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Locales have been studied as "topologies without points", mainly by tools ofcategory theory. While traditional topology presents a space as a set of pointswith specified neighborhoods, localic topology presents a space as a lattice ofneighborhoods of unspecified points. Points are derivable as perfect ideals ofneighborhoods.  Another convenient fiction are the objects in category theory. Thecategorical view of mathematical structures focuses on morphisms and theircompositions. Objects tell which morphisms are composable. They are the blackboxes enclosing structures accessible only along morphisms. Since we cannottell when such structures identical identical, the statements that involveobjects' identities have been called evil. Object prohibitions have a moralaspect.  A holiday quest for a categorical approach to categories as black boxes ledto catales as categories reduced to partial semigroups of morphisms, analogousto locales as spaces reduced to lattices of neighborhoods. Just like points canbe reconstructed as perfect filters of neighborhoods in locales, objects can bereconstructed as the top elements of the lattices of idempotents in catales.

Catales: From topologies without points to categories without objects