Why do we say morphisms preserve the structure/operations?

[Tosh5457]

What is exactly is a structure? Is a ring a structure?
And what does that mean exactly, that morphisms preserve the structure and operations?

 

[Stephen Tashi]

What is exactly is a structure? Is a ring a structure?
And what does that mean exactly, that morphisms preserve the structure and operations?

I think you are asking about a metphorical description of category theory - i.e. a poetic description of moprhisms in non technical language. So "structure" may not have an exact mathematical definition in that context.

As to whether a ring "is a structure" or "has structure", I don't think "structure" has a precise definition in that context either. (There is the further question of whether "a ring" means a specific ring or whether it only means the abstraction defined by the axioms of a ring.)

A page with interesting links about category theory is http://www.j-paine.org/make_category_theory_intuitive.html. However, I can't say any of those links have turned on a light bulb for me yet. Perhaps you can explain some of them to me!

 

[lurflurf]

A morphism is like a renaming. Morphisms preserve the structure means if we go though and change all the names the statements should remain true.

 

[lavinia]

To me srtucture is any mathematical property such as being an abelian group or a smooth manifold. Structure is preserved by a map if F(structure) = structure(F) by which is meant things like

F(a + b) = f(a) + f(b) or is f is a smooth map then so is foF.

A morphism may not be an isomorphism i.e. may not have a structure preserving inverse so I would not think of it as a renaming, For instance the morphism that maps an abelian group to the trivial group is not a renaming.

One can formalize the idea of a morphism by defining an abstract category which has objects and morphisms between them. One has the category whose objects are abelian groups and whose morphisms are group homomorphisms or the category of smooth manifolds and smooth maps or the category of commutative rings and ring homomorphisms.

 

[Vargo]

structure = salient properties of object under study in the context of the subject at hand. http://en.wikipedia.org/wiki/Mathematical_structure

morphism is a word from category theory which is a framework for what you are asking.

simple examples:

Euclidean geometry: Two figures are equivalent if you can map one onto the other via a similarity. In the context of Euclidean geometry it does not matter what scale my drawings are or their exact location or orientation in space.

Topology: Morphisms are continuous maps. i.e. maps that preserve "closeness" in a certain sense.

Algebra: Morphisms preserve the operations. e.g. T(ax+by)=aTx+bTy for linear transformations.