|logic||category theory||type theory|
|true||terminal object/(-2)-truncated object||h-level 0-type/unit type|
|false||initial object||empty type|
|proposition||(-1)-truncated object||h-level 1-type/h-prop|
|classical type theory?||internal logic||classical type theory?, logic programming?|
|cut elimination?||counit||beta reduction|
|disjunction||coproduct ((-1)-truncation of)||sum type (bracket type of)|
|implication||internal hom||function type|
|negation||internal hom into initial object||function type into empty type|
|universal quantification||dependent product||dependent product type|
|existential quantification||dependent sum ((-1)-truncation of)||dependent sum type (bracket type of)|
|equivalence||path space object||identity type|
|equivalence class||quotient||quotient type|
|induction||colimit||inductive type, W-type, M-type|
|higher induction||higher colimit||higher inductive type|
|completely presented set||discrete object/0-truncated object||h-level 2-type/preset/h-set|
|set||internal 0-groupoid||Bishop set/setoid|
|universe||object classifier||type of types|
|modality||closure operator monad||modal type theory, monad (in computer science)|
In formal logic such as type theory a term is an entity/expression of the formal language which is of some type . One writes to express this (a typing judgement). The semantics of terms in Set is: elements of a set, where one writes . One also says is an inhabitant of the type and that is an inhabited type if it has a term.
A term may depend on free variables that are themselves terms of some other type . For instance may be a term of type (the integers) which depends on a variable term also of type the integers. The notation for this in the metalanguage is
x : X \vdash z : Z \,.
Generally here also the type itself may depend on the variable (hence the term may be of different type dependending on its free variables) in which case one says that is a term of -dependent type.
f : B \to A
x\colon B \vdash f(x)\colon A \,.