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I’ve written a stub on Cole’s theory of spectrum which for the time being consists largely of references and links. Further curation or correction would be appreciated!
Thanks! I suppose this is the abstract concept of spectrum also considered in Structured Spaces.
I changed some wording in the definition section ever so slightly.
Thanks! I’ve added some further material from the account in Johnstone 1977. There Johnstone attributes the observation of the right adjointness property of spec to Hakim. Cole’s construction yields some standard examples. I don’t know enough about Hakim’s or Lurie’s work to see how Cole’s stuff is precisely related to theirs. I’ve added a link to Structured Spaces though.
Thanks.
We have an old stubby entry with the sweet name locally algebra-ed topos which is effectively about this concept of “$T$-modelled topos”, up so some evident variants. For the time being I have made T-modelled topos redirect to that. Eventually one might split off a separate entry, or rename that existing entry, and in any case expand further (maybe by copying the relevant section from Cole’s theory of spectra).
I have added some more hyperlinks to the entry, in the first paragraphs.
Regarding the entry title: a more canonical title choice might be “Cole spectrum” or maybe “spectrum of a model over a geometric theory” or the like.
I have no problems with renaming as the title is a bit heavy. I was kind of afraid that ’Cole spectrum’ suggests for one thing too much a concrete spectrum for some particular algebraic theory and too established a term, but I sure can live with it. Maybe something like ’Cole spectra’ could be an alternative. Given that these words appear in Cole’s title they are probably good search-engineering. So whoever has a clear preference should go ahead and rename!
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