??so consider a slice topos of the object classifier... say over object x ... ??then... ???this is the classifying topos for ... ??algebras of the free "substitution" monoid on x ... ??or something?? ???what about generalizing this somehow to non-free such monoids ?? ... ??or something?? ....

no wait, that's not correct... try some examples ...

x = "the object square", for example ... ???so a model should be... ???an object equipped with ... ???a pair of points??

x = "the object to its own power" ...??"object equipped with endomorphism" ???/ ???or something????

??compare this to some other hopefully straightforward interpretation of "theory of object equipped with endomorphism" ... ??....

??actually, now this (...) whole idea is seeming wrong ... ??simply because of exponentiation not really being part of geometric doctrine ?? ... ??and i sort of almost knew that already or should have... ??but i think that i was influenced by this alleged "minimalistic syntax" idea... about which i'm now a bit puzzled... how can you get "interesting structure" using just (??...) "adding a generic point of a given object" (or something...) ???.....

???for example, how to get classifiyng topos for "dynamical system" using the minimalistic syntax?? ...??maybe ask todd?? ...


??something about ... ??burroni monad ... as _not_ an example of a substitution monoid in object classifier over topos of directed graphs ... ??... ??or something ???...