In set theory, there are many ways to say that given the relation or correspondence , holds of elements and . In allegorical set theories like SEAR, it is a foundational concept. In categorical set theories like ETCS with elements, we say that holds of elements and if the ordered pair is in the image of the function . In material set theories like ZFC, we say that holds of elements and if the ordered pair is in .
In set theory, a one-to-one correspondence is a relation or correspondence between sets and for which for every element , there is a unique element such that holds, and for every element , there is a unique element such that holds. Equivalently, it is an anafunction between sets and such that for every element , there is a unique element such that holds.
In type theory, given types and , a one-to-one correspondence is a correspondence for which for every element , there exists a unique element such that , and for every element , there exists a unique element such that . Equivalently, it is an anafunction such that for every element , the dependent type is a contractible type.
Given a Tarski universe , the type of all -small one-to-one correspondences between and is given by the type
The univalence axiom for is given by the equivalence of types
By the principle of unique choice, given every one-to-one correspondence, there is a bijection . However, in the absence of the principle of unique choice, bijections and one-to-one correspondences are not necessarily the same; this is the same phenomena as the phenomena between functions and anafunctions. However, frequently in material set theories, functions are defined to be anafunctions, and so bijections are similarly defined to be one-to-one correspondences.
Something similar occurs in dependent type theory, where by the principle of unique choice, given every one-to-one correspondence, there is a function with contractible fibers , which becomes a bijection in the context of axiom K or uniqueness of identity proofs. However, in contrast to material set theory, in dependent type theory, there is always a distinction between a function and an anafunction , and so there is always a distinction between a function with contractible fibers and a one-to-one correspondence.
Last revised on May 17, 2024 at 18:08:15. See the history of this page for a list of all contributions to it.