What is a “structure-preserving” map?

• "X does something whilst preserving a certain structure"
• "There exist deformations of Y that preserve certain properties"
• "∃ function ƒ such that P, whilst respecting Q"

This common mathematical turn of phrase sounds vague, even when the speaker has something quite clear in mind.

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Smeet Bhatt brought up this unclarity in a recent question on Quora. Following is my answer:

It depends on the category. The idea of isomorphism varies across categories. It’s like if I ask you if two things are “similar” or not.

Think about a children’s puzzle where they are shown wooden blocks in a variety of shapes & colours. All the blocks that have the same shape are `shape-isomorphic`. All the blocks of the same colour are `colour-isomorphic`. All the blocks are wooden so they’re `material-isomorphic`.

In common mathematical abstractions, you might want to preserve a property like

after some transformation `φ`. It’s the same idea: "The same in what way?"

As John Baez & James Dolan pointed out, when we say two things are "equal", we usually don’t mean they are literally the same. `x=x` is the most useless expression in mathematics, whereas more interesting formulæ express an isomorphism:

• Something is the same about the LHS and RHS”.
• "They are similar in the following sense".

Just what the something is that’s the same, is the structure to be preserved.

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A related idea is that of equivalence-class. If I make an equivalence class of all sets with cardinality 4, I’m talking about “their size is equivalent”.

Of course the set  is quite different to the set  in other respects. Again it’s about "What is the same?" and "What is different?" just like on Sesame Street.

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Two further comments: “structure” in mathematics usually refers to a tuple or a category, both of which mean “a space" in the sense that not only is there a set with objects in it, but also the space or tuple or category has mappings relating the things together or conveying information about the things. For example a metric space is a tuple . (And: having a definition of distance implies that you also have a definition of the topology (neighbourhood relationships) and geometry (angular relationships) of the space.)

In the case of a metric space, a structure-preserving map between metric spaces would not make it impossible to speak of distance in the target space. The output should still fulfill the metric-space criteria: distance should still be a meaningful thing to talk about after the mapping is done.

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I’ve got a couple drafts in my 1500-long queue of drafts expositing some more on this topic. If I’m not too lazy then at some point in the future I’ll share some drawings of structure-preserving maps (different “samenesses”) such as the ones Daniel McLaury mentioned, also on Quora:

• Category: Structure-preserving mapsInvertible, structure-preserving maps

• Groups: (group) homomorphism; (group) isomorphism
• Rings: (ring) homomorphism; (ring) isomorphism
• Vector Spaces: linear transformation, invertible linear transformation
• Topological Spaces: continuous map; homeomorphism
• Differentiable Manifolds: differentiable map; diffeomorphism
• Riemannian Manifolds: conformal map; conformal isometry

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