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Undoubtedly, these terms play essential roles in (pure) mathematics. My problem is that I have feelings what they mean in different fields, such as, differential geometry (abstract manifolds vs. embedded ones), algebraic geometry (more down to earth, the study of Riemann surfaces and algebraic curves) when our objects can be embedded differently and each embedding gives us (I think) extrinsic properties rather than intrinsic ones, and intrinsic properties do not have anything to do with embeddings.

What I would like to know are as follows;

  • Can they be defined, precisely?

  • How can one recognize which properties are coming from intrinsic properties and which are not?

I would appreciate any comments in helping me understand them better.

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closed as off topic by Andy Putman, Chris Godsil, Ehsan M. Kermani, Will Jagy, Bill Johnson Feb 1 '12 at 17:42

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up vote 5 down vote accepted

Intrinsic properties are those which are invariant under isomorphism, whatever that notion happens to mean in the category under consideration.

Edit: I guess I would say also that an extrinsic property of an object is not a property of the object itself but a property of the object together with some other data, for example the object together with a map to or from some other specified object.

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@MTS: Thanks that's helpful. – Ehsan M. Kermani Feb 1 '12 at 4:21
An isomorphism is any invertible morphism in the category. So in the category of rings and ring homomorphisms, an isomorphism is a ring isomorphism. In the category of topological spaces and continuous maps it is a homeomorphism. Etc. – MTS Feb 1 '12 at 4:41

Premise: The adjectives intrinsic and extrinsic come from the Latin intrinsecus ("inner") and extrinsecus ("outer"), from the adverbs intra resp. extra, and the p.p. secutus of the verb sequor ("follow").

In the context of category theory, if we want to play the game of "non-philological (i.e. a posteriori) etymology", it is tempting to refer follow to arrows. I would therefore say that intrinsic is a categorical property of an object, which is stated by means of its only structure and self-maps, while extrinsic is a categorical property of an object which also depends from other objects and maps with different domains or co-domains. In this sense, compactness is an intrinsic property of topological spaces, while e.g. the homotopy extension property, or being an ANR, are extrinsic properties; in fact, most universal properties are. (But, I repeat, this is just a suggestion).

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