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The real spectrum functor is an analog of Spec for partially ordered commutative rings and real closed fields in place of commutative reals and algebraically closed fields. I was hoping that there would also be a nice Spec-like functor for not-necessarily-commutative rings, but this paper seems fairly discouraging about that. Are there other functors that people have studied which are like Spec but from other categories (particularly varieties of algebras, or the category of models of a first-order theory)? More generally, what properties should a functor satisfy for us to consider it Spec-like, and what can then be said about Spec-like functors in general?

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    $\begingroup$ There is a fair menagerie of "spectrum" functors in Peter Johnstone's book Stone Spaces. $\endgroup$
    – Mike Shulman
    Commented Aug 22, 2016 at 4:46
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    $\begingroup$ There also is an approach by Yves Diers based on multiadjoints - basically, if there is not a unique universal arrow but every connected component holds one, then some spectrum-like construction is possible. The usual spectrum occurs as every homomorphism to a local ring factors uniquely through exactly one of the localizations at primes. I believe Diers also considered some noncommutative examples $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Aug 22, 2016 at 6:31
  • $\begingroup$ @მამუკაჯიბლაძე, source? $\endgroup$
    – Alex Mennen
    Commented Aug 22, 2016 at 16:03
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    $\begingroup$ The most to the point one is probably Some spectra relative to functors (JPAA 1981). Johnstone wrote "A syntactic approach to Diers' localizable categories", it is in the 1977 Durham Symposium volume "Applications of sheaves" (Springer LNM 753, 466-478). Will try to find some more. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Aug 22, 2016 at 17:47
  • $\begingroup$ If you allow Spec to take values in some category of "spaces", one answer is that the dual of a category of ring-like objects can often be regarded as a category of "spaces" directly (e.g. affine schemes)-- so that the identity functor is "Spec-like". For example, if $\mathcal{V}$ is a Cauchy complete symmetric monoidal semiadditive category, then the opposite of the category of monoids in $\mathcal{V}$ is extensive, so "space-like" in at least a weak sense. $\endgroup$
    – Tim Campion
    Commented Aug 25, 2016 at 13:00

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