# What is the precise definition of the prefix “co” in mathematics?

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.

• – Mauro ALLEGRANZA Feb 2 at 14:52
• 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. – Klaas van Aarsen Feb 2 at 14:59
• Yes; I think that the moden ones are named so by analogy, to connote a couple of "complementar" concepts. – Mauro ALLEGRANZA Feb 2 at 15:00
• "A comathematician is a device for turning cotheorems into ffee." Some comathematician, I don't know . . . – Shaun Feb 2 at 15:05
• +1 for the "coconut" non-example/ – John Hughes Feb 2 at 15:06