??? model = co-simplicial object .... (?? in cocomplete category ???) ...

?? formula = simplicial set .... ??though "basic formula" = simplex ??? ....

?? co-yoneda morphism into basic formula = ... simplicial set over walking j-simplex ...

??? "simplicial set with 0-simplexes bi-colored, with all edges going from earlier color to later color" ... ????

?? "geometric realization scheme that makes this simplicial set look like the edge, via the hopefully obvious projection" ... ???...

?? bipartite graph vs set-pair family ???

???try taking bipartite graphjs as the "things to be geometrically realized" and set-pair families as the "geometric realization schemes" ....

?? notate a set-pair family something like this: {(2,0),(1,3),(1,1)}

??notate a bi-partite graph something like ... ???a rectangular matrix i guess???

2 5
0 1
2 3

???so what's the geometric realization / tensor product in this case?

??sew in 2 211's abd 3 031's ???....

??? not much "glue" to hold things together here ??????? ..... ??? .... ???? ....

???some conceptual mistake here ??? ..... ????

????hmmmmm....ok, i guess that i'm supposed to do more glueing than i was realizing at first ... ???when you "glue in an edge from a 211 to a 031" (??? ...), .... ????you're actually identifying the 2+0 to a single point, and also the 1+3 and the 1+1 .... ???? ??? fits naive intuition better if ... the "inclusion of the model source into the model arrow" is in fact a genuine "inclusion" ... injective and non-surjective .... ????? ???maybe should start with such an example first ???....

(?? again, "model" pun here ... ???? "staunton chess pieces" .... ??? ??as "abstract concrete (???) models of idealized abstract chess pieces" ??? ... ??? ... ???perhaps chess position as sort of thing that can get "realized using staunton realization scheme" ... ?? ...)

??? "classical-valued" model vs non-such .... ????..... ????...

???so ... ???now we seem to be looking for ... ???a set-pair family where ... ???? well, amounts to "2 disjoint subsets with non-total union" ... ??? ... say, {(1,0),(0,0),(0,1),(0,1)} .... ?????

?? then geometric realization of that 2-by-3 matrix above is ... ???1 point for each of the two sources... 2 points for each of the 3 targets ... ?????and 1 point for each of the 13 arrows... ??again, still (??...) not much "glue" to, for example, allow discernment of which source and which target an arrow is glued to .... ?????....

??anyway, seems to be a total of... 21 points in the geometric realization here ??? ....

???idea that what slightly _less_ naive intuition calls for is .... ????? yes, having the points of the model source not bump into each other outright when included into the model arrow, and not bump (outright ...) into the points of the model target either... but ... ??that "bumping into each other" (in both cases just alluded to...) should in fact happen in a more "subtle", "weak" way... (??discernible only in "variable" environment ???? ...) .... amounting to a sort of delicate but strong "glue" ... ????? ..... ??? ??? vague sense of interesting "homotopy-theoretic" ideas here .... ???? .....


(???? elevating finite limits theory to geometric theory, vs elevating arbitrary colimits theory to geometric theory ?????????? .....)

(??? possibility of getting those "naive intuition" constraints on geometric realization scheme to be enforced as axioms expressed under this or some extended doctrine .... ???? ... ????)