$\mathbf{SpFi}$ is the category of spaces with filters: an object is a pair $(X,\mathcal{F}) $, $X$ a compact Hausdorff space and $\mathcal{F}$ a filter of dense open subsets of $X$. A morphism $f\: (Y,\mathcal{G}) \rightarrow (X,\mathcal{F}) $ is a continuous function $f\: Y\rightarrow X$ for which $f^{-1}(F) \in \mathcal{G}$ whenever $F\in \mathcal{F}$. This category arises naturally from considerations in ordered algebra, e.g., Boolean algebra, lattice-ordered groups and rings, and from considerations in general topology, e.g., the theory of the absolute and other covers, locales, and frames, though we shall specifically address only one of these connections here in an appendix. Now we study the categorical monomorphisms in $\mathbf{SpFi}$. Of course, these monomorphisms need not be one-to-one. For general $\mathbf{SpFi}$ we derive a criterion for monicity which is rather inconclusive, but still permits some applications. For the category $\mathbf{LSpFi}$ of spaces with Lindelöf filters, meaning filters with a base of Lindelöf, or cozero, sets, the criterion becomes a real characterization with several foci ($C(X) $, Baire sets, etc.), and yielding a full description of the monofine coreflection and a classification of all the subobjects of a given $(X,\mathcal{F}) \in \mathbf{LSpFi}$. Considerable attempt is made to keep the discussion “topological,” i.e., within $\mathbf{SpFi}$, and to not get involved with, e.g., frames. On the other hand, we do not try to avoid Stone duality. An appendix discusses epimorphisms in archimedean $\ell $-groups with unit, roughly dual to monics in $\mathbf{LSpFi}$.
Usually, an abelian $\ell $-group, even an archimedean $\ell $-group, has a relatively large infinity of distinct $a$-closures. Here, we find a reasonably large class with unique and perfectly describable $a$-closure, the class of archimedean $\ell $-groups with weak unit which are “$\mathbb Q$-convex”. ($\mathbb Q$ is the group of rationals.) Any $C(X,\mathbb Q)$ is $\mathbb Q$-convex and its unique $a$-closure is the Alexandroff algebra of functions on $X$ defined from the clopen sets; this is sometimes $C(X)$.