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Given a notion "A" in mathematics, in many cases "coA" is also defined. Here are some common examples:

  1. sine and cosine;
  2. tangent and cotangent;
  3. secant and cosecant;
  4. function and cofunction;
  5. morphism and comorphism;
  6. functor and cofunctor;
  7. domain and codomain;
  8. limit and colimit;
  9. set and coset;
  10. product and coproduct;
  11. fibration and cofibration;
  12. homology and cohomology;
  13. homotopy and cohomotopy;
  14. prime and coprime;
  15. vector and covector;

and the list goes on. My question is, what is the generally accepted meaning of the prefix "co"? Given a mathematical notion "A", when is "coA" also defined? Also, is "cocoA" always the same as "A"? (Here I am only asking about mathematical terminologies, so "coconut" does not count.)

Edit: Among all the examples listed above, the pair puzzles me the most is "set and coset". A coset is defined in the context of a subgroup of a group. I am wondering if there is any reason to call it a coset.

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    $\begingroup$ See cosine : etymology. $\endgroup$ – Mauro ALLEGRANZA Feb 2 at 14:52
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    $\begingroup$ In most cases it's from the latin prefix 'com-', which means (usually) "together with". In the case of the trig functions it comes from the more specific latin 'complement' meaning it applies to another (complementary) side of the triangle. $\endgroup$ – Klaas van Aarsen Feb 2 at 14:59
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    $\begingroup$ Yes; I think that the moden ones are named so by analogy, to connote a couple of "complementar" concepts. $\endgroup$ – Mauro ALLEGRANZA Feb 2 at 15:00
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    $\begingroup$ "A comathematician is a device for turning cotheorems into ffee." Some comathematician, I don't know . . . $\endgroup$ – Shaun Feb 2 at 15:05
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    $\begingroup$ +1 for the "coconut" non-example/ $\endgroup$ – John Hughes Feb 2 at 15:06

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