Caving is a vertical sport (it can be classified as such), and often it's like climbing in reverse (depends on the cave system). For running, it depends on what they mean by running - mountain/trail running (similar to hiking which can be grouped under mountaineering), night running (with a head torch, similar or identical to ones used by mountaineers), or just normal running.
From Petzl's site (
http://www.petzl.com/company/expertise?language=en):"Products are organized into two major families:
- products for verticality: harnesses, helmets, belay devices, ascenders/rope clamps, crampons, ice axes…
- headlamps: compact headlamps, powerful headlamps, LED lighting
They are designed for two main areas.
- Sport: mountaineering, climbing, caving, canyoning, via ferrata, trail running...
- Professional: rope access, arborism, rescue, intervention, public works and structures, hands-free lighting…"
(Corrected spelling of Petzl, again)