nLab
orthogonality

Contents

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

In inner product spaces

Two elements x,yx,y in an inner product space V,,V, \langle -,-\rangle are orthogonal to each other, xy,x \perp y, if x,y=0\langle x,y\rangle = 0

In category theory

Definition

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.

Examples

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)

Revised on June 30, 2013 09:28:25 by Tim Porter (95.147.236.113)