Most general definition of homomorphism and isomorphism

Question

What is the most general definition of homomorphism and isomorphism? It is clear what they mean when there is an algebraic structure to be preserved, but what about when there is no such structure? Are homomorphism and isomorphism still relevant concepts?

For example, the statement "the real numbers are isomorphic to the equivalence class of Cauchy sequences in the completion of the rational numbers" means what exactly? Does such a statement presuppose the normal definitions of + and * on the reals and on Cauchy sequences?

In other words, are homomorphism and isomorphism very general concepts that can be used without specifying the operations, or not?

Answer 1

The most general definition (and the one I prefer) is the pure logic definition of homomorphism between structures; once this is well understood, it can be applied to any more specific topic like topology and analysis.

A structure is a set containing several sets that gives context to logical formulas:

1. A domain of discourse. This is a set of all the elements belonging to the structure. If we are talking about a group, this is the set of all elements in the group.
2. A signature. This is a set of the symbols used in formulas. For example, groups can have the signature $\langle G, \cdot, e \rangle$. Note, however, that signatures only provide the syntax. Technically, we also need to specify that $\cdot$ is a function from $G \times G \to G$ and $e$ is a constant symbol.
3. An interpretation of the signature. This is where we assign meaning to the symbols in the signature. We have to specify what set $G$ is, which function $ \cdot $ is, and which element of $G$ the symbol $e$ represents. In our group example, we can choose $G$ to be $\mathbb{Z}$, $\cdot$ is the addition operation, and $e$ can be the number $0$.

When a structure satisfies a set of axioms, we call the structure a model. Because our signature is the signature of group theory and our interpretation satisfies the axioms of group theory, we say that our structure is a group (a model of group theory).

Thus when we talk about a homomorphism between two structures of the same signature, we simply mean a function from one domain of discourse to another that preserves every single relation and function in the signature. An embedding is a homomorphism that is also an injection. An isomorphism is a homomorphism that is also a bijection.

In group theory, there is only one relation in the structure, so homomorphisms only have to preserve the group operation. Between rings, there are now two operations that must be preserved. Between structures of a signature with one billion relations, all one billion relations must be preserved under the homomorphism.

It happens to be the case that when a homomorphism exists from a group $G$ to another structure $H$, $H$ is necessarily a group as well. This is a theorem. But it is not generally the case: if structure $A$ is a model of theory $T$, and $f$ is a homomorphism from $A \to B$, $B$ is not necessarily a model of theory $T$. However, if $f$ is an isomorphism, then $B$ is necessarily a model of theory $T$. In fact, every first order logic sentence that is true in $A$ must be true in $B$ as well (this the meaning behind the phrase 'up to isomorphism').

As you can see, the terms isomorphism and homomorphism are meaningless without specifying the structures in question, however many if not most statements are made without explicitly mentioning the operations because they are assumed by the context. For example, when we say that "the real numbers are isomorphic to the equivalence class of Cauchy sequences in the completion of the rational numbers", we are implicitly talking about the field of real numbers. If you want to talk about homomorphisms between less canonical structures, you need to be more explicit to be understood.

Answer 2

Another candidate for the most general definition is that a homomorphism is just a map between objects of any category, and an isomorphism an invertible map. This is enormously abstract and goes beyond algebra to encompass essentially every type of mathematical object. For that reason, the logic-based answer might be better for your purposes at this point-but in particular for isomorphism the categorical notion is conceptually quite valuable.