Franklin's Notes


Subobject

In category theory , the notion of a subobject is a way of generalizing the notions of subsets and inclusion in the category of sets $\mathsf{Set}$. If $c\in\mathsf{C}$ is an object of a category $\mathsf{C}$, then a subobject of $c$ is defined to be a monic morphism $f:x\to c$, where $x$ is any other object of $\mathsf{C}$. Further, if $f:x\to c$ and $g:y\to c$ are subobjects of $c$, we say that $f\subset g$ if and only if there exists some $h$ such that $f=g\circ h$. (Note that if such $h$ exists, it is also monic, and therefore defines a subobject of $y$.) Under this definition, $\subset$ is a generalization of the inclusion relation in $\mathsf{Set}$.

In $\mathsf{Set}$, the inclusion relation satisfies $3$ important properties:

1. Reflexivity: $A\subset A$ for all $A$.
2. Transitivity: if $A\subset B$ and $B\subset C$, then $A\subset C$.
3. Antisymmetry: if $A\subset B$ and $B\subset A$, then $A=B$.

Notice that inclusion is reflexive, that is, $f\subset f$ for all subobjects $f:x\to c$, because $f=f\circ 1_x$. Additionally, it is transitive: if we have $3$ subobjects $f:x\to c$, $g:y\to c$, and $h:z\to c$ such that $f\subset g\subset h$, then there exist (mono)morphisms $j:x\to y$ and $k:y\to z$ such that $g\circ j = f$ and $h\circ k = g$. But this implies that $h\circ (k\circ j) = f$, and therefore $f\circ h$, implying transitivity.

What about antisymmetry? In general, $f\subset g$ and $g\subset f$ do not imply that $f=g$. For instance if $g:y\to c$ is a subobject and $i:x\to y$ is an isomorphism between $y$ and some distinct (but isomorphic) object $x$, then $f=g\circ i:x\to c$ is a subobject of $c$ as well. Further, $f\subset g$ since $f=g\circ i$, and $g\subset f$ because $g=f\circ i^{-1}$. In this case, we say that $f$ and $g$ are isomorphic subobjects and write $f\cong g$, so that we have a limited form of antisymmetry: if $f\circ g$ and $g\circ f$, then $f\cong g$. We can also remedy the failure of antisymmetry by considering equivalence classes of subobjects (with a bit of additional bookkeeping required to ensure that the other properties of inclusion are well-defined after modding out by isomorphism). This construction is similar to taking the skeleton of a category.

Using this construction, we can consider the poset of subobjects $(\text{Sub}(c),\subset)$ of some object $c\in\mathsf{C}$, where the elements of the poset are equivalence classes of isomorphic subobjects of $c$. That is, if $[f]$ denotes the equivalence class consisting of all subobjects of $c$ isomorphic to $f:x\to c$, then In particular, for $c\in\mathsf{Set}$, the poset $(\text{Sub}(c),\subset)$ is isomorphic (as a partial order) to the poset $(\mathcal{P}(c),\subset)$ of subsets of $c$.

category-theory

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