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In the literature I've read, it is often said that to a Hecke eigenclass, one would like (and sometimes succeeds) to "associate" or show the existence of a Galois representation such that the Hecke and Frobenius eigenvalues match up. My question is if there exists a good category of Hecke eigenclasses such that this association becomes a functor to the category of Galois representations?

Does follow from the assumptions that we put on the compatibility of the Hecke eigenclass and the Galois representations? A reason why I believe this is not crazy to hope for is that for instance in the global setting, in $\text{GL}_2$, by the work of many people (including Eichler, Shimura, Deligne, Langlands, Carayol etc) we know that the Langlands correspondence is given by $$\pi\mapsto \text{Hom}_{ \text{Gl}_2(\mathbb{A}_f) }(\pi, \text{colim}_N H^1_c(Sh(N),\overline{\mathbb{Q}_p})$$ which is a (contravariant) functor from some subcategory of $\text{Gl}_2(\mathbb{A}_f)$-representations to $G_\mathbb{Q}$-representations.

In particular, do we expect/know this in the mod $p$ setting, i.e. is the construction of Scholze in here functorial in the Hecke-eigenclass?

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    $\begingroup$ In the classical case, there is the conjectural Langlands group, e.g., see mathoverflow.net/q/74698/6518 $\endgroup$
    – Kimball
    Commented Apr 11, 2022 at 18:29
  • $\begingroup$ I don’t know a direct answer to your question but it feels somewhat unnatural - the global Langlands correspondence is most naturally a statement on the level of vector spaces (automorphic forms vs “functions on” (eg deformation rings of) Langlands parameters..), not of categories. OTOH Local Langlands and global geometric Langlands are (conjectural) statements about equivalences of categories (while local geometric Langlands concerns an equivalence of 2-categories).. $\endgroup$
    – David Ben-Zvi
    Commented Apr 12, 2022 at 17:08
  • $\begingroup$ You can also up the categorical level by varying the group, so you have a category of groups and homomorphisms - or more generally a “Morita” category of groups and joint actions (eg bi- Hamiltonian actions). This setup captures Langlands functoriality and the theory of periods (the relative Langlands program) $\endgroup$
    – David Ben-Zvi
    Commented Apr 12, 2022 at 17:16
  • $\begingroup$ @DavidBen-Zvi I mean yes... but...the correspondence $$\pi\mapsto \text{Hom}_{ \text{Gl}_2(\mathbb{A}_f) }(\pi, \text{colim}_N H^1_c(Sh(N),\overline{\mathbb{Q}_p})$$ is so... tempting to think of as a functor? Well, it is one for sure, no? That's a large reason for why I want there to be a functor in general.. $\endgroup$
    – curious math guy
    Commented Apr 12, 2022 at 17:40
  • $\begingroup$ I agree there are functors involved, but the question is how rich the categories are (eg what do you take for a category of automorphic representations), of which I'm ignorant.. if you're talking about a category say of irreducible (or more generally semisimple) reps of some group, there's not a lot of structure there beyond matching sets. $\endgroup$
    – David Ben-Zvi
    Commented Apr 12, 2022 at 22:46

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