  Soft robotics is an emerging field of research where the robot body is
composed of compliant and soft materials. It allows the body to bend, twist,
and deform to move or to adapt its shape to the environment for grasping, all
of which are difficult for traditional hard robots with rigid bodies. However,
the theoretical basis and design principles for soft robotics are not
well-founded despite their recognized importance. For example, the control of
soft robots is outsourced to morphological attributes and natural processes;
thus, the coupled relations between a robot and its environment are
particularly crucial. In this paper, we propose a mathematical foundation for
soft robotics based on category theory, which is a branch of abstract math
where any notions can be described by objects and arrows. It allows for a
rigorous description of the inherent characteristics of soft robots and their
relation to the environment as well as the differences compared to conventional
hard robots. We present a notion called the category of mobility that well
describes the subject matter. The theory was applied to a model system and
analysis to highlight the adaptation behavior observed in universal grippers,
which are a typical example of soft robotics. This paper paves the way to
developing a theoretical background and design principles for soft robotics.
