Point-surjective morphism

In category theory, a point-surjective morphism is a morphism ${\displaystyle f:X\rightarrow Y}$ that "behaves" like surjections on the category of sets.

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

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

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

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

${\displaystyle \epsilon \circ \langle f\circ x,a\rangle =g\circ a}$

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

Equivalently,^[4] one could think of the morphism ${\displaystyle f:X\rightarrow B^{A}}$ as the transpose of some other morphism ${\displaystyle {\tilde {f}}:X\times A\rightarrow B}$. Then the isomorphism between the hom-sets ${\displaystyle \mathrm {Hom} (X\times A,B)\cong \mathrm {Hom} (X,B^{A})}$ allow us to say that ${\displaystyle f}$ is weakly point-surjective if and only if ${\displaystyle {\tilde {f}}}$ is weakly point-surjective.^[5]

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

For this reason, morphisms ${\displaystyle a:1\rightarrow A}$ can serve as a "generalization" of elements of a set, and are sometimes called global elements.

With that correspondence, the definition of point-surjective morphisms closely resembles that of surjective functions. A function (morphism) ${\displaystyle f:X\rightarrow Y}$ is said to be surjective (point-surjective) if, for every element ${\displaystyle y\in Y}$ (for every morphism ${\displaystyle y:1\rightarrow Y}$), there exists an element ${\displaystyle x\in X}$ (there exists a morphism ${\displaystyle x:1\rightarrow X}$) such that ${\displaystyle f(x)=y}$ ( ${\displaystyle f\circ x=y}$).

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