?? trying to understand example of toric projective plane … ??image factorization of 1+1 -> l …. ??? confusion about which half of facotrization is causing the problem …. ???? ….. trying to identify extra axiom needed …. ??? …..

?? roberts … 2-group … generalize …. other stuff …. main theme … ??? ….response …

?? co/homology …. ??? derived functors … of …. ??? ?? operation vs homomorphism … ???   … leray-serre …. ??? … product along for the ride …. ??? ….

?? co-yoneda morphism …. slice yoga …. ?? of presheaf topos over representable … ??? … ?? level slip ??? … “potential coverings” (?? …) …. ?? then grothendieck topology on site category of such …. ??? …. ?? …. formula / model confusion ….. ????? …..non-/distributivity … and / or no limits in doctrine at all …. ???? …..

?? but …  when colimits already exist then co-yoneda morphism reduces to just plain morphism … invertiblization of morphisms, in cocomplete world …. ??? …..

?? morphisms vs monomorphisms …. vs cocomplete cat vs topos …. ??? …. ?? is it “x gets inverted if its monic part does” or is it “getting inverted implies monicness”, or …. ???? ….

?? sums, and pullbacks along summands … ??? weird to have colimits before limits ??? ….

?? being 2-group as mere property of toric dimensional theory ?? … ??? invertibleness of morphisms giving covariance of inverse …. ???? ….