As far as I know there are basically three different approaches to the tangent space -all of them coming with advantages and disadvantages:
- Though the oldest one precisely represents the derivative on manifolds it is very technical due to this. Thus it is taken out of discussion in most differential geometry lectures and found rarely in literature as in Spivak's comprehensive introduction to differential geometry.
- Though the more intuitive approach about tangent curves reveals what we think of what it means to be tangent it is almost chanceless the deal with it.
- The probably best approach -for now at least- certainly is the one by derivations. Though it masks our association of tangent space and derivative it is the one covering its algebraic properties.
- Surely, there's more approaches as the one by germs, however, most of them seem to fit inside one of these classes as variants as the diagram below could should suggest.
If one compares the algebraic approach to the analogeous problem of tensor products it might be the suggested starting point for a universal property of the tangent space. (Despite this, there's already the universal property given by Spivak in his works appendix. That, however, seems way to technical, giving rise to clarifications.)
So my hope:
Does somebody know how to work out the universal property of tangent spaces or is somebody familiar enough with Spivak's comprehensive introduction to differential geometry so to clarify his elaboration?
