[clump 1 ...

?? idea of .... ???? bicocontinuous extension of cartesian product having extra left adjoint acting as categorified comultiplication for tensor product acting as categorified multiplication ?? ....

?? basic object of study in toric geometry as sym mon cat which is also topos with essential diagonal with "the essentialness" (?? ...) acting as comult for the tensor product as mult .... ??? .... ??? _is_ this at all parsing (?? ...) now ??? .... ?? affine case ??? ....

?? mayhem ... ??? same category acting as both measures and functions .... ??? .... ?? topos as topos _of_ categorified functions on categorified space .... ?? ... toric quasicoherent sheaves as ... ??? categorified measures on ... ??? symmetric monoidal cat, in affine case ??? .... ????? ..... ?? if diagonal of topos is non-essential (?? when _does/n't_ this happen, if n/ever ?? .... ??? .... ?? confusion about relationship to "totally distributive" and/or non-/affine ?? ... ??? ....), then ..... ????? somewhat hopeless to think of the topos objects as measures ??? ....

???? "measure vs op-measure" here ???? ..... ???? ..... ???? ....

??? cartesian product in topos as _the_ left adjoint of the "diagonal" geometric morphism ... (?? consider decategorified analog ... ?? ....) .... ?? and that left adjoint is backwards to the diagonal arrow itself, so without essentialness of diagonal, topos objects seem to be behaving as functions rather than as measures ... ??? but now .... with essentialness of diagonal (?? so as to have topos objects able to behave as measures, so as to get a "comultiplication" left adjoint from diagonal, so as to use this as an actual comultiplication with bimonoid compatibility with tensor product .... ????? ...) then we should be getting an additional left adjoint, right ?? ... ?? but where / what is that extra left adjoint ?? ... is it something that we already had but was only a right adjoint, or is it itself new and its right adjoint was previously existing ??? .... ??? essentialness means there's an extra .... hmmm, i was going to say extra right adjoint ... ?? but maybe that's wrong, and moreover good that it's wrong ???? .... ?? the correct idea being that there's an extra _left_ adjoint, which makes "the" left adjoint of the geometric morphism into a right adjoint as well ... ???? ..... ??? so we're hoping that .... left adjoint of cocontinuous extension (??? hmmm, seems like i've been saying "bicocontinuous extension of cocontinuous" for "cocontinuous extension of bicocontinuous" a bit ...) of cartesian product really does exist and act as comultiplication for tensor product as multiplication, even / especially in non-affine case .... ??? .... ??? and we're still pretty confused about "fourier dual" of this idea .... ????? ......

??? antidote / synthesis to .... ??? .... level slips (?? ...) involving ... "bimonoid compatibility with diagonal comultiplication as roughly corresponding to living sharply on basis of point-like elts wrt diagonal comultiplication" ... ???? pun (?? or _is_ it just common sense ??? ....) on "diagonal" / "diagonalized" here ???? .... ?? when cartesian product of representables is/n't representable .... ??? ....

?? toric geometry as generalized fourier duality and todd's question about what accidental topos classifies ... ??? .....

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[clump 2 ....

?? correspondences ... and / or "partial maps" .... ?? vs alternative tensor product ??? ..... monoids wrt these various alternatives .... ???? .... ??? stacky vs non-stacky case ??? ... ?? funny relationship to toric geometry ???? ...... non-affine case .... ????? ...

?? post-doctrinal logic ... ?? weirdness thereof ?? ... ?? lack of automatic "agglomeration process" .... "sketch" .... ??? .... ?? progression from .... ?? theory where model cat is nicely closed under certain agglomeration processes (?? "positiveness" .... ??? ... ??? ....) to where model cat is not but theory 2-cat is ... ?? to case where not even (...) theory 2-cat is .... ??? .... ??? ....

]


?? "vicarious empathy" .... ?? ... (?? "even if you can't experience real empathy ...")

maxwell ... electromagnetism ... "restoration of symmetry between electricity and magnetism" ... analogy here .... ?? extra left adjoint to bicocontinuous extension of tensor product ??? ..... ?? problems ??? .... ?? usual one of back to danger of getting monoidal topos in non-affine case .... ???? .... ?? dual categorified bialgebra ... ??? .... ?? good (?? ... ?? "representative" ?? ... ???? ....) categorified linear functional .... ????? boolean alg ... opposite category .... commutativity ..... ???? .....

?? idea of ordinary quasicoherent sheaves on toric variety as closer than toric quasicoherent sheaves to achieving maxwell symmetry ideal ?? .... ?? but maybe also losing distinctive toric quality ... ??? i mean, in that non-toruic examples may exhibit same pattern ... ???? ..... .... ??? bug / feature ... ?? ....

??? 3/4 ??? .... ???

?? level slip (??? ...) concerning ... ?? automaticness (?? ...) of compatibility with diagonal comultiplication .... from ordinary commutative monoid to bialg of measures and/or cateogrified bialg of categorified measures .... ??? ..... ??? ....