what is that?
No Capture Chess

Since you can't capture anything, you can't check the king. So it's impossible to mate.

Hmm.... even if you do play regularly it still draws, although maybe that's because of the high playing level:
Lc0 vs Lc0 - Nibbler - No capture - <15 second per move:
Maybe to make this game playable, you have to have at least one way to capture the king at all times. (You can't block off the square the access the king either. So i'll define it as: You cannot move such that the resulting position is where, no matter how many extra moves your opponent gets, your opponent cannot capture your king. (In the endgame that might be impossible. If it is, normal no-capture rules apply.))
This also prevents 3-fold repetition if you can avoid it, since the opponent cannot take your king with any amount of extra moves.
Lc0 vs Lc0
Can we have No Capture Chess?