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orthogonal morphisms (Definition)

A morphism $ f:A\to B$ in a category $ \mathcal{C}$ is said to be orthogonal to a morphism $ g:C\to D$ in $ \mathcal{C}$, written

$\displaystyle f\perp g$
if whenever we have a commutative diagram
$\displaystyle \xymatrix@+=3pc{A \ar[r]^f \ar[d] & B \ar[d] \\ C \ar[r]_g & D}$
there is a unique morphism $ h:B\to C$ such that the diagram
$\displaystyle \xymatrix@+=3pc{A \ar[r]^f \ar[d] & B \ar[d] \ar@{.>}[dl]\vert h \\ C \ar[r]_g & D}$
is commutative also. If $ f\perp g$, we sometimes call the ordered pair $ (f,g)$ a diagonally polar pair.

For example, in Set, the category of sets, any surjective function is orthogonal to an injective function. To see this, suppose $ f:A\to B$ is surjective and $ g:C\to D$ injective, with $ y\circ f= g\circ x$, where $ x:A\to C$ and $ y:B\to D$ are functions. For any $ b\in B$, there is some $ a\in A$ such that $ f(a)=b$ since $ f$ is surjective. Define $ h:B\to C$ by $ h(b)=x(a)$. Now, if there is $ c\in A$ such that $ b=f(a)=f(c)$, then $ g(x(a))= y(f(a))=y(b)=y(f(c))=g(x(c))$. Since $ g$ is injective, $ x(a)=x(c)$. This shows that $ h$ is a well-defined function. It is clear that $ h\circ f=x$ and $ g\circ h=y$. Now, if $ e:B\to C$ is another such a function, then $ g(e(b))=y(b)=g(h(b))$, so that $ e(b)=h(b)$ since $ g$ is injective. This shows that $ h$ is uniquely defined.

Here are some basic properties of the orthogonality relation on morphisms:

  • If either $ f$ or $ g$ is an isomorphism, then $ f\perp g$.
  • If $ f\perp f$, then $ f$ is an isomorphism.
  • If $ f\perp g$ and $ f\perp h$, then $ f\perp (h\circ g)$. Similarly, $ g\perp f$ and $ h\perp f$ imply $ (h\circ g)\perp f$. Of course, both statements make sense provided that $ h\circ g$ exists.

More generally, if $ \mathcal{F}$ and $ \mathcal{G}$ are two classes of morphisms in a category $ \mathcal{C}$, we say that $ \mathcal{F}$ is orthogonal to $ \mathcal{G}$, or that $ (\mathcal{F},\mathcal{G})$ is a diagonally polar pair, written $ \mathcal{F}\perp \mathcal{G}$, if $ f\perp g$ for every $ f$ in $ \mathcal{F}$ and every $ g$ in $ \mathcal{G}$.

For every class $ \mathcal{X}$ of morphism, the largest class of morphisms in $ \mathcal{C}$ such that $ \mathcal{X}$ is orthogonal to is denoted by $ \mathcal{X}_*$, and the largest class of morphisms that is orthogonal to $ \mathcal{X}$ is denoted by $ \mathcal{X}^*$.

Based on the properties of $ \perp$ above, below are some properties of $ ^*$ and $ _*$:

  • $ \mathcal{X} \subseteq \mathcal{Y}_*$ iff $ \mathcal{Y}\subseteq \mathcal{X}^*$. Equivalently, if $ \mathscr{M}$ is the class of all subclasses of morphisms of $ \mathcal{C}$, then $ (-^*,-_*)$ is a Galois connection between $ (\mathscr{M},\subseteq)$ and $ (\mathscr{M},\supseteq)$.
  • A morphism is in both $ \mathcal{X}^*$ and $ \mathcal{X}_*$ iff it is an isomorphism.
  • Both $ \mathcal{X}^*$ and $ \mathcal{X}_*$ are closed under $ \circ$.
  • Given that $ m=m_1\circ m_2$ exists in $ \mathcal{C}$ and $ m_2\in \mathcal{X}_*$, then $ m \in \mathcal{X}_*$ iff $ m_1\in \mathcal{X}_*$.
  • If $ f\in \mathcal{X}_*$, then the pullback of $ f$ along any morphism is again in $ \mathcal{X}_*$.

Bibliography

1
F. Borceux Basic Category Theory, Handbook of Categorical Algebra I, Cambridge University Press, Cambridge (1994)



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Other names:  diagonally polar pair
Also defines:  orthogonal

Attachments:
properties of orthogonality on morphisms (Derivation) by CWoo
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Cross-references: pullback, closed under, Galois connection, subclasses, iff, classes, imply, isomorphism, orthogonality relation, properties, clear, well-defined, injective function, function, surjective, category of sets, ordered pair, commutative, diagram, commutative diagram, category, morphism
There are 6 references to this entry.

This is version 15 of orthogonal morphisms, born on 2008-10-14, modified 2008-10-16.
Object id is 11171, canonical name is OrthogonalMorphisms.
Accessed 317 times total.

Classification:
AMS MSC: 18A32 (Category theory; homological algebra :: General theory of categories and functors :: Factorization of morphisms, substructures, quotient structures, congruences, amalgams)
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