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Some time ago, I prepared a detailed (hopefully correct) summary on using the AFT for proving the cocompleteness of varieties of algebras (by providing a left adjoint for the diagonal functor).
From glancing in Borceux (and in algebra over a Lawvere theory), I’ve seen that this can be proved in a different way for algebras over Lawvere theories (which I’m still not familiar with). The question is, in view of the fact that there is already an $n$Lab entry proving cocompleteness in a more modern language, is there a point in adding such a summary for varieties of algebras? I can think of some reasons to do so, but I prefer not to clutter the $n$Lab with entries that are considered useless by everyone else.
Note also, that the summary in question assumes the foundations of CWM, and from the discussion in Categories Work I understood that these are not widely accepted by the set theory experts here in $n$Lab.
Thanks,
Yaron
You should go ahead and add that stuff somewhere. If you fear an existing entry my be cluttered up, just put a link there saying something like
“A discussion of cocompleteness using xyz can be found here.”
and create a dedicated entry for your discusison.
There is lots of space on the nLab. Any discussion that is considered useful by anyone can have its place somewhere here.
Thanks. I’ll do that.
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