Even formulating the problem for needs in representation theory is non-trivial, e.g. to see that constructible sheaves on $X\times X$ is monoidal under convolution. The problem is how to formulate (upper-*, lower-!) base-change in a homotopy-smart way. Fortunately, this problem has been solved: Francis-Gaitsgory suggest a solution in their paper on chiral algebras using categories of correspondences. The idea is obviously rich enough to carry over to any sheaf theoretic setting. But that format hasn't been published yet, though I understand there is forthcoming work of Gaitsgory-Rozenblyum.

I think this is exactly what people call "non-Abelian Lubin-Tate theory."

If f^* has a left adjoint, then f^* is left exact. I.e., f is flat.

My comment was about the geometric side of things. One might hope for a geometric story in wild ramification as well (as conjecturally there is one in the characteristic zero/D-modules case).

For b: certainly you mean in the tamely ramified case, yes?

It might be clarifying to note that the $0$-space of $HA$ is $A$ considered as a discrete space, and that there's not much choice other than Eilenberg-Maclane for how to turn $A$ into a spectrum.

What work of Klein are you thinking of?

@Mariano: As you probably know, your local unitality is the same as my topology above! One won't be able to get this category as (unital) modules over a (unital) ring, and every answer in this thread finds some way to circumvent it :)

You have a forgetful functor to abelian groups: let $A$ be the endomorphisms of this functor considered as a topological ring via the projective limit topology. Your category will be equivalent to discrete modules over this ring since the forgetful functor is exact, faithful, and commutes with colimits. You can see topologies are necessary already in the case of graded vector spaces, where the ring is $\mathcal{O}_{\mathbb{G}_m}^{\vee}$ equipped with the algebra structure coming from the coalgebra structure on $\mathbb{G}_m$.

Right, and if the automorphism groups are trivial then it's an algebraic space.

The map to the moduli of pointed curves (without tangent vectors) seems to be trivially representable -- am I missing something?

And one should note that Anon's example is a special case of this when $G$ is the semi-direct product of $\mathbb{G}_m$ and $\mathbb{G}_a$.

Though if you don't use basepoints (as in Dold-Thom), the limiting result is (weakly) contractible.

@Donu: Indeed, but then the question really is: why $\ell$-adic sheaves at all, instead of some other coefficients? (And this is more or less the question Ryan has answered). @Jan: A good way to get a feel for this definition is to work out what it means for the spectrum of a field, to see that it's the same (modulo equivalence) as continuous representations of the Galois group of the field into a finite extension of $\mathbb{Q}_{\ell}$.

What is a more naive definition?

There is a deep conjecture here which is explained nicely in the paper of Kontsevich-Zagier.

There's a general notion of G-bundle with flat connection for an algebraic group G, in particular for G unipotent. However, they form a groupoid and not an abelian category, so perhaps it's not what you want. Can you clarify what you're looking for? Does some author use this phrase? Or else can you give examples of what you want to axiomatize?

1. Yes. 2. It's called the Weil restriction. Certainly it exists if $\pi$ is finite flat. Perhaps it always exists (at least as an algebraic space) if $\pi$ is proper flat.