symmetric monoidal (∞,1)-category of spectra
This can be thought of as:
We think of as a subset (in fact -vector subspace) of by identifying with . is equipped with a -linear involution , called complex conjugation, that maps to . Concretely, . Complex conjugation is the nontrivial field automorphism of which leaves invariant. In other words, the Galois group is cyclic of order two and generated by complex conjugation. also has an absolute value:
notice that the absolute value of a complex number is a nonnegative real number, with
Most concepts in analysis can be extended from to , as long as they do not rely on the order in . Sometimes even works better, either because it is algebraically closed or because of Goursat's theorem. Even when the order in is important, often it is enough to order the absolute values of complex numbers. See ground field for some of the concepts whose precise definition may vary with the choice of or (or even other possibilities).
The complex numbers form a plane, the complex plane. Indeed, a map given by sending to the standard real-valued coordinates on this plane is a bijection. Much of complex analysis can be understood through differential topology by identifying with , using either and or and . (For example, Cauchy's integral theorem is Green's/Stokes's theorem.)
It is often convenient to use the Alexandroff compactification of , the Riemann sphere . One may think of as ; functions valued in but containing ‘poles’ may be taken to be valued in , with whenever is a pole of .
|Lorentzian spacetime dimension||spin group||normed division algebra||brane scan entry|
|the real numbers|
|the complex numbers|
|the quaternions||little string|
|the octonions||heterotic/type II string|