Questions taking place in the category of locales, which is given by the opposite of the category of frames. Also appropriate for questions about pointless topology.

A locale is just a frame, that is, a poset which, when viewed as a category, has all small (meaning indexed over a set) coproducts (joins), and all finite products (meets), and such that there is an infinite distributive property:

$x \times \big( \coprod_{i \in I} y_i \big) \leq \coprod_{i \in I} ( x \times y_i )$.

The prototypical example of a frame is the frame of open sets on a topological space.

A morphism from a locale $A$ to a locale $B$ is a frame morphism from $B$ to $A$. That is, a poset morphism from $B$ to $A$ which preserves finite meets and small joins. The motivation for this is the observation that, given topological spaces $X$ and $Y$, with frames of opens given respectively by $A$ and $B$, a map from $X$ to $Y$ is continuous if and only if the inverse-image map on the powersets takes opens in $Y$ to opens in $X$. That is, it is a frame morphism from $B$ to $A$. Thus the category of locales is kind of like the category of topological spaces, but with the points forgotten, making them 'pointless'.

The definitive source on matters of pointless topology is generally considered to be Peter Johnstone.