Electrolysis of water(Electron transfer)?

kevin_tee

I'm still little bit not sure about the process of electrolysis.
According to this equation...

Cathode (reduction): 2 H2O(l) + 2e− → H2(g) + 2 OH-(aq)

Does the electrons from the cathode went in side H20 and break apart into H2 and 2OH-

or

Does H2O break apart into H+ and OH- and then electrons went into 2H+ to make H2 ?

Thank very much