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orthogonal morphisms
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(Definition)
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A morphism in a category
is said to be orthogonal to a morphism in
, written
if whenever we have a commutative diagram
there is a unique morphism such that the diagram
is commutative also. If , we sometimes call the ordered pair 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 is surjective and injective, with
, where and are functions. For any , there is some such that since is surjective. Define by
. Now, if there is such that
, then
. Since is injective, . This shows that is a well-defined function. It is clear that
and
. Now, if is another such a function, then
, so that since is injective. This shows that is uniquely defined.
Here are some basic properties of the orthogonality relation on morphisms:
- If either
or is an isomorphism, then .
- If
, then is an isomorphism.
- If
and , then
. Similarly, and imply
. Of course, both statements make sense provided that exists.
More generally, if
and
are two classes of morphisms in a category
, we say that
is orthogonal to
, or that
is a diagonally polar pair, written
, if for every in
and every in
.
For every class
of morphism, the largest class of morphisms in
such that
is orthogonal to is denoted by
, and the largest class of morphisms that is orthogonal to
is denoted by
.
Based on the properties of above, below are some properties of and :
-
iff
. Equivalently, if
is the class of all subclasses of morphisms of
, then is a Galois connection between
and
.
- A morphism is in both
and
iff it is an isomorphism.
- Both
and
are closed under .
- Given that
exists in
and
, then
iff
.
- If
, then the pullback of along any morphism is again in
.
- 1
- F. Borceux Basic Category Theory, Handbook of Categorical Algebra I, Cambridge University Press, Cambridge (1994)
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"orthogonal morphisms" is owned by CWoo.
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(view preamble | get metadata)
| Other names: |
diagonally polar pair |
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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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Pending Errata and Addenda
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