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I have read the following:

They all seem to be attempting to scratch the same itch, and I'm unsatisfied with the answers. It seems to me that in order for someone to answer questions like "do morphisms have to be structure preserving?" we would first need to agree on what structure is.

Question

Is structure in the sense that we might say "FOO is a structure preserving morphism in the category of BAR" formalized? For example, is there consensus on a definition that generalizes the structure preserved by continuous maps in the Top category, the structure preserved by group homomorphisms in the Grp category, and the structure preserved by smooth maps in the Man category?

The wiki article for mathematical structure seems a bit vague.

user20672
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    There is no all-encompassing definition, and certainly not one that has wide acceptance. I read that Bourbaki tried to give such a definition but it was not successful. – Zhen Lin Mar 19 '21 at 03:07
  • I found the Wiki to be pretty clear. Are you familiar with formal languages? If so, a structure is a set along with interpretations for constant, predicate and function symbols found in the language. For more information, consult any book on Mathematical Logic. This will not be defined in a book on Category Theory. – John Douma Mar 19 '21 at 03:07
  • @JohnDouma I don't see constant, predicate, or function symbols mentioned anywhere in the wiki page. Do you have a recommendation for a book on Mathematical Logic? – user20672 Mar 19 '21 at 03:11
  • I learned this stuff from "A Mathematical Introduction to Logic" by Herb Enderton. Shoenfeld's "Mathematical Logic" is also very good. – John Douma Mar 19 '21 at 03:16
  • Thanks. If you ever get bored, maybe you could add a less vague definition of mathematical structure to the wiki page ;). I'm curious to know how the above-mentioned structures are described in terms of the definition you reference. – user20672 Mar 19 '21 at 03:20

1 Answers1

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An "algebraic structure" can be defined as a set together with a collection of formal logical sentences satisfied by the set. These sentences are usually called the axioms of the structure, and structure-preserving maps may be defined as those such that the sentences remain true when applied to the image set of the morphism.

I'm sure you've read about examples, such as the thin category's generalization of the poset, wherein morphisms are not structure-preserving. Such things' ubiquity is why the idea that morphisms are "structure-preserving" is little more than a pedagogical aid; the question of how morphisms act on structure is perhaps the definition of standard abstract algebra.

Working directly with such a definition of "algebraic structure" according to such a definition has a storied history; see for example universal algebra. The basic idea of category theory is to understand such algebraic structure without looking at structures themselves, only the maps between classes of structures. In my opinion, category theory is by far the best way to do such meta-analysis. It actually has predictive power, allowing one to relate wildly distant algebraic structures to one another (an example I'm familiar with is categorical quantum mechanics connecting Hilb to linear logics), whereas other approaches feel more ad-hoc. Furthermore, it is truly universal: one of the first categories one reads about is Cat, which describes the structure of category theory in terms of itself. Furthermore, some current researchers believe categories may end up being used to describe all of mathematics, even formal logic.

TL;DR: while one can can rigorously define structure, the (informal) idea behind category theory is that you don't need to.

Duncan W
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    Your first paragraph is misleading at best. An algebraic structure is not a set together with some axioms, it's a set together with some distinguished operations. Axioms are useful to specifying kinds of algebraic structures (groups, monoids, etc.) but they're not part of the data of a structure. Similarly, a structure-preserving map is one that preserves the distinguished operations, not the axioms. For example, not every surjective function between groups is a group homomorphism, despite the fact that the domain and the image both satisfy the group axioms. – Alex Kruckman Mar 19 '21 at 13:26
  • Thank you Duncan. This is absolutely closer to what I was thinking – user20672 Mar 19 '21 at 15:57
  • @AlexKruckman Such semantics honestly are a good example of why category theory is expressly constructed to try and avoid defining structure as much as possible. I define an "axiom" as an arbitrary logical sentence (notable imprecision here--think we disagree on the meaning of "logical") that holds on a structure. Deciding which such sentences are to be given the rank of "structure important enough to be preserved" is sufficiently arbitrary, and sufficiently variable, that developing theory that applies given some such definition of structure-preserving is about the best we can do. – Duncan W Mar 19 '21 at 20:51
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    I too am a big fan of category theory, and categorical logic in particular. I'm just pointing out that your first paragraph is either wrong, or else you're using the words "algebraic structure", "formal logical sentences", or "axioms of the structure" in a sufficiently nonstandard way so that the paragraph will be misleading. – Alex Kruckman Mar 19 '21 at 21:12
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    Trying to understand what you meant: Let $G$ be a group. By the "formal logical sentences" satisfied by the set, do you mean all sentences of the form $a\cdot b = c$ where $a,b,c\in G$? If so, I agree that the set of all these sentences determines $G$ uniquely, and a function $G\to H$ which preserves the truth of these sentences is exactly a group homomorphism. But the set of these sentences is called the "diagram of $G$", not the "axioms of $G$". The issue of using established terminology correctly is not just semantic quibbling, it's crucial for clear communication. – Alex Kruckman Mar 19 '21 at 21:18
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    Thanks @AlexKruckman I understand your concern. I know what Duncan was trying to say, so it doesn't bother me. This has honestly been a very productive answer for me, in as much as it has given me a few leads to follow. At the moment, it does seem the idea I'm trying to wrap my head around lies in the overlap of universal algebra and category theory. I need to do some more reading when I have time to understand what others have already done in this area. – user20672 Mar 20 '21 at 15:10