proposition | set | object | type |

predicate | family of sets | display morphism | dependent type |

proof | element | generalized element | term/program |

cut rule | | composition of classifying morphisms / pullback of display maps | substitution |

introduction rule for implication | | counit for hom-tensor adjunction | lambda |

elimination rule for implication | | unit for hom-tensor adjunction | application |

cut elimination for implication | | one of the zigzag identities for hom-tensor adjunction | beta reduction |

identity elimination for implication | | the other zigzag identity for hom-tensor adjunction | eta conversion |

true | singleton | terminal object/(-2)-truncated object | h-level 0-type/unit type |

false | empty set | initial object | empty type |

proposition, truth value | subsingleton | subterminal object/(-1)-truncated object | h-proposition, mere proposition |

logical conjunction | cartesian product | product | product type |

disjunction | disjoint union (support of) | coproduct ((-1)-truncation of) | sum type (bracket type of) |

implication | function set (into subsingleton) | internal hom (into subterminal object) | function type (into h-proposition) |

negation | function set into empty set | internal hom into initial object | function type into empty type |

universal quantification | indexed cartesian product (of family of subsingletons) | dependent product (of family of subterminal objects) | dependent product type (of family of h-propositions) |

existential quantification | indexed disjoint union (support of) | dependent sum ((-1)-truncation of) | dependent sum type (bracket type of) |

logical equivalence | bijection set | object of isomorphisms | equivalence type |

| support set | support object/(-1)-truncation | propositional truncation/bracket type |

| | n-image of morphism into terminal object/n-truncation | n-truncation modality |

equality | diagonal function/diagonal subset/diagonal relation | path space object | identity type/path type |

completely presented set | set | discrete object/0-truncated object | h-level 2-type/set/h-set |

set | set with equivalence relation | internal 0-groupoid | Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation |

| equivalence class/quotient set | quotient | quotient type |

induction | | colimit | inductive type, W-type, M-type |

higher induction | | higher colimit | higher inductive type |

- | | 0-truncated higher colimit | quotient inductive type |

coinduction | | limit | coinductive type |

| preset | | type without identity types |

| set of truth values | subobject classifier | type of propositions |

domain of discourse | universe | object classifier | type universe |

modality | | closure operator, (idempotent) monad | modal type theory, monad (in computer science) |

linear logic | | (symmetric, closed) monoidal category | linear type theory/quantum computation |

proof net | | string diagram | quantum circuit |

(absence of) contraction rule | | (absence of) diagonal | no-cloning theorem |

| | synthetic mathematics | domain specific embedded programming language |