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

In category theory


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

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

there exists a unique diagonal filler making both triangles commute:

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

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

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


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

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

  • The orthogonal subcategory problem for a class of morphisms Σ\Sigma in a category CC asks whether the full subcategory Σ \Sigma^\perp of objects XX orthogonal to Σ\Sigma is a reflective subcategory. Here we define fXf \perp X to mean f!:X1f \perp !: X \to 1.

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

type of subspace WW of inner product spacecondition on orthogonal space W W^\perp
isotropic subspaceWW W \subset W^\perp
coisotropic subspaceW WW^\perp \subset W
Lagrangian subspaceW=W W = W^\perp(for symplectic form)
symplectic spaceWW ={0}W \cap W^\perp = \{0\}(for symplectic form)

Last revised on February 17, 2021 at 14:13:15. See the history of this page for a list of all contributions to it.