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There are several important facts that I first heard about here on MO. One of the most enlightening of these is that $\mathscr D$-modules on a scheme $X$ may be viewed as sheaves on the groupoid of infinitesimally near points of $X$.

Recently I also learned about several versions of $q$-analogs of $\mathscr D$-modules. The naïve idea is just to replace the differential operators with $q$-difference operators. There seem to be many subtleties involved, so the actual definitions seem to be different; one well-developed version seems to be deformation quantization modules introduced by Kashiwara & Schapira.

However what I want to ask is very naïve anyway. It is related to my previous, even more lightweight question Multiplicative infinitesimals in q-analogs?Multiplicative infinitesimals in q-analogs? where I asked how to model algebraically something that is "infinitesimally close to 1" (as opposed to "infinitesimally close to 0", which is modelled by nilpotents).

Namely, what I want to ask is this: is there an analog of the infinitesimal path groupoid with the property that $q$-analogs of $\mathscr D$-modules could be viewed as sheaves on it?

There are several important facts that I first heard about here on MO. One of the most enlightening of these is that $\mathscr D$-modules on a scheme $X$ may be viewed as sheaves on the groupoid of infinitesimally near points of $X$.

Recently I also learned about several versions of $q$-analogs of $\mathscr D$-modules. The naïve idea is just to replace the differential operators with $q$-difference operators. There seem to be many subtleties involved, so the actual definitions seem to be different; one well-developed version seems to be deformation quantization modules introduced by Kashiwara & Schapira.

However what I want to ask is very naïve anyway. It is related to my previous, even more lightweight question Multiplicative infinitesimals in q-analogs? where I asked how to model algebraically something that is "infinitesimally close to 1" (as opposed to "infinitesimally close to 0", which is modelled by nilpotents).

Namely, what I want to ask is this: is there an analog of the infinitesimal path groupoid with the property that $q$-analogs of $\mathscr D$-modules could be viewed as sheaves on it?

There are several important facts that I first heard about here on MO. One of the most enlightening of these is that $\mathscr D$-modules on a scheme $X$ may be viewed as sheaves on the groupoid of infinitesimally near points of $X$.

Recently I also learned about several versions of $q$-analogs of $\mathscr D$-modules. The naïve idea is just to replace the differential operators with $q$-difference operators. There seem to be many subtleties involved, so the actual definitions seem to be different; one well-developed version seems to be deformation quantization modules introduced by Kashiwara & Schapira.

However what I want to ask is very naïve anyway. It is related to my previous, even more lightweight question Multiplicative infinitesimals in q-analogs? where I asked how to model algebraically something that is "infinitesimally close to 1" (as opposed to "infinitesimally close to 0", which is modelled by nilpotents).

Namely, what I want to ask is this: is there an analog of the infinitesimal path groupoid with the property that $q$-analogs of $\mathscr D$-modules could be viewed as sheaves on it?

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მამუკა ჯიბლაძე
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$q$-crystals - is there such a thing?

There are several important facts that I first heard about here on MO. One of the most enlightening of these is that $\mathscr D$-modules on a scheme $X$ may be viewed as sheaves on the groupoid of infinitesimally near points of $X$.

Recently I also learned about several versions of $q$-analogs of $\mathscr D$-modules. The naïve idea is just to replace the differential operators with $q$-difference operators. There seem to be many subtleties involved, so the actual definitions seem to be different; one well-developed version seems to be deformation quantization modules introduced by Kashiwara & Schapira.

However what I want to ask is very naïve anyway. It is related to my previous, even more lightweight question Multiplicative infinitesimals in q-analogs? where I asked how to model algebraically something that is "infinitesimally close to 1" (as opposed to "infinitesimally close to 0", which is modelled by nilpotents).

Namely, what I want to ask is this: is there an analog of the infinitesimal path groupoid with the property that $q$-analogs of $\mathscr D$-modules could be viewed as sheaves on it?