To any space $X$ you can associate its de Rham space $X_{dR}$. Vector bundles on $X_{dR}$ are the same thing as vector bundles on $X$ with a flat connection.

Can anything like this be said for meromorphic connections?

For instance, a naive idea is that there might literally be a space $X_{mdR}$ whose vector bundles are vector bundles on $X$ with a flat meromorphic connection. Then there would be maps $\eta_{dR}\to X_{mdR}\to X_{dR}$, where $\eta$ is the generic point of $X$ (on which all meromorphic bundles are holomorphic), suggesting that maybe $X_{mdR}$ could be constructed as some sort of thickening of $X_{dR}$ in $\eta_{dR}$.

I'm aware that Deligne has a structure theorem for regular meromorphic connections, so maybe it's more reasonable to restrict to the regular meromorphic case.

To any space $X$ you can associate its de Rham space $X_{dR}$. Vector bundles on $X_{dR}$ are the same thing as vector bundles on $X$ with a flat connection.

Can anything like this be said for meromorphic connections?

For instance, a naive idea is that there might literally be a space $X_{mdR}$ whose vector bundles are vector bundles on $X$ with a flat meromorphic connection. Then there would be maps $\eta_{dR}\to X_{mdR}\to X_{dR}$, where $\eta$ is the generic point of $X$ (on which all meromorphic bundles are holomorphic), suggesting that maybe $X_{mdR}$ could be constructed as some sort of intermediate object between $\eta_{dR}$ and $X_{dR}$.

I'm aware that Deligne has a structure theorem for regular meromorphic connections, so maybe it's more reasonable to restrict to the regular meromorphic case.

To any space $X$ you can associate its de Rham space $X_{dR}$. Vector bundles on $X_{dR}$ are the same thing as vector bundles on $X$ with a flat connection.

Can anything like this be said for meromorphic connections?

For instance, a naive idea is that there might literally be a space $X_{mdR}$ whose vector bundles are vector bundles on $X$ with a flat meromorphic connection. Then there would be maps $X_{dR}\to X_{mdR}\to \eta_{dR}$, where $\eta$ is the generic point of $X$ (on which all meromorphic bundles are holomorphic), suggesting that maybe $X_{mdR}$ could be constructed as some sort of thickening of $X_{dR}$ in $\eta_{dR}$.

I'm aware that Deligne has a structure theorem for regular meromorphic connections, so maybe it's more reasonable to restrict to the regular meromorphic case.

To any space $X$ you can associate its de Rham space $X_{dR}$. Vector bundles on $X_{dR}$ are the same thing as vector bundles on $X$ with a flat connection.

Can anything like this be said for meromorphic connections?

For instance, a naive idea is that there might literally be a space $X_{mdR}$ whose vector bundles are vector bundles on $X$ with a flat meromorphic connection. Then there would be maps $\eta_{dR}\to X_{mdR}\to X_{dR}$, where $\eta$ is the generic point of $X$ (on which all meromorphic bundles are holomorphic), suggesting that maybe $X_{mdR}$ could be constructed as some sort of thickening of $X_{dR}$ in $\eta_{dR}$.

I'm aware that Deligne has a structure theorem for regular meromorphic connections, so maybe it's more reasonable to restrict to the regular meromorphic case.