Point-surjective morphism#Definition#Weak point-surjectivity

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

File:Weak point-surjectivity.svg

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

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

:$\backslash epsilon\; \backslash circ\; \backslash langle\; f\; \backslash circ\; x,\; a\; \backslash rangle\; =\; g\; \backslash circ\; a$

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

Equivalently,{{cite web |last1=Reinhart |first1=Tobias |last2=Stengle |first2=Sebastian |title=Lawvere's Theorem |url=https://www.uibk.ac.at/mathematik/algebra/staff/fritz-tobias/ct2021_course_projects/lawvere.pdf |website=Universität Innsbruck}} one could think of the morphism $f:\; X\; \backslash rightarrow\; B^A$ as the transpose of some other morphism $\backslash tilde\{f\}:\; X\; \backslash times\; A\; \backslash rightarrow\; B$. Then the isomorphism between the hom-sets $\backslash mathrm\{Hom\}(X\backslash times\; A,B)\; \backslash cong\; \backslash mathrm\{Hom\}(X,B^A)$ allow us to say that $f$ is weakly point-surjective if and only if $\backslash tilde\{f\}$ is weakly point-surjective.{{cite web |last1=Frumin |first1=Dan |last2=Massas |first2=Guillaume |title=Diagonal Arguments and Lawvere's Theorem |url=https://groupoid.moe/pdf/diagonal_argument.pdf |access-date=9 February 2024}}

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

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

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 $f\; :\; A\; \backslash rightarrow\; B$.

{{Reflist}}

{{Category theory}}

Category:Category theory

Category:Morphisms