns them something from the second, but because it is a bijection, we can actually go backwards too. If we go through each fantasy book in the second shelf, then because every second shelf book was accounted for in the bijection, there’ll definitely be a corresponding partner from the first shelf, and because no two first shelf books were sent to the same book in the second, there’ll be only one first shelf partner. So we can use the bijection to construct another function, this time from the second shelf to the first. This new function is known in math as the inverse function, or inverse for short.
We can actually then see that the inverse is itself a bijection too. Therefore, we can produce another inverse based on it. However, this “inverse of the inverse” really turns out to be just the original, so we just have two functions here that are inverses of each other.
We can now see that these three conditions are all the same:
- The number of books is the same.
- We can form a bijection from the first shelf to the second.
- We can form an inverse function from the second shelf to the first.
We have so far just formed functions for the books in fantasy, but since the number of books in each section is the same, we can in the same way make a bijection for each section. If we then put these functions together, we can make an overall bijection that sends every book in the first shelf to a book in the second. This function would also only send a book to something in the same section, so it would “respect the structure” of organizing by section. Such a function is referred to in math as structure-preserving. The inverse of this overall function would similarly be structure-preserving as well. So with this terminology, we can see that these two conditions are really the same:
- The bookshelves are isomorphic.
- We can form a structure-preserving bijection whose inverse is also structure-preserving.
We call this kind of function by the noun form of “isomorphic,” or isomorphism.
This applies to the general situation as well, where we have two sets of data and some structure on them. There, the sets being isomorphic is the same as being able to form a structure-preserving bijection whose inverse is also structure-preserving, and we call this kind of function an isomorphism.
Notice that the nature of the isomorphism depends on the structure you’re placing. For example, maybe the bookshelves are isomorphic if you organized them by genre, but not by age level (e.g., children’s versus adult.)
Also, for some structures that mathematicians study, you can have a bijection that is structure-preserving but where the inverse is not, so it is generally important to remember that the inverse must also be structure-preserving for it to be an isomorphism.
Some Implications and Applications of Isomorphisms
OK, so we know what an isomorphism is. What implications can we derive from knowing that we have two sets and an isomorphism between them?
Well, let’s say that you only really care about looking at them through the structure you’ve imposed on them — you don’t actually care about any differences between the sets that might show up beyond that structure. (For the bookshelves, we just care about the number of books in each section, not about what the books are.) Then, as far as you’re concerned, you can basically treat the sets as “the same!”
This is what mathematicians do. If they only care about a certain structure, and they are able to prove two sets to be isomorphic, then they only need to study one of them, and they immediately know that the results translate over to the other. Often times, one of the sets will be easier to study or more familiar to work with, so this is a very useful thing to be able to do. This type of application makes isomorphisms very important across math, allowing us to make connections between different subfields.
This idea, that isomorphism connotes what we can think of as “structural equality,” is so useful that mathematicians will frequently use expressions like “equal up to isomorphism.” Sometimes they even just drop “up to isomorphism” and say “equal” when they really mean “isomorphic.”
Now, we mentioned in the beginning of the page that isomorphisms are typically introduced in a subject called abstract algebra, and indeed that is one of their biggest applications. But what is this subject about?
Well, we can define abstract algebra to be the study of certain kinds of structures called algebraic structures. As we discussed, as soon as we define some kind of structure, that implies a corresponding concept of isomorphism. Thus, every algebraic structure has a corresponding concept of isomorphism. It turns out that algebraic structures can be applied to a wide variety of different kinds of sets, and often times these sets can be found to be isomorphic. Furthermore, it is very often the case that one of the sets is much easier to work with than the other. This becomes immensely useful, and it allows us to copy results and form connections that would be hard to notice otherwise between different kinds of mathematical objects. In fact, some of the most important theorems from abstract algebra are literally just statements that two seemingly unalike sets are actually isomorphic.
Of course, the reach of these applications extends far beyond math. In general, math is the foundational language of science, and science in turn underlies the engineering and building of all the technology that we have come to rely on so heavily throughout our lives. Advancements in abstract algebra and other areas of math progress through this pipeline to enable and affect everything that we interact with all the time. In fact, your ability to view this page is a consequence of a chain of innovation in areas like electronics, computer hardware, and computer security, and all these subjects depend on applications of isomorphisms in fundamentally important ways.
Going Further
This is only an introduction. If you study math further, you will eventually learn the technical definitions of algebraic structures and other structures that we study, which in turn determine how isomorphisms are defined. There is a lot of interesting theory and knowledge out there about these concepts, and we are not even scratching the surface of it here.
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Thanks to Alex and Kanika for suggestions and contributions to this.
Written August 24, 2022. Last updated January 30, 2023.
