In category theory we are largely concerned with mappings between objects. These could be the mappings between the objects within a category (e.g. connections between members of a set that makeup the category) or these could be higher level, such as mappings between categories themselves.
The mapping between objects within a category are morphisms, and the mappings between categories are functors.
My questions is, can morphisms and functors be partial? In other words, could there be a "degraded" morphism between any 2 objects, such that there is "some" connection but it's not as good as it could be.
This would be analogous to a communication channel, with input, channel and output (or source and sink), where the communication is not perfect (say noise in the channel) but the message can still be understood.
"Restriction categories" address your question almost entirely: