(joint w/ D. Halpern-Leistner)

Given two algebraic stacks X and Y, when is Hom(X, Y)
representable by an algebraic stack and how do you prove it?
The original "Weil restriction" (i.e., X finite) constructions
were explicit, and one still has explicit constructions in the
case that X is projective via Quot-schemes. The more general
cases where X is a proper scheme, or a proper stack, seems to
inevitably require an appeal to Artin's representability
theorem aided by Grothendieck existence. The classical
literature does this in case X is a proper stack with finite
diagonal [Olsson] or X a proper stack but with restrictions on
Y [Aoki, Lieblich]; but one also finds the ingredients needed
to prove the case X = BG_{m} and Y = BG -- and here X is
certainly not a proper stack.

In this talk, we'll give a common generalization of these results. To do this, we will offer up new definitions of "proper" and "projective" for stacks so that the implications

(1) "projective" => "proper"

(2) flat and "proper" => "mapping stack exists Hom(X, Y) exists for most reasonable Y"

hold and where now "proper" is phrased in terms of sheaves and
is not much stronger than just the conclusion of the
Grothendieck existence theorem. We'll give a laundry-list of
examples of "projective" stacks, including BG_{m}, A^{1}/G_{m}, and
more generally X/G where X is projective-over-affine and
H^{0}(O_{X})^{G} = k; and, all stacks with enough vector bundles and
admitting a projective good moduli space, and quotients X/G
where X is projective-over-affine with G acting on the
affinization.

Beyond definitions, we'll also explain the salient points of a few key proofs: J. Lurie's beautiful Tannakian argument that essentially proves the implication (2); a "derived h-descent" result that gives a different view on classical Chow's lemma arguments for showing that ordinary proper implies "proper"; and some words on the rest.