A morphism $f:A\to B$ in a category $\mathcal{C}$ is said to be orthogonal to a morphism $g:C\to D$ in $\mathcal{C}$ , written $$f\perp g$$ if whenever we have a commutative diagram

For example, in Set, the category of sets, any surjective function is orthogonal to an injective function. To see this, suppose $f:A\to B$ is surjective and $g:C\to D$ injective, with $y\circ f= g\circ x$ , where $x:A\to C$ and $y:B\to D$ are functions. For any $b\in B$ , there is some $a\in A$ such that $f(a)=b$ since $f$ is surjective. Define $h:B\to C$ by $h(b)=x(a)$ . Now, if there is $c\in A$ such that $b=f(a)=f(c)$ , then $g(x(a))= y(f(a))=y(b)=y(f(c))=g(x(c))$ . Since $g$ is injective, $x(a)=x(c)$ . This shows that $h$ is a well-defined function. It is clear that $h\circ f=x$ and $g\circ h=y$ . Now, if $e:B\to C$ is another such a function, then $g(e(b))=y(b)=g(h(b))$ , so that $e(b)=h(b)$ since $g$ is injective. This shows that $h$ is uniquely defined.

Here are some basic properties of the orthogonality relation on morphisms:

• If either $f$ or $g$ is an isomorphism, then $f\perp g$ .
• If $f\perp f$ , then $f$ is an isomorphism.
• If $f\perp g$ and $f\perp h$ , then $f\perp (h\circ g)$ . Similarly, $g\perp f$ and $h\perp f$ imply $(h\circ g)\perp f$ . Of course, both statements make sense provided that $h\circ g$ exists.

More generally, if $\mathcal{F}$ and $\mathcal{G}$ are two classes of morphisms in a category $\mathcal{C}$ , we say that $\mathcal{F}$ is orthogonal to $\mathcal{G}$ , or that $(\mathcal{F},\mathcal{G})$ is a diagonally polar pair, written $\mathcal{F}\perp \mathcal{G}$ , if $f\perp g$ for every $f$ in $\mathcal{F}$ and every $g$ in $\mathcal{G}$ .

For every class $\mathcal{X}$ of morphism, the largest class of morphisms in $\mathcal{C}$ such that $\mathcal{X}$ is orthogonal to is denoted by $\mathcal{X}_*$ , and the largest class of morphisms that is orthogonal to $\mathcal{X}$ is denoted by $\mathcal{X}^*$ .

Based on the properties of $\perp$ above, below are some properties of $^*$ and $_*$ :

• $\mathcal{X} \subseteq \mathcal{Y}_*$ iff $\mathcal{Y}\subseteq \mathcal{X}^*$ . Equivalently, if is the class of all subclasses of morphisms of $\mathcal{C}$ , then $(-^*,-_*)$ is a Galois connection between and .
• A morphism is in both $\mathcal{X}^*$ and $\mathcal{X}_*$ iff it is an isomorphism.
• Both $\mathcal{X}^*$ and $\mathcal{X}_*$ are closed under $\circ$ .
• Given that $m=m_1\circ m_2$ exists in $\mathcal{C}$ and $m_2\in \mathcal{X}_*$ , then $m \in \mathcal{X}_*$ iff $m_1\in \mathcal{X}_*$ .
• If $f\in \mathcal{X}_*$ , then the pullback of $f$ along any morphism is again in $\mathcal{X}_*$ .

Bibliography

1

F. Borceux Basic Category Theory, Handbook of Categorical Algebra I, Cambridge University Press, Cambridge (1994)