Timeline for What Spec-like functors are there?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Aug 30, 2016 at 20:59 | comment | added | 54321user | From the point of view of the functor of points, you should just consider functors $\textbf{Spec}(X) := \text{Hom}_\mathcal{C}(X,-)$. | |
| Aug 25, 2016 at 13:02 | comment | added | Tim Campion | See here (that post talks about commutative monoids in $\mathcal{V}$, but noncommutative monoids in $\mathcal{V}$ work equally well, as long as $\mathcal{V}$ is enriched in commutative monoids). | |
| Aug 25, 2016 at 13:00 | comment | added | Tim Campion | 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. | |
| Aug 22, 2016 at 17:47 | comment | added | მამუკა ჯიბლაძე | 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. | |
| Aug 22, 2016 at 16:03 | comment | added | Alex Mennen | @მამუკაჯიბლაძე, source? | |
| Aug 22, 2016 at 6:31 | comment | added | მამუკა ჯიბლაძე | 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 | |
| Aug 22, 2016 at 4:46 | comment | added | Mike Shulman | There is a fair menagerie of "spectrum" functors in Peter Johnstone's book Stone Spaces. | |
| Aug 22, 2016 at 4:25 | history | edited | Alex Mennen |
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| Aug 22, 2016 at 4:18 | history | asked | Alex Mennen | CC BY-SA 3.0 |