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?

• Morphism is "more" than just a mapping, see here. – Dietrich Burde May 22 '16 at 15:49
• @dietrich Could you elaborate? Thanks. – zoyb May 22 '16 at 15:51
• A morphism is a mapping that preserves some kind of structure inherent to the objects between which you map. – ÍgjøgnumMeg May 22 '16 at 15:51
• @ed to use a metaphor in real(or physical) world to illustrate my understanding of your comment: Mapping emphasized on the process, like how to process a apple into apple juice, the squeeze process is mapping; while morphism emphasized on the objects, like all relationship or differences between apple and apple juice. Is this the sense you want to make? – zoyb May 22 '16 at 16:02

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.

• Thanks first. I'm still confusing about the concept of group structure. Could you help me by explaining what is group structure, and could you explain what does ϕ(g∗h)=ϕ(g)⋄ϕ(h). mean? – zoyb May 23 '16 at 4:12
• A group is an algebraic structure often used to encode and study symmetries of objects, among other things. I suggest learning about groups before trying to understand morphisms as it will give you a better understanding of why a morphism is slightly different from a mapping – ÍgjøgnumMeg May 23 '16 at 12:25

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 ..