Let $D$ be a small category. Does the category of diagrams $\mathsf{Top}^{D^{\text{op}}}$ have a classifier of (strong?) subobjects? I tried following the "sieve construction" for the category of presheaves, but I don't see what topology to put on the set of sieves on an object in $D$ (or perhaps this won't work anyway).


I'm also wondering whether the existence of (strong?) subobject classifier has anything at all to do with the existence of a (strong) subobject classifier in $\mathsf{Top}$, or whether this is more or less unrelated.



Subobject classifier in $\mathsf{Top}^{D^{\text{op}}}$?
If any of these categories had a subobject classifier, every monomorphism would be regular, so that's not happening.


The indiscrete two-point space is a strong subobject classifier in $\mathsf{Top}.$ Similarly, the subobject classifier for $\mathsf{Set}^{D^{\mathrm{op}}},$ equipped with the indiscrete topology on its values, is a strong subobject classifier for $\mathsf{Top}^{D^{\mathrm{op}}}.$ This is because the forgetful functor to presheaves induces an isomorphism between strong subobjects of a topological presheaf and arbitrary subobjects of a set-valued presheaf.


