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As far as I know, there are four possible ways to generalize algebraic geometry by 'simply' replacing the basic category of rings with something similar but more general:

$\bullet$ In the approach by Toen-Vaquié we fix a nice symmetric monoidal category $C$, also called a relative context. An affine scheme is defined to be an algebra object in $C$, and an arbitrary scheme is a certain presheaf on affine schemes. We optain the category $\mathrm{Sch}(C)$ of schemes relative to $C$.

$\bullet$ In Durov's theory a generalized ring is an algebraic monad which is commutative in a certain sense. Then affine schemes are defined to be the spectra of generalized rings and arbitrary schemes are optained by gluing. This results in the category $\mathrm{genSch}$.

$\bullet$ In his book Categories of commutative algebras Yves Diers considers Zariski categories, which seem to axiomatize familiar properties of categories of commutative algebras. If $\mathcal{A}$ is such a Zariski category, then one can develope commutative algebra internal to $\mathcal{A}$, construct affine schemes and then by gluing also schemes as usual. We optain the category $\mathrm{Sch}(\mathcal{A})$.

$\bullet$ In derived algebraic geometry one replaces the category of rings with the category of simplicial rings (but I don't really know enough about that, yet).

My question is: What are the connections between these 'generalized algebraic geometries'?

Fortunately there is a map of $\mathbb{F}_1$-land which draws connections between all these various approaches to schemes over $\mathbb{F}_1$. For example monoid schemes à la Deitmar/Kato are in the intersection of Toen-Vaquiè and Durov. Note, however, that the theories mentioned above are far more general.

Specifically, one might ask the following questions: Is the category of generalized rings a Zariski category and is Durov's theory (say, with the unary localization theory) a special case of the one by Yves Diers? What is the relationship between Toen-Vaquié schemes relative to the symmetric monoidal category of simplicial rings and derived schemes? If $C$ is a relative context, is then the category of algebra objects in $C$ a Zariski category and do the corresponding schemes coincide? Probably not because Diers never mentions monoids as an example, but perhaps it's the other way round? Of course, many more questions are out there ...

Probably I'm not the first one with this question, therefore I've also put the "reference-request" tag. It would be great if there is some paper like "Mapping AG-land".

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Connections between various generalized algebraic geometries (Toen-Vaquié, Durov, Diers, Lurie)?

As far as I know, there are four possible ways to generalize algebraic geometry by 'simply' replacing the basic category of rings with something similar but more general:

$\bullet$ In the approach by Toen-Vaquié we fix a nice symmetric monoidal category $C$, also called a relative context. An affine scheme is defined to be an algebra object in $C$, and an arbitrary scheme is a certain presheaf on affine schemes. We optain the category $\mathrm{Sch}(C)$ of schemes relative to $C$.

$\bullet$ In Durov's theory a generalized ring is an algebraic monad which is commutative in a certain sense. Then affine schemes are defined to be the spectra of generalized rings and arbitrary schemes are optained by gluing. This results in the category $\mathrm{genSch}$.

$\bullet$ In his book Categories of commutative algebras Yves Diers considers Zariski categories, which seem to axiomatize familiar properties of categories of commutative algebras. If $\mathcal{A}$ is such a Zariski category, then one can develope commutative algebra internal to $\mathcal{A}$, construct affine schemes and then by gluing also schemes as usual. We optain the category $\mathrm{Sch}(\mathcal{A})$.

$\bullet$ In derived algebraic geometry one replaces the category of rings with the category of simplicial rings (but I don't really know enough about that, yet).

My question is: What are the connections between these 'generalized algebraic geometries'?

Fortunately there is a map of $\mathbb{F}_1$-land which draws connections between all these various approaches to schemes over $\mathbb{F}_1$. For example monoid schemes à la Deitmar/Kato are in the intersection of Toen-Vaquiè and Durov. Note, however, that the theories mentioned above are far more general.

Specifically, one might ask the following questions: Is the category of generalized rings a Zariski category and is Durov's theory (say, with the unary localization theory) a special case of the one by Yves Diers? What is the relationship between Toen-Vaquié schemes relative to the symmetric monoidal category of simplicial rings and derived schemes? If $C$ is a relative context, is then the category of algebra objects in $C$ a Zariski category and do the corresponding schemes coincide? Of course, many more questions are out there ...

Probably I'm not the first one with this question, therefore I've also put the "reference-request" tag. It would be great if there is some paper like "Mapping AG-land".
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