User:Rovigo/Sandbox

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This is Rovigo's sandbox page for practicing the use of mathematical notation on Wikipedia.

1 Subobjects (Category Theory)
2 Motivation
3 Definition
- 3.1 Ordering of Subobjects
- 3.2 Isomorphism of Subobjects
4 Comments
5 See also

[edit] Subobjects (Category Theory)

In mathematics, specifically in the field of category theory, the notion of subobject is an attempt to abstract the properties of 'substructures' found all over mathematics.

[edit] Motivation

Consider the familiar notion of subset

Inclusions
Image of a set-monomorphism is isomorphic to its source.
Ordering of subsets.

[edit] Definition

In an arbitrary category $\mathcal{C}$ one speaks of 'a subobject of $D$ ', for some $\mathcal{C}$ -object $D$ . In its simplest formulation such a subobject is merely a $\mathcal{C}$ -monomorphism:

$m \colon A \rightarrowtail D$

[edit] Ordering of Subobjects

Just as subsets of a set X are ordered by inclusion, so it is desirable to impose an ordering on subobjects of D in an arbitrary category $\mathcal{C}$ .

$g \sqsubseteq f \quad \mbox{iff} \quad g = f \circ k \quad \mbox{for some} \quad k \colon B \rightarrowtail A$

Diagram

Note that $k$ will itself be a subobject of $A$ in this situtation, ie $k$ is itself a monomorphism.

It is easily verified that this relation is both reflexive and transitive but, in general, $\sqsubseteq$ will pre-order the set of monomorphisms into $D$ , not partially order it, ie it fails to be antisymmetric.

[edit] Isomorphism of Subobjects

The ordering described in the previous section yields a pre-order on coterminous monomorphisms.

Converting the preorder $\sqsubseteq$ into an equivalence relation $\backsimeq$ is achieved by quotienting the preordered set of coterminous monomorphisms into equivalence classes such that monomorphisms $f$ , $g$ are deemed equivalent iff they factor through one another. Precisely: