# 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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