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Point-surjective morphism

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In category theory, a point-surjective morphism is a morphism that "behaves" like surjections on the category of sets.

The notion of point-surjectivity is an important one in Lawvere's fixed-point theorem,[1][2] and it first was introduced by William Lawvere in his original article.[3]

Definition

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Point-surjectivity

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In a category with a terminal object , a morphism is said to be point-surjective if for every morphism , there exists a morphism such that .

Weak point-surjectivity

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If is an exponential object of the form for some objects in , a weaker (but technically more cumbersome) notion of point-surjectivity can be defined.

A morphism is said to be weakly point-surjective if for every morphism there exists a morphism such that, for every morphism , we have

where denotes the product of two morphisms ( and ) and is the evaluation map in the category of morphisms of .

Equivalently,[4] one could think of the morphism as the transpose of some other morphism . Then the isomorphism between the hom-sets allow us to say that is weakly point-surjective if and only if is weakly point-surjective.[5]

Relation to surjective functions in Set

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Set elements as morphisms from terminal objects

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In the category of sets, morphisms are functions and the terminal objects are singletons. Therefore, a morphism is a function from a singleton to the set : since a function must specify a unique element in the codomain for every element in the domain, we have that is one specific element of . Therefore, each morphism can be thought of as a specific element of itself.

For this reason, morphisms can serve as a "generalization" of elements of a set, and are sometimes called global elements.

Surjective functions and point-surjectivity

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With that correspondence, the definition of point-surjective morphisms closely resembles that of surjective functions. A function (morphism) is said to be surjective (point-surjective) if, for every element (for every morphism ), there exists an element (there exists a morphism ) such that ( ).

The notion of weak point-surjectivity also resembles this correspondence, if only one notices that the exponential object in the category of sets is nothing but the set of all functions .

References

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  1. ^ Lawvere, Francis William (1969). "Diagonal arguments and Cartesian closed categories". Category Theory, Homology Theory and their Applications II (Lecture Notes in Mathematics, vol 92). Berlin: Springer.
  2. ^ Lawvere, William (2006). "Diagonal arguments and cartesian closed categories with author commentary". Reprints in Theory and Applications of Categories (15): 1–13.
  3. ^ Abramsky, Samso (2015). "From Lawvere to Brandenburger–Keisler: Interactive forms of diagonalization and self-reference". Journal of Computer and System Sciences. 81 (5): 799–812. arXiv:1006.0992. doi:10.1016/j.jcss.2014.12.001.
  4. ^ Reinhart, Tobias; Stengle, Sebastian. "Lawvere's Theorem" (PDF). Universität Innsbruck.
  5. ^ Frumin, Dan; Massas, Guillaume. "Diagonal Arguments and Lawvere's Theorem" (PDF). Retrieved 9 February 2024.