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A scheme is equivalent to a functor $\mathcal{F}:\textbf{AffSchemes} \rightarrow \textbf{Set}$ such that it admits a cover of affine schemes and is a sheaf of rings on the Zariski site.

Suppose $\mathcal{C}$ is a subcategory of $\textbf{AffSchemes}$ and $\mathcal{T}:\mathcal{C}\rightarrow \textbf{Set}$ is a functor

Question, can we find a condition on $\mathcal{T}$ and $\mathcal{C}$ which tells us if $\mathcal{T}$ extends to a functor $\mathcal{F}:\textbf{AffSchemes} \rightarrow \textbf {Set}$ and is represented by an scheme.

Example for example we can consider the category of Noetherian complete local rings (which is in fact the one that interests me. $\underline{\text{EDIT}}$: it is a bit more, I'm interested in Noetherian complete local rings with a fixed residue field).

Question Same question for algebraic spaces and stacks.

A scheme is equivalent to a functor $\mathcal{F}:\textbf{AffSchemes} \rightarrow \textbf{Set}$ such that it admits a cover of affine schemes and is a sheaf of rings on the Zariski site.

Suppose $\mathcal{C}$ is a subcategory of $\textbf{AffSchemes}$ and $\mathcal{T}:\mathcal{C}\rightarrow \textbf{Set}$ is a functor

Question, can we find a condition on $\mathcal{T}$ and $\mathcal{C}$ which tells us if wether $\mathcal{T}$ extends to a functor $\mathcal{F}:\textbf{AffSchemes} \rightarrow \textbf {Set}$ and is represented by an scheme.

Example for example we can consider the category of Noetherian complete local rings (which is in fact the one that interests me. $\underline{\text{EDIT}}$: it is a bit more, I'm interested in Noetherian complete local rings with a fixed residue field).

Question Same question for algebraic spaces and stacks.

A scheme is equivalent to a functor $\mathcal{F}:\textbf{AffSchemes} \rightarrow \textbf{Set}$ such that it admits a cover of affine schemes and is a sheaf of rings on the Zariski site.

Suppose $\mathcal{C}$ is a subcategory of $\textbf{AffSchemes}$ and $\mathcal{T}:\mathcal{C}\rightarrow \textbf{Set}$ is a functor

Question, can we find a condition on $\mathcal{T}$ and $\mathcal{C}$ which tells us if $\mathcal{T}$ extends to a functor $\mathcal{F}:\textbf{AffSchemes} \rightarrow \textbf {Set}$ and is (a scheme) represented by an element of $\mathcal{C}$.

Example for example we can consider the category of Noetherian complete local rings (which is in fact the one that interests me. $\underline{\text{EDIT}}$: it is a bit more, I'm interested in Noetherian complete local rings with a fixed residue field).

Question Same question for algebraic spaces and stacks.

A scheme is equivalent to a functor $\mathcal{F}:\textbf{AffSchemes} \rightarrow \textbf{Set}$ such that it admits a cover of affine schemes and is a sheaf of rings on the Zariski site.

Suppose $\mathcal{C}$ is a subcategory of $\textbf{AffSchemes}$ and $\mathcal{T}:\mathcal{C}\rightarrow \textbf{Set}$ is a functor

Question, can we find a condition on $\mathcal{T}$ and $\mathcal{C}$ which tells us if $\mathcal{T}$ extends to a functor $\mathcal{F}:\textbf{AffSchemes} \rightarrow \textbf {Set}$ and is represented by an scheme.

Example for example we can consider the category of Noetherian complete local rings (which is in fact the one that interests me. $\underline{\text{EDIT}}$: it is a bit more, I'm interested in Noetherian complete local rings with a fixed residue field).

Question Same question for algebraic spaces and stacks.

A scheme is equivalent to a functor $\mathcal{F}:\textbf{AffSchemes} \rightarrow \textbf{Set}$ such that it admits a cover of affine schemes and is a sheaf of rings on the Zariski site.

Suppose $\mathcal{C}$ is a subcategory of $\textbf{AffSchemes}$ and $\mathcal{T}:\mathcal{C}\rightarrow \textbf{Set}$ is a functor

Question, can we find a condition on $\mathcal{T}$ and $\mathcal{C}$ which tells us if $\mathcal{T}$ extends to a functor $\mathcal{F}:\textbf{AffSchemes} \rightarrow \textbf {Set}$ and is (a scheme) represented by an element of $\mathcal{C}$.

Example for example we can consider the category of Noetherian complete local rings (which is in fact the one that interests me).

Question Same question for algebraic spaces and stacks.

A scheme is equivalent to a functor $\mathcal{F}:\textbf{AffSchemes} \rightarrow \textbf{Set}$ such that it admits a cover of affine schemes and is a sheaf of rings on the Zariski site.

Suppose $\mathcal{C}$ is a subcategory of $\textbf{AffSchemes}$ and $\mathcal{T}:\mathcal{C}\rightarrow \textbf{Set}$ is a functor

Question, can we find a condition on $\mathcal{T}$ and $\mathcal{C}$ which tells us if $\mathcal{T}$ extends to a functor $\mathcal{F}:\textbf{AffSchemes} \rightarrow \textbf {Set}$ and is (a scheme) represented by an element of $\mathcal{C}$.

Example for example we can consider the category of Noetherian complete local rings (which is in fact the one that interests me. $\underline{\text{EDIT}}$: it is a bit more, I'm interested in Noetherian complete local rings with a fixed residue field).

Question Same question for algebraic spaces and stacks.