It's another long, boring math post. If you are afraid of math I suggest you drop everything immediately, run in the opposite direction as fast as you can, cover your ears, and shout "LALALALA" as loud as you can while ignoring the funny looks everyone is giving you as a lot of abstract math is up ahead. Or you could just ignore this post, it's your choice. Disclaimer that I probably should of attached with my other math posts: Ok, I'm really just making up words as I go along to describe concepts for a theory I've thought up. I'm not just sticking to established theory. Chances are I will misuse official terminology, unintentionally describe theory that has already been discovered, or say nonsense. First things first, before I explain what morphisms are I'm going to explain what sets are. A set is, like everything else described in these posts, a mathematical object. They represent a collection of other ( some would say one could include itself, but that brings up so logical problems a la Russell's Paradox, unless that old dead white dude wasn't named Russell) objects where every object in existence either belongs to the set, or it does not. These objects that belong to the set are called the elements of the set. Thor total number of elements of a set is referred to the cardinality of the set, and this number is often infinite. Anyway, the important part about sets is they have certain objects in them, and each object can only be in an object once or not at all as sets simply measure whether or not objects belong or not. Now, if all the objects in set A also belong to set B, than set A is considered to be a subset of set B. Set B may be a subset of set A as well, but only if the two sets are identical to each-other, meaning they are really the same set. That's all there is to a set. Now, there is another type of objects called categories. These are thought of as types of objects, so really a category is simply a glorified set, so I guess technically you could arbitrary define any set as a category, but generally you can reason what sets would logically work as category. The set of all sets, assuming such a thing can even exist (see Russell's paradox), would make a good category. Other categories include numbers, matrices, and morphisms, which I will discuss in a moment. Generally categories are not thought of as being sets after they are declared to be categories. As sets can have subsets, categories can have subcategories, where I guess all categories are a subcategory of objects. Now for the important part. Morphisms are a broad category, a very broad and important category which I consider to be the basis of all mathematics, as I believe all other objects can be created with morphisms. Now what is a morphism? A morphism is an object that transforms a bunch of other objects into another object. The objects beings transformed are the input, or arguments, and the object created is the output. The number of objects a morphism transforms at a time, or the number of arguments of the morphism is called it's artity or something like that. Now this is probably pretty confusing, so I well give an example. + represents addition, which is a binary morphism. Binary means it has an artity of two, which means it takes two objects, which happen to both be numbers, and addition outputs a number which is an object, which also happens to be a number. The first two numbers are addition's arguments. In the expression 2+3 2 is the first argument of addition here, 3 is the second, and 5 is the output. Generally, a morphism is represented with a symbol, for example f, its arguments are represented with other symbols, for example $ and ;, and the morphism transforming the objects is represented by first placing the symbol, then placing an opening parenthese [ ( ], followed by all the arguments in order from first argument to last with commas [ , ] between them, and finished with a closing parenthese [ ) ], though they are often denoted with other notations. If we represent addition as f, two as $, and three as ;, then using this format f($,;) would represent two plus three, or we could use the normal symbols so +(2,3) is two plus three, or we could just say 2+3 like a normal person. When f($,;) is used it no longer just denotes a morphism and some objects, it denotes the output, which here is five. Keep in mind that this is not the only notation, and if you cleverly define objects you can cleverly define morphisms to prove all sorts of things. The simplest type of of morphism is the unary morphism ( well, technically it's the nonary morphism, with artity 0, but those are pointless as they just immediately transform to another object), which has one argument. This means it takes an object, and transforms it to another object. Usually transformations using them/ their output are denoted f(x), the notation mentioned in the last paragraph is used far more often in unary functions than binary operations. Now unary morphisms have some special properties. The first is that they have a set associated with them called it's domain. The domain is a set of all values were the morphism is defined. So f(x) denotes some output if x belongs to f's domain, otherwise the expression is meaningless. Likewise, it has another associated set called it's domain. This is a set of all possible outputs of the morphism. It should be noted when a unary morphism's range is it's domain, and also when it is true that for every object on the morphism's range there is exactly one object in the domain which will transform into the first object when the morphism is applied. If a unary morphism has those noted properties than it will have interesting resulting properties. The other important property of unary morphisms is that every morphism with a finite artity can be constructed from unary morphisms, which I will now prove through mathematical induction. First, we can represent a binary morphism with a pseudo-binary morphism which is actually a unary morphism. The domain of this morphism is all objects which are valid for the first argument for the binary morphism, while it's range is a set of unary morphisms. This pseudo-binary operation is denoted f2, and it's defined so that f2(x2)(x1) = f(x2,x1). The process used to create f2 and reduce f's artity by one after we input a number is called currying Now we can define any fn for any n so fn(xn) = f(n-1) for any j-nary morphism and any f1 so f1(x1) = f(xj,x[j-1],...,x1), proving the validity of the process for any operation with artity j so fj(xj)(x[j-1])...(x1) = f(xj,x[j-1],...,x1). The last thing I'd like to discuss for now is operations. Operations are morphisms that have been bound to a category, and all of it's arguments and it's output are elements of that category. For example, addition and multiplication are both binary operations of numbers. There are a few operations I'd like to discuss. The first is union, and it's an operation of sets. It's output is a set whose elements belong to ANY of the inputs. The second is also an operation of sets, and it's called union. It's output is a set whose elements belong to ALL of the inputs. The last operation I'd like to discuss is composition, and it's a binary operation of unary operations. Since I don't have the symbol here I'll denote it with •, and I'll use the standard notation used with addition. It's defined so (f•g)(x) = f(g(x)). It's extremely important to remember than it does NOT mean that f•g = f(g). TL;DR; Morphisms MORPH stuff, hence the name.