Point-surjective morphism

Not to be confused with the similar notion of epimorphism.

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

$ϵ\circ ⟨f\circ x,a⟩=g\circ a$

where $⟨-,-⟩:A\to B\times C$ denotes the product of two morphisms ($A\to B$ and $A\to C$) and $ϵ:B^A\times A\to B$ is the evaluation map in the category of morphisms of $\mathbf{C}$. Equivalently, one could think of the morphism $f:X\to B^A$ as the transpose of some other morphism $\tilde{f}:X\times A\to B$. Then the isomorphism between the hom-sets $\mathrm{H}\mathrm{o}\mathrm{m}(X\times A,B)\cong \mathrm{H}\mathrm{o}\mathrm{m}(X,B^A)$ 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 $a:1\to A$ 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 $a:1\to A$ can be thought of as a specific element of $A$ itself. For this reason, morphisms $a:1\to A$ 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) $f:X\to Y$ is said to be surjective (point-surjective) if, for every element $y\in Y$ (for every morphism $y:1\to Y$), there exists an element $x\in X$ (there exists a morphism $x:1\to X$) 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 $f:A\to B$.