If you want to run over snow, you need a water proof running shoe and an alu crampon, I have a combo that weighs : 400 grs -shoe 200gr +crampon 200 grs.
You have two feet total just under 800grs.
If you want to run up ice it's 1 kg for one foot, 2 kg for Bi peds. This is the lightest combo with a back crampon§
If you want to run up steep mountains, this combo comes out at 1.5 kg, 3kg for Bi peds. This is the lightest boot crampon combo that was commercially possible, the boot and the axe are no longer manafactured. Why? Because you the customer dont know what you got till it's gone! And reviewers in mags and on the web dont know their exit from their entry portal.
The last combo might just be ok in a karakorum summer§