?? cauchy completion of commutative monoid ??? ...

didn't notice at first how non-trivial this might be ??? todd more or less pointed this out when i brought up the idea of trying to recover a toric variety from its accidental topos alone ....

(??non-toric analog here ??? ? ?? stuff simon willerton told me about ... ???"correspondences" between algebraic varieties, and situations where there aren't too many, so that you can actually recover the variety from its mere abelian category of quasicoherent sheaves ??? ... ???? ....)

??so what _is_ the cauchy completion of a semi-lattice like, for example ??? ..... ???how about the "half-decategorification" of such?? ???

??very vaguely reminds me, again, of ... ?? cocone category of cofan, and of things that that reminds me of ... ??? ..... ??closed category ... ?? ....

?? ??? _other_ parsing of "cauchy-completion of semi-lattice" ??? .... ????? ....