Editing Subobject (section)

==Definitions==
An appropriate categorical definition of "subobject" may vary with context, depending on the goal. One common definition is as follows.

In detail, let ''<math>A</math>'' be an object of some category.  Given two [[monomorphism]]s

:<math>u: S \to A \ \text{and} \ v: T\to A</math>

with [[codomain]] ''<math>A</math>'', we define an equivalence relation by <math>u \equiv v</math>  if there exists an isomorphism <math>\phi: S \to T</math> with <math>u = v \circ \phi</math>.

Equivalently, we write <math>u \leq v</math> if <math>u</math> [[Mathematical jargon#factor through|factors through]] ''<math>v</math>''—that is, if there exists <math>\phi: S \to T</math> such that <math>u = v \circ \phi</math>.  The binary relation <math>\equiv</math> defined by

:<math>u \equiv v \iff u \leq v \ \text{and} \ v\leq u </math>

is an [[equivalence relation]] on the monomorphisms with codomain ''<math>A</math>'', and the corresponding [[equivalence class]]es of these monomorphisms are the '''subobjects''' of ''<math>A</math>''.

The relation ≤ induces a [[partial order]] on the collection of subobjects of <math>A</math>.

The collection of subobjects of an object may in fact be a [[proper class]]; this means that the discussion given is somewhat loose. If the subobject-collection of every object is a [[Set (mathematics)|set]], the category is called ''well-powered'' or, rarely, ''locally small'' (this clashes with a different usage of the term [[Locally small category|locally small]], namely that there is a set of morphisms between any two objects).

To get the dual concept of '''quotient object''', replace "monomorphism" by "[[epimorphism]]" above and reverse arrows. A quotient object of ''A'' is then an equivalence class of epimorphisms with domain ''A.''

However, in some contexts these definitions are inadequate as they do not concord with well-established notions of subobject or quotient object. In the category of topological spaces, monomorphisms are precisely the injective continuous functions; but not all injective continuous functions are subspace embeddings. In the category of rings, the inclusion <math>\mathbb{Z} \hookrightarrow \mathbb{Q}</math> is an epimorphism but is not the quotient of <math>\mathbb{Z}</math> by a two-sided ideal. To get maps which truly behave like subobject embeddings or quotients, rather than as arbitrary injective functions or maps with dense image, one must restrict to monomorphisms and epimorphisms satisfying additional hypotheses. Therefore, one might define a "subobject" to be an equivalence class of so-called "regular monomorphisms" (monomorphisms which can be expressed as an equalizer of two morphisms) and a "quotient object" to be any equivalence class of "regular epimorphisms" (morphisms which can be expressed as a coequalizer of two morphisms)