  One of the most fundamental open problems in Incidence Geometry, posed by
Tits in the 1960s, asks for the existence of so-called "locally finite
generalized polygons" | that is, generalized polygons with "mixed parameters"
(one being finite and the other not). In a more specialized context, another
long-standing problem (from the 1990s) is as to whether the endomorphism ring
of any translation generalized quadrangle is a skew field (the answer of which
is known in the finite case). (The analogous problem for projective planes, and
its positive solution, the "Bruck-Bose construction," lies at the very base of
the whole theory of translation planes.) In this short note, we introduce a
category, representing certain very specific embeddings of generalized
polygons, which surprisingly controls the solution of both (apparently entirely
unrelated) problems.
