nLab orthogonality

(This entry describes two distinct notions, one in the theory of inner product spaces, and the second in a more purely category theoretic context.)

In inner product spaces

Two elements $x,y$ in an inner product space, $(V, \langle -,-\rangle)$, are orthogonal or normal vectors, denoted $x \perp y,$ if $\langle x,y\rangle = 0$.

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

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

there exists a unique diagonal filler making both triangles commute:

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

Given a class of maps $E$, the class $\{m | e\perp m \;\forall e\in E\}$ is denoted $E^{\downarrow}$ or $E^\perp$. Likewise, given $M$, the class $\{e | e\perp m \;\forall m\in M\}$ 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 ${}^\perp(Mono)$, where $Mono$ is the class of monomorphisms. (If the category has equalizers, then every map in ${}^\perp(Mono)$ is epic.) Dually, a strong monomorphism is a monomorphism in $(Epi)^\perp$.

• The orthogonal subcategory problem for a class of morphisms $\Sigma$ in a category $C$ asks whether the full subcategory $\Sigma^\perp$ of objects $X$ orthogonal to $\Sigma$ is a reflective subcategory. Here we define $f \perp X$ to mean $f \perp !: X \to 1$.

  The orthogonal subcategory problem is related to localization. Suppose $\Sigma^\perp$ is indeed a reflective subcategory; let $r: C \to \Sigma^\perp$ be the reflector (the left adjoint to the inclusion $i: \Sigma^\perp \to C$). Certainly $r$ sends arrows in $\Sigma$ to isomorphisms in $\Sigma^\perp$. Indeed, if $f: A \to B$ belongs to $\Sigma$, then the inverse to $r(f): r(A) \to r(B)$ is the unique arrow extending $1_{r(A)}$ along $r(f): r(A) \to r(B)$ to an arrow $g: r(B) \to r(A)$, using the fact that $r(A)$ belongs to $\Sigma^\perp$.

Related concepts

• injective object

type of subspace $W$ of inner product space	condition on orthogonal space $W^\perp$
isotropic subspace	$W \subset W^\perp$
coisotropic subspace	$W^\perp \subset W$
Lagrangian subspace	$W = W^\perp$	(for symplectic form)
symplectic space	$W \cap W^\perp = \{0\}$	(for symplectic form)

• orthogonal structure, orthogonal basis

• Grassmannian, Stiefel manifold

• orthogonal spectrum, coordinate-free spectrum, equivariant spectrum