Point-surjective morphism

In category theory, a point-surjective morphism is a morphism $$f : X \rightarrow Y$$ 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.

Point-surjectivity
In a category $$\mathbf{C}$$ with a terminal object $$1$$, a morphism $$f : X \rightarrow Y$$ is said to be point-surjective if for every morphism $$y : 1 \rightarrow Y$$, there exists a morphism  $$x : 1 \rightarrow X$$ such that  $$f \circ x = y$$.

Weak point-surjectivity
If $$Y$$ is an exponential object of the form $$B^A$$ for some objects $$A, B$$ in $$\mathbf{C}$$, a weaker (but technically more cumbersome) notion of point-surjectivity can be defined.

A morphism $$f : X \rightarrow B^A$$ is said to be weakly point-surjective if for every morphism $$g : A \rightarrow B$$ there exists a morphism $$x : 1 \rightarrow X$$ such that, for every morphism $$a : 1 \rightarrow A$$, we have


 * $$ \epsilon \circ \langle f \circ x, a \rangle = g \circ a$$

where $$ \langle -, - \rangle : A \rightarrow B \times C$$ denotes the product of two morphisms ($$A \rightarrow B$$ and $$A \rightarrow C$$) and $$ \epsilon : B^A \times A \rightarrow B $$ is the evaluation map in the category of morphisms of $$\mathbf{C}$$.

Equivalently, one could think of the morphism $$f: X \rightarrow B^A$$ as the transpose of some other morphism $$\tilde{f}: X \times A \rightarrow B$$. Then the isomorphism between the hom-sets $$\mathrm{Hom}(X\times A,B) \cong \mathrm{Hom}(X,B^A)$$ allow us to say that $$f$$ is weakly point-surjective if and only if $$\tilde{f}$$ is weakly point-surjective.

Set elements as morphisms from terminal objects
In the category of sets, morphisms are functions and the terminal objects are singletons. Therefore, a morphism $$a : 1 \rightarrow A$$ is a function from a singleton $$\{x\}$$ to the set $$A$$: since a function must specify a unique element in the codomain for every element in the domain, we have that $$a(x) \in A$$ is one specific element of $$A$$. Therefore, each morphism $$a : 1 \rightarrow A$$ can be thought of as a specific element of $$ A$$ itself.

For this reason, morphisms $$a : 1 \rightarrow A$$ 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 (morphism) $$f : X \rightarrow Y $$ is said to be surjective (point-surjective) if, for every element $$y \in Y$$ (for every morphism $$y : 1 \rightarrow Y$$), there exists an element $$x \in X$$ (there exists a morphism $$x: 1 \rightarrow X$$) such that $$f(x) = y $$ ( $$ f \circ x = y$$).

The notion of weak point-surjectivity also resembles this correspondence, if only one notices that the exponential object $$B^A$$ in the category of sets is nothing but the set of all functions $$f : A \rightarrow B$$.