Difference between revisions of "Subobject"
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''of an object in a category'' | ''of an object in a category'' | ||
− | A concept analogous to the concept of a substructure of a mathematical structure. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909801.png" /> be any [[Category|category]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909802.png" /> be a fixed object in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909803.png" />. In the class of all monomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909804.png" /> with codomain (target) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909805.png" />, one may define a pre-order relation (the relation of divisibility from the right): <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909806.png" /> precedes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909807.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909808.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909809.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098010.png" />. In fact, the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098011.png" /> is uniquely determined because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098012.png" /> is a monomorphism. The pre-order relation induces an equivalence relation between the monomorphisms with codomain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098013.png" />: The monomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098015.png" /> are equivalent if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098017.png" />. An equivalence class of monomorphisms is called a subobject of the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098018.png" />. A subobject with representative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098019.png" /> is sometimes denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098020.png" /> or by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098021.png" />. It is also possible to use Hilbert's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098022.png" />-symbol to select representatives of subobjects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098023.png" /> and consider these representatives as subobjects. In the categories of sets, groups, Abelian groups, and vector spaces, a subobject of any object is defined by the imbedding of a subset (subgroup, subspace) in the ambient set (group, space). However, in the category of topological spaces, the concept of a subobject is wider than that of a subset with the induced topology. | + | A concept analogous to the concept of a substructure of a mathematical structure. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909801.png" /> be any [[Category|category]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909802.png" /> be a fixed object in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909803.png" />. In the class of all monomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909804.png" /> with codomain (target) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909805.png" />, one may define a pre-order relation (the relation of divisibility from the right): <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909806.png" /> precedes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909807.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909808.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909809.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098010.png" />. In fact, the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098011.png" /> is uniquely determined because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098012.png" /> is a monomorphism. The pre-order relation induces an equivalence relation between the monomorphisms with codomain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098013.png" />: The monomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098015.png" /> are equivalent if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098017.png" />. An equivalence class of monomorphisms is called a subobject of the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098018.png" />. A subobject with representative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098019.png" /> is sometimes denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098020.png" /> or by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098021.png" />. It is also possible to use Hilbert's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098022.png" />-symbol to select representatives of subobjects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098023.png" /> and consider these representatives as subobjects. In the categories of sets, groups, Abelian groups, and vector spaces, a subobject of any object is defined by the imbedding of a subset (subgroup, subspace) in the ambient set (group, space). However, in the category of topological spaces, the concept of a subobject is wider than that of a subset with the [[induced topology]]. |
The pre-order relation between the monomorphisms with codomain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098024.png" /> induces a partial order relation between the subobjects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098025.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098026.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098027.png" />. This relation is analogous to the inclusion relation for subsets of a given set. | The pre-order relation between the monomorphisms with codomain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098024.png" /> induces a partial order relation between the subobjects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098025.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098026.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098027.png" />. This relation is analogous to the inclusion relation for subsets of a given set. |
Revision as of 15:26, 20 November 2014
of an object in a category
A concept analogous to the concept of a substructure of a mathematical structure. Let be any category and let
be a fixed object in
. In the class of all monomorphisms of
with codomain (target)
, one may define a pre-order relation (the relation of divisibility from the right):
precedes
, or
, if
for some
. In fact, the morphism
is uniquely determined because
is a monomorphism. The pre-order relation induces an equivalence relation between the monomorphisms with codomain
: The monomorphisms
and
are equivalent if and only if
and
. An equivalence class of monomorphisms is called a subobject of the object
. A subobject with representative
is sometimes denoted by
or by
. It is also possible to use Hilbert's
-symbol to select representatives of subobjects of
and consider these representatives as subobjects. In the categories of sets, groups, Abelian groups, and vector spaces, a subobject of any object is defined by the imbedding of a subset (subgroup, subspace) in the ambient set (group, space). However, in the category of topological spaces, the concept of a subobject is wider than that of a subset with the induced topology.
The pre-order relation between the monomorphisms with codomain induces a partial order relation between the subobjects of
:
if
. This relation is analogous to the inclusion relation for subsets of a given set.
If the monomorphism is regular (cf. Normal monomorphism), then any monomorphism equivalent to it is also regular. One can therefore speak of the regular subobjects of any object
. In particular, the subobject represented by
is regular. In categories with zero morphisms one similarly introduces normal subobjects. If
possesses a bicategory structure
, then a subobject
of an object
is called admissible (with respect to this bicategory structure) if
.
Comments
The notation used in this article (and elsewhere in this Encyclopaedia) is not standard. Most authors do not bother to distinguish notationally between a subobject and a monomorphism which represents it.
For references see Category.
Subobject. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subobject&oldid=15026