Does the existence of pullbacks of monomorphisms imply exisyence of arbitrary intersections of subobjects?

Question

I'm sorry for the silly doubt.

Assume that a category admits pullbacks of monomorphisms along any arrow. May I deduce the existence of arbitrary intersections in every subobject poset?

Answer 1

If you have pullbacks of monomorphisms then you have finitary meets (greatest lower bounds, infima, intersections – whatever you want to call them) of subobjects. For infinitary meets you need some other assumption.

For example, consider the filter of cofinite subsets of any infinite set. It is not hard to see that it is closed under finitary intersections in the powerset, so the filter has finitary meets. However, the filter has no lower bound, so the filter does not have infinitary meets. Thus, we have an example of a category with pullbacks of monomorphisms but does not have complete subobject lattices.

Answer 2

Disclaimer:
I still think the answer is: in general no. However, the following is not a complete counterexample, since the subobject-lattice is in fact complete (thanks to Zhen Lin for pointing this out!). I will keep the answer up for the time being, since it at least gives a meta reason, why the implication should fail in general.

Consider the category $\mathsf{Top}_O$ of topological spaces and open continuous maps. The forgetful functor $\mathsf{Top}_O\rightarrow \mathsf{Set}$ has a left adjoint given by discrete spaces, which means that monomorphisms are injective open continuous functions ie. inclusion of open subspaces. But obviously an arbitrary intersection of open subspaces need not be open anymore. This shows that the arbitrary intersection of open subobjects may not be realized as a categorical limit.