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Given a natural number and types and , a function is a -monic function if for all terms the fiber of over has an homotopy level of .
A equivalence is a -monic function. -monic functions are typically just called monic functions or inclusions.
The type of all -monic functions with domain and codomain is defined as
Given types and , is called a subtype of , and is called a supertype of , if is inhabited. We define the proposition of subtype inclusion as
where is the propositional truncation of . If and are sets, then is a subset of and is a superset of .
The type of all subtypes of in a universe is defined as