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edited Sep 15, 2021 at 17:17

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To 'turn an object $A$ of a category $\mathcal{C}$ into a set' you can consider the set of objects mapping into/out of that object from/to some other object; if you consider maps out of a terminal object ${\bf 1}\to A$ these are called global elements of the object.

If $\mathcal{C}$ is a well-pointed topos these global elements are sufficient to delineate between parallel arrows with domain $X$ and thusly contain 'all relevant information' about $X$ in the topos. As Paul pointed out in his answer and Andrej elaborated on in his comment, if you consider $\Omega^A$ as a set in this way by considering its set of global elements $\{f:{\bf 1}\to \Omega^A|f\in\mathcal{C}\}$, this set is isomorphic to $Sub(A)$ as a lattice.

You may also want to read about the subobject fibration, which deals with the notion of categorical subobjects in a more satisfying way imo; you can turn the subobjects of an object into a category in an obvious way, and it turns out these categories are also the fibers of the fibration obtained by turning the codomain fibration into an indexed category via the Grothendieck construction, postcomposing with the skeleton endo $2$-functor on $\mathfrak{Cat}$, then turning the resulting indexed category back into a fibration again via Grothendieck in the other direction.

To 'turn an object $A$ of a category $\mathcal{C}$ into a set' you can consider the set of objects mapping into/out of that object from/to some other object; if you consider maps out of a terminal object ${\bf 1}\to A$ these are called global elements of the object.

If $\mathcal{C}$ is a well-pointed topos these global elements are sufficient to delineate between parallel arrows with domain $X$ and thusly contain 'all relevant information' about $X$ in the topos. As Paul pointed out in his answer and Andrej elaborated on in his comment, if you consider $\Omega^A$ as a set in this way by considering its set of global elements $\{f:{\bf 1}\to \Omega^A|f\in\mathcal{C}\}$, this set is isomorphic to $Sub(A)$ as a lattice.

You may also want to read about the subobject fibration, which deals with the notion of categorical subobjects in a more satisfying way imo; you can turn the subobjects of an object into a category in an obvious way, and it turns out these categories are also the fibers of the fibration obtained by turning the codomain fibration into an indexed category via the Grothendieck construction, postcomposing with the skeleton endo $2$-functor on $\mathfrak{Cat}$, then turning the resulting indexed category back into a fibration again via Grothendieck.

To 'turn an object $A$ of a category $\mathcal{C}$ into a set' you can consider the set of objects mapping into/out of that object from/to some other object; if you consider maps out of a terminal object ${\bf 1}\to A$ these are called global elements of the object.

If $\mathcal{C}$ is a well-pointed topos these global elements are sufficient to delineate between parallel arrows with domain $X$ and thusly contain 'all relevant information' about $X$ in the topos. As Paul pointed out in his answer and Andrej elaborated on in his comment, if you consider $\Omega^A$ as a set in this way by considering its set of global elements $\{f:{\bf 1}\to \Omega^A|f\in\mathcal{C}\}$, this set is isomorphic to $Sub(A)$ as a lattice.

You may also want to read about the subobject fibration, which deals with the notion of categorical subobjects in a more satisfying way imo; you can turn the subobjects of an object into a category in an obvious way, and it turns out these categories are also the fibers of the fibration obtained by turning the codomain fibration into an indexed category via the Grothendieck construction, postcomposing with the skeleton endo $2$-functor on $\mathfrak{Cat}$, then turning the resulting indexed category back into a fibration via Grothendieck in the other direction.

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created Sep 15, 2021 at 17:12

10.1k
3
30
88

To 'turn an object $A$ of a category $\mathcal{C}$ into a set' you can consider the set of objects mapping into/out of that object from/to some other object; if you consider maps out of a terminal object ${\bf 1}\to A$ these are called global elements of the object.

If $\mathcal{C}$ is a well-pointed topos these global elements are sufficient to delineate between parallel arrows with domain $X$ and thusly contain 'all relevant information' about $X$ in the topos. As Paul pointed out in his answer and Andrej elaborated on in his comment, if you consider $\Omega^A$ as a set in this way by considering its set of global elements $\{f:{\bf 1}\to \Omega^A|f\in\mathcal{C}\}$, this set is isomorphic to $Sub(A)$ as a lattice.

You may also want to read about the subobject fibration, which deals with the notion of categorical subobjects in a more satisfying way imo; you can turn the subobjects of an object into a category in an obvious way, and it turns out these categories are also the fibers of the fibration obtained by turning the codomain fibration into an indexed category via the Grothendieck construction, postcomposing with the skeleton endo $2$-functor on $\mathfrak{Cat}$, then turning the resulting indexed category back into a fibration again via Grothendieck.