[Parse it as (locally compact)ly generated.]
I stumbled across this one whilst supervising an undergraduate thesis. Convenient categories for homotopy theory (e.g. CGWH) have been discussed here before. As an alternative to CGWH, Rainer Vogt proposed the category of locally compactly generated spaces; see also the recent German-language point-set topology textbook Grundkurs Topologie by Gerd Laures and Markus Szymik.
If (as is now usual) one means (compact Hausdorff)ly generated when one says compactly generated, then the category of compactly generated spaces is a full subcategory of the category of locally compactly generated spaces. Back in 1971, Vogt asked whether this inclusion is strict or not. Do we know the answer yet?