User:Rovigo/Sandbox

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

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.

Motivation
Consider the familiar notion of subset
 * Inclusions
 * Image of a set-monomorphism is isomorphic to its source.
 * Ordering of subsets.

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:


 * $$m \colon A \rightarrowtail D$$

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.

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:


 * $$f \backsimeq g \quad \mbox{iff} \quad f \sqsubseteq g \quad \mbox{and} \quad g \sqsubseteq f$$

Comments
In many category theory texts there is some fudging of what exactly is referenced by the term 'subobject'.


 * $$S$$ is a subobject of $$D$$.
 * $$m \colon A \rightarrowtail D$$ is a subobject.
 * Equivalence class of a monomorphism.

Really, it is the monomorphism itself that specifies the subobject not merely its source.