??well so for example, let's continue playing around with set-pair families construed as geometric realization schemes for bipartite graphs ... ??to get some of the flavor of various possibilities ... ???for example ... ??well, for a real simple example, realize the sources and targets as points, and realize the arrows as equations ... ???this as vaguely reminding me of something ?? ... ?? maybe various things ... ???....

anyway, this seems to give the set of "components" of the bipartite graph ...

i was going to give a more involved, random silly example ... something like realize a source as 5 points a,b,c,d,e and a target as 4 points f,g,h,j, and an arrow as identifying , say, the d of the source with the f of the target ... ??? ...

???let's see... ?? is this example (of a site category ...) the right example for ... ??? putting in axiom saying that a model co-span is a co-product co-span ??? ... ??perhaps yes ?? ... ??so then what does the (??) axiom look like ??? .... ??hmm, might be able to think of this particular example as simply a grothendieck topology ... ???...

???hmm, this example reallty does seem to relate to .... ??? some of those "naive intuition" constraints that we mentioned... about what a geometric realization scheme might be like ... ???

???co-product injections as in fact monic ???... ?? monicness being something expressible in _limits_ doctrine, rather than colimits ... ???? .... ??? ...

???so ... ???a geometric realization scheme here as ess just a co-product co-span
precisely in case .... ???? .... the co-yoneda morphism from walking source-target-pair to walking arrow gets geometrically realized as an isomorphism ... ???....

seems to hang together reasonably well so far ...

let me switch for a moment to a different very simple example ... site category terminal ... co-yoneda morphism from walking point-pair to walking point ... ??this co-yoneda morphism being non-monic ... ???? and ... ??so when is this co-yoneda morphism realized as an isomorphism?? ???precisely when the "model point" is initial??? .... ???so that this is "the theory of nothing" ??? .... ????

??how about ... co-yoneda morphism from walking nothing to walking point ... ???.... realized as isomorphism when model point is ... ???? initial ??? ... ??? ....

?? do formulas still form a topos ??? ..... .... ????...

didn't have time yet to check last few examples more carefully ...

??and in fact there's probably something a bit screwed up there ....

???but ... ???if quasitopologies really did involve non-monic co-yoneda morphisms in a key way, then wouldn't they call them "quasi-cribles" or "quasi-sieves" or something??? .... ??... so... ??do they??? ....

??? theory of object x with invertible codiagonal ... ????.... ???"non-disjoint coproduct" ... ???....