To any space $X$ you can associate its [de Rham space][1] $X_{dR}$. 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? 
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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.


  [1]: https://ncatlab.org/nlab/show/de+Rham+space