nLab
orthogonality

In inner product spaces

Two elements $x,y$ in an inner product space $V,⟨-,-⟩$ are orthogonal to each other, $x\perp y,$ if $⟨x,y⟩=0$

In category theory

Definition

Two morphisms $e:A\to B$ and $m:C\to D$ in a category are said to be orthogonal, written $e\perp m$, if $e$ has the left lifting property with respect to $m$, i.e. if in any commutative square

$$\begin{array}{ccc}A& \stackrel{e}{\to }& B\\ \downarrow & & \downarrow \\ C& \underset{m}{\to }& D\end{array}$$\array{ A & \overset{e}{\to} & B\\ \downarrow && \downarrow \\ C & \underset{m}{\to} & D}

there exists a unique diagonal filler making both triangles commute:

$$\begin{array}{ccc}A& \stackrel{e}{\to }& B\\ \downarrow & \swarrow & \downarrow \\ C& \underset{m}{\to }& D\end{array}$$\array{ A & \overset{e}{\to} & B\\ \downarrow & \swarrow & \downarrow \\ C & \underset{m}{\to} & D}

Given a class of maps $E$, the class $\{m\mid e\perp m\forall e\in E\}$ is denoted $E^{\downarrow }$ or $E^{\perp }$. Likewise, given $M$, the class $\{e\mid e\perp m\forall m\in M\}$ is denoted $M^{\uparrow }$ or ${}^{\perp }M$. These operations form a Galois connection on the poset of classes of morphisms in the ambient category. In particular, we have $({}^{\perp }(E^{\perp }){)}^{\perp }=E^{\perp }$ and ${}^{\perp }(({}^{\perp }M{)}^{\perp })={}^{\perp }M$.

A pair $(E,M)$ such that $E^{\perp }=M$ and $E={}^{\perp }M$ is sometimes called a prefactorization system. If in addition every morphism factors as an $E$-morphism followed by an $M$-morphism, it is an (orthogonal) factorization system.

Examples

• Of course, any orthogonal factorization system gives plenty of examples. The ur-example is that $e\perp m$ in Set (or actually, any pretopos) for any surjection $e$ and injection $m$.

• A strong epimorphism in any category is, by definition, an epimorphism in ${}^{\perp }(\mathrm{Mono})$, where $\mathrm{Mono}$ is the class of monomorphisms. (If the category has equalizers, then every map in ${}^{\perp }(\mathrm{Mono})$ is epic.) Dually, a strong monomorphism is a monomorphism in $(\mathrm{Epi}{)}^{\perp }$.

• The orthogonal subcategory problem for a class of morphisms $\Sigma$ in a category $C$ asks whether the full subcategory ${\Sigma }^{\perp }$ of objects orthogonal to $\Sigma$ is a reflective subcategory.

  This problem is related to the problem of localization. Suppose ${\Sigma }^{\perp }$ is indeed a reflective subcategory; let $r:C\to {\Sigma }^{\perp }$ be the reflector (the left adjoint to the inclusion $i:{\Sigma }^{\perp }\to C$).