Morphism

A morphism is the abstract representation of a relation between mathematical objects.

Usually, it is used to refer to functions mapping element of one set to another, but it may represent a more general notion of a relation in category theory.

Isomorphisms

To understand a morphism, it is easier to first understand the concept of an isomorphism. Two mathematical structures (say two groups) are called isomorphic if they are indistinguishable using the information of the language and theory under consideration.

Imagine you are the Count von Count. You care only about counting things. You don’t care what it is you count, you just care how many there are. You decide that you want to collect objects you count into boxes, and you consider two boxes equal if there are the same number of elements in both boxes. How do you know if two boxes have the same number of elements? You pair them up and see if there are any left over in either box. If there aren’t any left over, then the boxes are “bijective” and the way that you paired them up is a bijection. A bijection is a simple form of an isomorphism and the boxes are said to be isomorphic.

For example, the theory of groups only talks about the way that elements are combined via group operation (and whether they are the identity or inverses, but that information is already given by the information of how elements are combined under the group operation (hereafter called multiplication). 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 their multiplication acts the same way.