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dg.differential geometry – Is there any Lie groupoid construction on $Hom(mathcal{G}, mathcal{H})$ the place $mathcal{G}$ and $mathcal{H}$ are Lie groupoids? Answer

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dg.differential geometry – Is there any Lie groupoid construction on $Hom(mathcal{G}, mathcal{H})$ the place $mathcal{G}$ and $mathcal{H}$ are Lie groupoids?

We know that in common, there isn’t any flush manifold construction on $Hom(X,Y)$ the place $X$ and $Y$ are flush manifolds, however beneath inescapable good situations (behold https://ncatlab.org/nlab/display/manifold+construction+of+mapping+areas) we can provide flush construction on $Hom(X,Y)$.

Let $mathcal{G}$ and $mathcal{H}$ breathe two Lie groupoids. Now allow us to deem the class $Hom(mathcal{G}, mathcal{H})$ whose objects are homomorphisms of Lie groupoids and the morphisms are flush unaffected isomorphisms.

My questions are the next:

(1) Under what situations on $mathcal{G}$ and $mathcal{H}$, we now have a (canonical) Lie groupoid construction on $Hom(mathcal{G}, mathcal{H})$?

(2) Is $Hom(mathcal{G}, mathcal{H})$ at all times a diffelogical groupoid in common?

It would breathe too noble if somebody can imply some literatures on this path.

Thanks in forward

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