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A Weil Cohomology theory is a functor from the category of smooth projective varieties (over some fixed field $k$) to graded $K$-algebras (for some fixed field $K$) satisfying various axioms. For example, Betti,de Rham, $\ell$-adic and rigid cohomologies are all Weil cohomology theories.

In each of these settings one frequently encounters the concept of a $\textit{category category of coefficients}$ coefficients for a given variety. For example, in the $\ell$-adic setting the category of coefficients of a variety $X$ is the category of (not necessarily lisse) $\ell$-adic sheaves on $X$. In the Betti setting it is the category of constructible sheaves on $X$. In the de Rham setting it is the category of regular, holonomic $D$-modules on $X$. In the rigid setting a category of coefficients on $X$ is given by overholonomic arithmetic $F$-$D$-modules. For each theory there is a $\textit{constant}$ constant object for every variety and these give the usual Weil cohomology functor.

I have two questions (the second is related to the first):

1. What is the conceptual meaning of a category of coefficients of a given variety for a Weil cohomology theory? Is there a set of axioms such categories must obey? If so does anyone have a reference? Why, for example, do we take only regular holonomic $D$-modules and not all $D$-modules in the de Rham setting.?

2. In each of the above examples there are important subcategories of $\textit{smooth}$ smooth objects. For the above cases these are given by: lisse $\ell$-adic sheaves, local systems, integrable connections, and overconvergent $F$-isocrystals. What is the conceptual meaning of these privileged subcategories?

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# Coefficients of Weil Cohomology Theories

A Weil Cohomology theory is a functor from the category of smooth projective varieties (over some fixed field $k$) to graded $K$-algebras (for some fixed field $K$) satisfying various axioms. For example, Betti,de Rham, $\ell$-adic and rigid cohomologies are all Weil cohomology theories.

In each of these settings one frequently encounters the concept of a $\textit{category of coefficients}$ for a given variety. For example, in the $\ell$-adic setting the category of coefficients of a variety $X$ is the category of (not necessarily lisse) $\ell$-adic sheaves on $X$. In the Betti setting it is the category of constructible sheaves on $X$. In the de Rham setting it is the category of regular, holonomic $D$-modules on $X$. In the rigid setting a category of coefficients on $X$ is given by overholonomic arithmetic $F$-$D$-modules. For each theory there is a $\textit{constant}$ object for every variety and these give the usual Weil cohomology functor.

I have two questions (the second is related to the first):

1. What is the conceptual meaning of a category of coefficients of a given variety for a Weil cohomology theory? Is there a set of axioms such categories must obey? If so does anyone have a reference? Why for example, do we take only regular holonomic $D$-modules and not all $D$-modules in the de Rham setting.

2. In each of the above examples there are important subcategories of $\textit{smooth}$ objects. For the above cases these are given by: lisse $\ell$-adic sheaves, local systems, integrable connections, and overconvergent $F$-isocrystals. What is the conceptual meaning of these privileged subcategories?