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This is a follow-up to this question. There I asked for an intuition of what "structure preserving" means.

My question here is, is there a universally applicable method (given the objects that define a structure, and axioms defined on those objects) to find conditions on a map that makes it structure preserving (i.e. a homomorphism)?

For example,

  • monoid homomorphism. A map $f$ on monoids is a monoid homomorphism if $f(a\cdot b)=f(a)\cdot f(b)$ and $f(e)=e$.

  • topological homomorphism (homeomorphism). For any open set $U$, $f^{-1}(U)$ is also an open set.

While it is clear that these definitions are structure preserving if you first think hard about what monoids and topologies are, it is not clear to me how we could have derived that these are the correct definitions for the structures' homomorphisms, based on a universally applicable method.

user56834
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  • It's of course to some extent subjective. You might be interested in: https://en.wikipedia.org/wiki/Structure_(mathematical_logic)#Homomorphisms_and_embeddings – Rafi Feb 01 '19 at 18:42
  • @RafayAshary, are there exceptions to that definition from mathematical logic? – user56834 Feb 01 '19 at 18:59
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    @user56834 Yes, topological spaces are an exception. They don't fit into the framework of "function preserving some functions and relations" without an awful lot of squeezing. But there are basically no exceptions from algebra, which is where "structure-preserving map" is usually referring. It's arguable whether that's even an accurate way to think about continuous maps. In particular, there are multiple useful kinds of morphisms between topological spaces other than the continuous maps. – Kevin Carlson Feb 01 '19 at 20:15

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I expect there will be people who will even the following very general view on what amounts to a "conditions on a map that makes it structure preserving":

Given a definition of thing deemed the "structure" (in your example, those are groups and topologies), if you now consider a class that holds all instances of this kind as objects, then whenever you can extend this to a category with at least 3 arrows connecting 3 different objects within it, then these arrows are a "structure preserving maps".

By the axioms required of a category, those arrows here fulfil some algebraic relation rending one of the arrows $h$ expressible as $g\circ f$. As such, your object take on the character of an algebraic structure.

I see this as one "universally applicable method". Although addmittedly, for mere cardinals as object, this is quite a void notion of structure preservation.

As a side note, there's a very great 400 page history study of "structure" called Modern-Algebra-Rise-Mathematical-Structures by Leo Corry, covering "Algebra" between Galois and Grothendieck.

Here, since this covers over a century, you can see that "Algebra" as a subject is subject to evolution. And its conceptualizations were always subject to revision as well.

Nikolaj-K
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  • "then whenever you can extend this to a category with at least 3 arrows connecting 3 different objects within it, then these arrows are a "structure preserving maps"." I don't understand why you say this? why would 3 arrows imply structure preserving? e.g. in the case of topologies, the three arrows could be arbitrary non-continuous functions. – user56834 Feb 03 '19 at 19:06
  • @user56834 Note that I wrote "at least 3 arrows connecting 3 different objects. If you have an object $B$ and an arrow $f$ going into it, and another arrow $g$ going out of it, then by the laws that define a category, there is another arrow $g\circ f$, which we may call $h$. The arrow $h$ doesn't touch the object $B$ at all, but as $f$ followed by $g$ is the same thing as $h$, there's certain aspect of $h$ that we know can be found in $B$. – Nikolaj-K Feb 03 '19 at 23:08
  • Here, if $B$ holds any "structure", then some information about it is within the morphisms. Patches of the space, maybe. You may now ask where the "preservation" aspect is, but then I ask why the homomorphisms mapping to the trivial group would be "preserving" of anything either. You may also restrict yourself to the invertible arrows. And yeah, even if you fix the objects of a category to be topological spaces, this doesn't imply that the arrows need to be the continuous functions. The category with homotpies as arrows also have the same topological spaces as class of objects. – Nikolaj-K Feb 03 '19 at 23:15