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.

Orthogonality of morphisms to objects

A morphism $e:A\to B$ is said to be (left) orthogonal to an object $C$, written $e\perp C$ (equivalently : $C$ is right orthgonal to $e$), if for any morphism $A \to C$

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

there exists a unique morphism $B \to C$ making the following diagram commute:

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

If the category we are working with has a terminal object $1$, this is equivalent to saying that $e \perp !$ where $! : C \to 1$ is the unique morphism to the terminal object. Abusing notation, we often write $E^\perp$ for the collection of objects $C$ which are right orthogonal to each morphism in $E$. We sometimes refer to $E^\perp$ as the (right) orthogonal complement of $E$.

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 $(Epi)^\perp$.

• If $\mathcal{C}$ is a category, interesting full subcategories $\mathcal{D} \subseteq \mathcal{C}$ are often usefully characterized as the orthogonal complement of some collection of morphisms. In fact, if $\mathcal{D}$ is a reflective subcategory of $\mathcal{C}$, then we always have $\mathcal{D} = E_\ast^\perp$ where $E_\ast = \{ C \xrightarrow{\eta_C} iL C \mid C \in \mathcal{C}\}$ (where we write $i : D \to C$ for the inclusion, $L$ for the left adjoint, and $\eta : 1_{\mathcal{C}} \to iL$ for the unit of the adjunction), but typically we already have $\mathcal{D} = E^\perp$ for some much smaller collection $E \subseteq E_\ast$. It is often convenient to think about $\mathcal{D}$ in this way for some such well-chosen set $E$.

• Conversely, 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