When I started climbing in 1973, almost everyone wore socks with E.B.s or P.A.s, non-sticky rubber shoes that were smart accessory footwear to the thumbscrew, but of limited value on rock.

At Shoes for Crews’ closet-sized lab in its penthouse office, Chief Executive Matt Smith oversees a demonstration of the skid-proof rubber soles invented by his company. An executive slathers ...