|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-proposition, mere proposition|
|cut rule||composition of classifying morphisms / pullback of display maps||substitution|
|cut elimination for implication||counit for hom-tensor adjunction||beta reduction|
|introduction rule for implication||unit for hom-tensor adjunction||eta conversion|
|logical conjunction||product||product type|
|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, (idemponent) 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|
A function from a set to a set is determined by giving, for each element of , a specified element of . The process of passing from elements of to elements of is called function application. The set is called the domain of , and is called its codomain.
A function is sometimes called a total function to distinguish it from a partial function.
More generally, every morphism between objects in a category may be thought of as a function in a generalized sense. This generalized use of the word is wide spread (and justified) in type theory, where for and two types, there is a function type denoted and then the expression means that is a term of function type, hence is a function.
For more on this more general use of “function” see at function type.
The formal definition of a function depends on the foundations chosen.
In material set theory, a function is often defined to be a set of ordered pairs such that for every , there is at most one such that . The domain of is then the set of all for which there exists some such . This definition is not entirely satisfactory since it does not determine the codomain (since not every element of the codomain may be in the image); thus to be completely precise it is better to define a function to be an ordered triple where is the domain and the codomain.
In structural set theory, the role of functions depends on the particular axiomatization chosen. In ETCS, functions are among the undefined things, whereas in SEAR, functions are defined to be particular relations (which in turn are undefined things).
See the MathOverflow: what-are-maps-between-proper-classes