On an infinite limit of BGG categories $\mathcal O$

We study a version of the BGG category $\mathcal{O}$ for Dynkin Borel subalgebras of root-reductive Lie algebras $\mathfrak{g}$, such as $\mathfrak{gl}(\infty)$. We prove results about extension fullness and compute the higher extensions of simple modules by Verma modules. In addition, we show that our category $O$ is Ringel self-dual and initiate the study of Koszul duality. An important tool in obtaining these results is an equivalence we establish between appropriate Serre subquotients of category $O$ for $\mathfrak{g}$ and category $\mathcal{O}$ for finite dimensional reductive subalgebras of $\mathfrak{g}$.