What is the difference between Mapping and Morphism I wonder if there's differences between Mapping and Morphism. Although the terms are used in different context i.e. mapping for set theory and morphism for category theory, from my understanding they are both used to describe the relationship among objects, so what's the difference?
 A: Let $G$ and $H$ simply be sets filled with numbers. We can easily define mappings between $G$ and $H$. Let's say that $f : G \to H$ is given by $f(g) = g * g = g^2$. We could also say that $f(g) = g^3$ or $f(g) = \frac{57}{g}$. These are all "mappings". A "mapping" is simply a rule that one uses to take an element of $G$ and "mess around" with it to get something out. Imagine a machine, that has an input and an output. Nothing is particularly special about a mapping, it is simply a rule as stated above.
A $\textit{Morphism}$ is a much more interesting kind of map. Let's endow $G$ and $H$ with group structure and call them $(G, *)$ and $(H, \diamond)$. Now we ask the question: can we look at some mapping, $\phi$ say, that $\underline{\textbf{preserves}}$ the group structure? That is to say, if $\phi : G \to H$, then to be a $\textit{homomorphism}$ we require that for $g, h \in G$
$$\phi(g*h) = \phi(g) \diamond \phi(h).$$
Now, several properties of the structure follow from this. If $e_G$ is the identity in $G$ then $\phi(e_G) = e_H$ in $H$. In English, we can state that via a group homomorphism, identities map to identities. Other properties of the structure are also preserved under the morphism.
There are several morphisms that have different requirements depending on your context. That is to say, Ring Homomorphisms differ from Group Homomorphisms, Linear Transformations are actually the morphisms of the category of Vector Spaces, the morphisms of topological spaces are continuous maps etc. etc.
LET IT BE SAID, however, that the terms "mapping" and "function" are often used interchangeably by different people and the description given above is that of my own usage of the term.
A: A morphism is a concept introduced in the language of
categories to designate one element of the set  Hom (X, Y) where X and Y are two objects of said category.
So if we talk about the category of sets, a morphism is just a mapping.
if we talk about category of groups, a morphism is a group homomorphism ie mapping that complies with the laws of the groups in question.
Morphism of ring is ring homomorphism.
The morphism of topological space  is the  mapping that preserves the topology of concidered space,  ie  continuous mapping, areas so on ..
