general definition of a transformation

Question

I've always thought of a transformation as "the movement of a mathematical object through space", but I found this definition that seems to be better: "An operation that changes the status of a (mathematical) object to another state." Does anyone have a better definition that works for both dilations and isometries and also with more advanced math like linear transformations? Or, what would be the best way to combine these definitions? Is the "space" part important?

Answer 1

To have a general definition of a transformation, one needs to formulate the general definition of a mathematical object. Of course, most of the mathematical objects that a typical mathematician studies are just sets, with some "structure" and there is no precise definition of a structure. It can be a function (say a binary operation), a binary relation (say an order), a set (say a topology) etc. After putting a "structure", one wishes to have some axioms which list out the basic properties that a structure possesess. Having defined an object, one usually defines a transformation as a mapping of one object into another (of the same species) that respects/ is compatible with the structure.

But trying to define what a structure is, is really difficult. This issue is (sort of) circumvented by Category Theory, which serves as a unifying theory for Mathematics. A mathematical object (a group, a topological space, a scheme, a manifold, a partially ordered set, etc) for a Category Theorist is just a blob (dot/point). What Category Theory is interested in are the transformations between like-objects and these transformations are thought of as arrows from one object to another. So, a transformation is a primitive notion in category theory with one property, namely that it has a "direction". This is the most general definition of a transformation. A transformation has a domain and a codomain and you can compose transformations just as you compose maps.

Coming back to the case where mathematical objects are sets with some "structure", there's also a name for categories whose objects are actually sets with some structure. Such categories are called Concrete categories. A concrete category is one which is equipped with a faithful functor into the category of Sets (and maps). You can think of the faithful functor as a transformation of your concrete category into the category of sets (mapping objects to their underlying sets and arrows to set functions, keeping all the "category structure" intact) and such that distinct arrows from object A to object B are mapped onto distinct functions of the underlying sets.

Now, of course, this still doesn't address how would one go on to define a transformation from one mathematical object to another (with the same structure) in a way that would work for all instances of structure (say a vector space, group, topological space etc). But then again, for that you would need a precise definition of a "structure" in full generality.

At least, in the context when the structure is a collection of operations, one usually defines a transformation from one object to another as a map which preserves the ambient structure. Say, an algebraic structure is a set with two finitary operations (one of arity m and one of arity n). Take two such structures $(A, ._1,._2)$ and $(B, *_1, *_2)$. A transformation of $(A,._1,._2)$ into $(B, *_1,*_2)$ is a mapping $f : A \to B$ that makes the two diagrams commute :

$$\require{AMScd} \begin{CD} A^m @>{._1}>> A\\ @V{f \times f \times ... f}VV @V{f}VV \\ B @>{*_1}>> B \end{CD}$$

$$\require{AMScd} \begin{CD} A^n @>{._2}>> A\\ @V{f \times f \times ... f}VV @V{f}VV \\ B^n @>{*_2}>> B \end{CD}$$

On elements, this amounts to saying that

1. $f(._1(a_1, a_2, .... ,a_m)) = *_1(f(a_1), f(a_2),..,f(a_m))$.

2. $f(._2(a_1, a_2,..., a_n)) = *_2(f(a_1), f(a_2),...,f(a_n))$.

for all $a_i \in A$.