Given a homomorphism f:G→H between smooth algebraic groups, we get an induced homomorphism of algebraic stacks Bf:BG→BH, given by sending a G-torsor P over a scheme X to the H-torsor PxGH, whose (scheme-theoric) points are {(p,h)|p∈P,h∈H}/∼, where (pg,h)∼(p,f(g)h).


Is every morphism of algebraic stacks BG→BH of the form Bf? If not, what is an example of a morphism not of this form?



Does every morphism BG-->BH come from a homomorphism G-->H?
Depends on the base scheme and the topology being used. For example if you're working over a field k in the etale or the flat topology, and take the group G to be trivial, you're asking if H^1(k,H) is trivial, which is obviously false in general. This is, in a sense, the only obstruction: for any base scheme S, giving a map from BG to any stack Y (in stacks/S) is the same as specifying a point y of Y(S), and a homomorphism G -> Aut\_S(y). In particular, if BH(S) is connected (i.e., if H^1(S,H) = \*) then the answer to your question is positive.


