Point-surjective morphism

Not to be confused with the similar notion of epimorphism.

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.

Definition

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.

Relation to surjective functions in 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 $a:1\rightarrow A$ 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 $f:A\rightarrow B$.