Point-surjective morphism
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, and it first was introduced by William Lawvere in his original article.
Definition
Point-surjectivity
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
If is an exponential object of the form for some objects in, a weaker 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 is the evaluation map in the category of morphisms of.
Equivalently, 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.
Relation to surjective functions in Set">Category of sets">Set
Set elements as morphisms from terminal objects
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
With that correspondence, the definition of point-surjective morphisms closely resembles that of surjective functions. A function is said to be surjective if, for every element, there exists an element 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.