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edited Jan 25, 2023 at 18:51

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You might be interested in Cole's theory of spectrum. Besides the references given on that nlab page (which include several related theoretical frameworks, such as Lurie's theory of structured infinity-topoi), another good reference is AlexAxel Osmond's thesis.

In brief, the idea is something like the following. If $\mathcal B$ is a topos (e.g. $\mathcal B = BCRing = [CRing^{fp},Set]$, the classifying topos for commutative ring objects), one seeks to identify additional structure to define the category of "locally $\mathcal B$-structured topoi" (in our example, the additional structure picks out those commutative ring objects which are local, and those maps of ringed topoi which are local). Then one sets up a contravariant adjunction between points of $\mathcal B$ and locally $\mathcal B$-structured topoi (the functor from the former to the latter is the Spectrum construction in our example).

One nice thing about this sort of framework is that it also includes the example of the étale spectrum of a ring, by changing the additional data determining what it means to be "locally $\mathcal B$-structured".

You might be interested in Cole's theory of spectrum. Besides the references given on that nlab page (which include several related theoretical frameworks, such as Lurie's theory of structured infinity-topoi), another good reference is Alex Osmond's thesis.

In brief, the idea is something like the following. If $\mathcal B$ is a topos (e.g. $\mathcal B = BCRing = [CRing^{fp},Set]$, the classifying topos for commutative ring objects), one seeks to identify additional structure to define the category of "locally $\mathcal B$-structured topoi" (in our example, the additional structure picks out those commutative ring objects which are local, and those maps of ringed topoi which are local). Then one sets up a contravariant adjunction between points of $\mathcal B$ and locally $\mathcal B$-structured topoi (the functor from the former to the latter is the Spectrum construction in our example).

One nice thing about this sort of framework is that it also includes the example of the étale spectrum of a ring, by changing the additional data determining what it means to be "locally $\mathcal B$-structured".

You might be interested in Cole's theory of spectrum. Besides the references given on that nlab page (which include several related theoretical frameworks, such as Lurie's theory of structured infinity-topoi), another good reference is Axel Osmond's thesis.

In brief, the idea is something like the following. If $\mathcal B$ is a topos (e.g. $\mathcal B = BCRing = [CRing^{fp},Set]$, the classifying topos for commutative ring objects), one seeks to identify additional structure to define the category of "locally $\mathcal B$-structured topoi" (in our example, the additional structure picks out those commutative ring objects which are local, and those maps of ringed topoi which are local). Then one sets up a contravariant adjunction between points of $\mathcal B$ and locally $\mathcal B$-structured topoi (the functor from the former to the latter is the Spectrum construction in our example).

One nice thing about this sort of framework is that it also includes the example of the étale spectrum of a ring, by changing the additional data determining what it means to be "locally $\mathcal B$-structured".

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created Jan 25, 2023 at 17:49

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You might be interested in Cole's theory of spectrum. Besides the references given on that nlab page (which include several related theoretical frameworks, such as Lurie's theory of structured infinity-topoi), another good reference is Alex Osmond's thesis.

In brief, the idea is something like the following. If $\mathcal B$ is a topos (e.g. $\mathcal B = BCRing = [CRing^{fp},Set]$, the classifying topos for commutative ring objects), one seeks to identify additional structure to define the category of "locally $\mathcal B$-structured topoi" (in our example, the additional structure picks out those commutative ring objects which are local, and those maps of ringed topoi which are local). Then one sets up a contravariant adjunction between points of $\mathcal B$ and locally $\mathcal B$-structured topoi (the functor from the former to the latter is the Spectrum construction in our example).

One nice thing about this sort of framework is that it also includes the example of the étale spectrum of a ring, by changing the additional data determining what it means to be "locally $\mathcal B$-structured".