Did you get the joke in the picture to the left? If not, you will do so in a few minutes. I was recently reading an article and I came across the terms mentioned in the title. From the looks of it, they are very close to each other, right? In many fields within mathematics, we talk about objects and the maps between them. Now you may ask why we would want to do that? Well, transformation is one of the most fundamental things in any field. For example, how do we transform a line into a circle, or fuel into mechanical energy, or words into numbers? There are infinitely many types of transformations that can exist. Obviously, we cannot account for every single type of transformation that can possibly exist. So we limit ourselves to only the interesting ones. So what exactly is it all about? How does it even relate to the title of this blog post?
First of all, what is a “morphism”?
In mathematics, a morphism is basically a structure preserving map. Okay now what is this “structure” and why do we need to preserve it? To get a perspective on this, let’s take a step back and talk about set theory. Everybody is familiar with set theory and everybody understands the basics here, right? A set is a collection of distinct objects, and set theory aims to study the properties of these sets. Sets need to follow certain rules, and that’s why we call them sets. For example, a set cannot have two elements that are exactly the same. If that happens, then it ceases to be a set.
In set theory, we have something called “functions”. These functions are the building blocks of set theory and they transform one set into another. For example, let’s consider the set of natural numbers N. Let’s define a function ‘f’ which takes each value and computes the square of that number. Here, we can see that every single squared value will belong also to the set N. Here, we say that the function ‘f’ maps the set N to N while preserving the structure. All the rules hold true! This is an example of a morphism. In the field of set theory, a morphism is just a function. Morphisms are actually more general and they appear in various forms in different fields. We can say that all functions in set theory are morphisms, but not all morphisms are functions.
What is a homomorphism?
The term “homomorphism” applies to structure-preserving maps in some domains of mathematics, but not others. So technically, homomorphisms are just morphisms in algebra, discrete mathematics, groups, rings, graphs, and lattices. A structure-preserving map between two groups is a map that preserves the group operation. To give a fairly rudimentary example, let’s consider a group G with a some numbers. In G, if you take any two numbers and add them, the resulting number also belongs to G. It’s nice and contained! Now if you apply a map to this group and transform into something else, the resultant group should also obey the addition property we just talked about. This is called preserving the group operation in this particular case. The concept of groups in mathematics is way more involved than what we just discussed. This is just to give you an idea of what we are talking about.
Now if such a map that is also a bijection, it is called an isomorphism. Bijection is like a two-way street i.e. you get to go back and forth between two groups. Those maps typically preserve whatever structure the objects carry. Mathematically, we can express it in the following manner:
f(A . B) = f(A) . f(B)
What this means is that applying a function to the result of the operator ‘.’ will be the same as the result of applying the function individually and then applying the operator. To draw a simple analogy, let’s say A and B are 3 and 4 respectively, and the operation is “addition”. Now let’s define ‘f’ to be a function that multiplies a given value by 2. Let’s compute the values:
f(3 . 4) = f(3 + 4) = f(7) = 14 f(3) . f(4) = 6 + 8 = 14
As we can see here, the value remains the same. This holds true for all values. Hence we can say that the structure is preserved.
What is a homeomorphism?
When we talk about homeomorphisms, we talk about continuous deformations that leave a geometric structure intact. This term is used almost exclusively in the field of mathematical topology. To give an example, we can think of a topological space as a geometric object. Homeomorphism refers to continuous stretching and bending of the object into a new shape. If we take an ellipse, we can continuously stretch and bend it to make it look like a rectangle. We don’t have to break the shape in any way. Hence we say that an ellipse and a rectangle are homeomorphic to each other.
On the other hand, you cannot change a rectangle to make it look like a line (straight or curved) without breaking it. Hence we say that a rectangle and a line are not homeomorphic to each other. So in our situation here, a structure-preserving map between two topological spaces is something that sends points that are close to one another in the first space to points that are close to one another in the second space. Now if the same map has an inverse that can send the points in the second space back to first (basically a bijection), then we say that it is a homeomorphism.
What’s the difference and how are these terms related to isomorphism?
In case you didn’t get the coffee cup and doughnut joke earlier, look at this picture. It’s a popular joke in the mathematics community! Coming back to our discussion, the term isomorphism is used more broadly, because category theory applies to so many objects. A morphism basically refers to any kind of mapping, and it can occur in various scenarios. For example, a morphism between groups is a homomorphism; a morphism between topological spaces is a continuous mapping; a morphism between vector spaces is a linear mapping; a morphism between sets is a bijection, and so on. In each category, an isomorphism is a morphism with a two-sided inverse. So technically, we can say that homeomorphisms are isomorphisms in the category of topological spaces, while homomorphisms are morphisms in the category of groups. Homeomorphisms are a special type of continuous maps, and homomorphisms are not continuous maps.
Isomorphism is a specific type of homomorphism. It refers to a homomorphism which happens to be invertible and whose inverse is itself a homomorphism. For example, a map taking all the elements from one group to the unit element of some other group is a perfectly legitimate homomorphism, but it’s very far from being an isomorphism.