Let $\mathcal{I}$ be the class of all functions $\iota: S \rightarrow X$ inserting a subset $S \subseteq X$ into a larger set.
The presence of the family $\mathcal{I}$ underlies the subset relation $\subseteq$ between sets. For example, $S$ is subset of $X$ if and only if there exists a morphism $\iota \in \mathcal{I}$ going from $S$ to $X$($S \xrightarrow{\iota} X$). $\mathcal{I}$ also underlies set intersection, set union.
Is there a way to define an analogous class $\mathcal{I}^\prime$ of monomorphism in another topos $\mathcal{E}$ in a way that allows us to do "set like" reasoning in $\mathcal{E}$?
I would like to define $\mathcal{I}^\prime$ in a way that gives me a sensible notion of intersection and union between two objects $X, Y \in \mathcal{E}$.
I would like a family $\mathcal{I}^\prime$ that satisfies at least the conditions below:
- every morphism in the family $\mathcal{I}^\prime$ is mono;
- if $X \in \mathcal{E}$ and $S$ is a subobject of $X$, then there exists a unique $m: M \rightarrow X$ contained in $\mathcal{I}^\prime$ and $S$. Said in other words, every subobject contain a unique morphism of $\mathcal{I}^\prime$
- $\mathcal{I}^\prime$ is closed under identity, that is, for every $X \in \mathcal{E}$, the identity morphism $X \xrightarrow{1_X} X$ is in $\mathcal{I}^\prime$;
- $\mathcal{I}^\prime$ is closed under composition, that is, if $X \xrightarrow{f} Y$ and $Y \xrightarrow{g} Z$ belong to $\mathcal{I}^\prime$, then $X \xrightarrow{f;g} Z$ also belongs to $\mathcal{I}^\prime$
- let $f, g, h$ be morphisms in $\mathcal{E}$, as in the triangle below. If $g$ and $h$ belong to $\mathcal{I}^\prime$, then $X \xrightarrow{f;g} Z$ also belongs to $\mathcal{I}^\prime$.
All of those four properties are satisfied in the categories of Sets by the family $\mathcal{I}$.