??so as preparation for trying to understand weighted homotopy limits and weighted homotopy colimits of spectrums (or something like that...), let me try to remind myself about certain aspects of weighted limits and weighted colimits of vector spaces (if i ever actually understood those aspects...)...

let's consider weighted limits and colimits where the k-enriched "diagram scheme" is the unit k-enriched category (aka "the walking object") ... ??so the weight is a k-object w and the diagram is a c-object x with c some k-enriched category... and the weighted limit is something like "x raised to the power of the external exponent w" or something like that... k-enriched right-universal property of x^w as... ??...

y -> x^w
----------
w -> [y,x]

??or something??

??is there some nice relationship between "spanier-whitehead duality" (or something...) and "verdier duality" ??? ....

hmm... googling on these two together seems to give a lot of hits connected with someone named "roy joshua" ...

hmmm... checking notebook357 from when simon willerton was here... and some wikipedia articles... some things look promising (??some idea from simon about "derived category as motivated by duality" or something... ??...) but also some confusion between various types of "duality" ... serre ... verdier ... poincare ... and so forth ... ???....