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The work of Street "Two-dimensional sheaf theory" (JPAA 23 1982 251-270) is relevant I believe.

He has notions of 1-separated and 2-separated, and a 2-analog of the $\_^+$ functor which he calls $L$, with natural transformation $l:\text{identity}\to L$, such that

• $P$ is 1-separated iff $l_P:P\to LP$ is mono;
• for every $P$, $LP$ is 1-separated;
• $P$ is 2-separated iff $l_P:P\to LP$ is $\textit{chronic}$;
• for every 1-separated $P$, $LP$ is 2-separated;
• $P$ is a stack iff $l_P:P\to LP$ is an isomorphism;
• for every 2-separated $P$, $LP$ is a stack.

Here a morphism $f:P\to Q$ is called chronic when the functor $\hom(X,f):\hom(X,P)\to\hom(X,Q)$ is full, faithful and injective on objects for every $X$.

Comparing this with the properties of $\_^+$ seemingly leads to the conclusion that separatedness naturally splits into two notions, something like "$2/3$-separatedness" and "$4/3$-separatedness".

See also the (later) Question regarding 2-mathematics: Can you stackify a 2-functor without prestackifying it first? here on MO.

Update

After comments by David Roberts I must confess I no longer believe it is relevant. However I still don't understand why it is irrelevant, so let me leave it for a while. Sorry.

The work of Street "Two-dimensional sheaf theory" (JPAA 23 1982 251-270) is relevant I believe.

He has notions of 1-separated and 2-separated, and a 2-analog of the $\_^+$ functor which he calls $L$, with natural transformation $l:\text{identity}\to L$, such that

• $P$ is 1-separated iff $l_P:P\to LP$ is mono;
• for every $P$, $LP$ is 1-separated;
• $P$ is 2-separated iff $l_P:P\to LP$ is $\textit{chronic}$;
• for every 1-separated $P$, $LP$ is 2-separated;
• $P$ is a stack iff $l_P:P\to LP$ is an isomorphism;
• for every 2-separated $P$, $LP$ is a stack.

Here a morphism $f:P\to Q$ is called chronic when the functor $\hom(X,f):\hom(X,P)\to\hom(X,Q)$ is full, faithful and injective on objects for every $X$.

Comparing this with the properties of $\_^+$ seemingly leads to the conclusion that separatedness naturally splits into two notions, something like "$2/3$-separatedness" and "$4/3$-separatedness".

See also the (later) Question regarding 2-mathematics: Can you stackify a 2-functor without prestackifying it first? here on MO.

Update

After comments by David Roberts I must confess I no longer believe it is relevant. However I still don't understand why it is irrelevant, so let me leave it for a while. Sorry.

The work of Street "Two-dimensional sheaf theory" (JPAA 23 1982 251-270) is relevant I believe.

He has notions of 1-separated and 2-separated, and a 2-analog of the $\_^+$ functor which he calls $L$, with natural transformation $l:\text{identity}\to L$, such that

• $P$ is 1-separated iff $l_P:P\to LP$ is mono;
• for every $P$, $LP$ is 1-separated;
• $P$ is 2-separated iff $l_P:P\to LP$ is $\textit{chronic}$;
• for every 1-separated $P$, $LP$ is 2-separated;
• $P$ is a stack iff $l_P:P\to LP$ is an isomorphism;
• for every 2-separated $P$, $LP$ is a stack.

Here a morphism $f:P\to Q$ is called chronic when the functor $\hom(X,f):\hom(X,P)\to\hom(X,Q)$ is full, faithful and injective on objects for every $X$.

Comparing this with the properties of $\_^+$ seemingly leads to the conclusion that separatedness naturally splits into two notions, something like "$2/3$-separatedness" and "$4/3$-separatedness".

See also the (later) Question regarding 2-mathematics: Can you stackify a 2-functor without prestackifying it first? here on MO.

The work of Street "Two-dimensional sheaf theory" (JPAA 23 1982 251-270) is relevant I believe.

He has notions of 1-separated and 2-separated, and a 2-analog of the $\_^+$ functor which he calls $L$, with natural transformation $l:\text{identity}\to L$, such that

• $P$ is 1-separated iff $l_P:P\to LP$ is mono;
• for every $P$, $LP$ is 1-separated;
• $P$ is 2-separated iff $l_P:P\to LP$ is $\textit{chronic}$;
• for every 1-separated $P$, $LP$ is 2-separated;
• $P$ is a stack iff $l_P:P\to LP$ is an isomorphism;
• for every 2-separated $P$, $LP$ is a stack.

Here a morphism $f:P\to Q$ is called chronic when the functor $\hom(X,f):\hom(X,P)\to\hom(X,Q)$ is full, faithful and injective on objects for every $X$.

Comparing this with the properties of $\_^+$ seemingly leads to the conclusion that separatedness naturally splits into two notions, something like "$2/3$-separatedness" and "$4/3$-separatedness".

See also the (later) Question regarding 2-mathematics: Can you stackify a 2-functor without prestackifying it first? here on MO.

Update

After comments by David Roberts I must confess I no longer believe it is relevant. However I still don't understand why it is irrelevant, so let me leave it for a while. Sorry.

The work of Street "Two-dimensional sheaf theory" (JPAA 23 1982 251-270) is relevant I believe.

He has notions of 1-separated and 2-separated, and the 2-analog of the $\_^+$ functor which he calls $L$, with natural transformation $l:\text{identity}\to L$, such that

• $P$ is 1-separated iff $l_P:P\to LP$ is mono;
• for every $P$, $LP$ is 1-separated;
• $P$ is 2-separated iff $l_P:P\to LP$ is $\textit{chronic}$;
• for every 1-separated $P$, $LP$ is 2-separated;
• $P$ is a stack iff $l_P:P\to LP$ is an isomorphism;
• for every 2-separated $P$, $LP$ is a stack.

Here a morphism $f:P\to Q$ is called chronic when the functor $\hom(X,f):\hom(X,P)\to\hom(X,Q)$ is full, faithful and injective on objects for every $X$.

Comparing this with the properties of $\_^+$ seemingly leads to the conclusion that separatedness naturally splits into two notions, something like "$2/3$-separatedness" and "$4/3$-separatedness".

See also the (later) Question regarding 2-mathematics: Can you stackify a 2-functor without prestackifying it first? here on MO.

The work of Street "Two-dimensional sheaf theory" (JPAA 23 1982 251-270) is relevant I believe.

He has notions of 1-separated and 2-separated, and a 2-analog of the $\_^+$ functor which he calls $L$, with natural transformation $l:\text{identity}\to L$, such that

• $P$ is 1-separated iff $l_P:P\to LP$ is mono;
• for every $P$, $LP$ is 1-separated;
• $P$ is 2-separated iff $l_P:P\to LP$ is $\textit{chronic}$;
• for every 1-separated $P$, $LP$ is 2-separated;
• $P$ is a stack iff $l_P:P\to LP$ is an isomorphism;
• for every 2-separated $P$, $LP$ is a stack.

Here a morphism $f:P\to Q$ is called chronic when the functor $\hom(X,f):\hom(X,P)\to\hom(X,Q)$ is full, faithful and injective on objects for every $X$.

Comparing this with the properties of $\_^+$ seemingly leads to the conclusion that separatedness naturally splits into two notions, something like "$2/3$-separatedness" and "$4/3$-separatedness".

See also the (later) Question regarding 2-mathematics: Can you stackify a 2-functor without prestackifying it first? here on MO.