Capability ideas appear within computer systems in several guises, sometimes as hardware constructs and sometimes as software. Here are those guises which come to mind just now.
Memory Maps
The hardware memory map has been the most significant contribution by hardware designers to capability practice. The early hardware contribution came about in the Burroughs machines where a hardware memory descriptor provided access to an array. The descriptor controlled access and also allowed the operating systems for those machines to provide the benefits of virtual memory, divorcing the physical array address from the application logic.

The memory maps of modern machines also provide capability discipline as did the Burroughs machines, albeit with different tradeoffs. Modern maps are seldom deployed with specific arrays in mind. Large linear virtual memories have come into style. When it is desirable to share an array between programs, one may locate the array at a virtual address that the hardware conveniently shares.

Language & Interpreter
An object reference in Java is a pretty effective surrogate for a capability. Access to both data and procedures is controlled by object references that may be transferred and stored.
Language & Compiler
Some translators of Java byte codes to machine language presumably preserve security and thus preserve capability discipline.
Software implementations of capability discipline, to my knowledge, involve a privileged mode kernel which interprets capabilities. Machine language is that native language of these systems. Any compiler and its language may be supported thereby.
One may reason about the security of hardware partly by viewing wires as static capabilities.
These styles are often mixed. A capability kernel is likely to use the hardware memory map as the ultimate access control to data.
Today the capability like features of a particular system are likely to appear at the foundation of the system and at higher levels hierarchical directory systems are likely to be the site of access control. In a pure capability system capabilities are manifest at the top as well as the bottom. Some designers feel that until the capability aspects of the system are hidden the design is incomplete. I think that if the capability aspects of the system are hidden, the system will lose its advantages, and be unconventional as well.