Intrinsic vs. Extrinsic - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-20T04:45:50Z http://mathoverflow.net/feeds/question/87201 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87201/intrinsic-vs-extrinsic Intrinsic vs. Extrinsic Ehsan M. Kermani 2012-02-01T03:55:24Z 2012-02-01T16:01:20Z <p>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.</p> <p>What I would like to know are as follows;</p> <ul> <li><p>Can they be defined, precisely?</p></li> <li><p>How can one recognize which properties are coming from intrinsic properties and which are not?</p></li> </ul> <p>I would appreciate any comments in helping me understand them better.</p> http://mathoverflow.net/questions/87201/intrinsic-vs-extrinsic/87203#87203 Answer by MTS for Intrinsic vs. Extrinsic MTS 2012-02-01T04:08:43Z 2012-02-01T04:08:43Z <p>Intrinsic properties are those which are invariant under isomorphism, whatever that notion happens to mean in the category under consideration.</p> <p>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.</p> http://mathoverflow.net/questions/87201/intrinsic-vs-extrinsic/87221#87221 Answer by Pietro Majer for Intrinsic vs. Extrinsic Pietro Majer 2012-02-01T12:08:05Z 2012-02-01T16:01:20Z <p>Premise: The adjectives <em>intrinsic</em> and <em>extrinsic</em> come from the Latin <em>intrinsecus</em> ("inner") and <em>extrinsecus</em> ("outer"), from the adverbs <em>intra</em> resp. <em>extra</em>, and the p.p. <em>secutus</em> of the verb <em>sequor</em> ("follow"). </p> <p>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 <em>follow</em> to arrows. I would therefore say that <em>intrinsic</em> is a categorical property of an object, which is stated by means of its only structure and self-maps, while <em>extrinsic</em> 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). </p>