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David Roberts ♦
David Roberts ♦
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It is a diffeomorphism, since there is an equivalence of bicategories $DifferentiableStacks \simeq LieGroupoids[W^{-1}]$ where the RHS is the bicategorical localisation of the usual 2-category of Lie groupoids a la Pronk. Because of the special nature of the domain of your $\theta$, namely it is a Lie groupoid with one object (call it $\mathbf{B}G$; note the boldface B!), then in fact $$ LieGroupoids[W^{-1}](\mathbf{B}G,\mathbf{B}H) \simeq LieGroupoids(\mathbf{B}G,\mathbf{B}H) $$ is an equivalence of categories. The latter category is isomorphic to the category whose objects are homomorphisms $G\to H$ and whose 2-arrows are elements of $H$, acting by conjugation on homomorphisms. Tracing what happens to the quasi-inverse $\phi\colon BH\to BG$ to $B\theta$ through these equivalences of categories one gets a homomorphism $\psi\colon H\to G$ such that $\psi\circ \theta$ is conjugate to the identity map on $G$, and $\theta\circ \psi$ is conjugate to the identity map on $H$. This is enough to know that $\theta$ is a diffeomorphism, since we can pre- and post-compose $\psi$ with the inner automorphisms to get an inverse for $\theta$.

It is a diffeomorphism, since there is an equivalence of bicategories $DifferentiableStacks \simeq LieGroupoids[W^{-1}]$ where the RHS is the bicategorical localisation of the usual 2-category of Lie groupoids a la Pronk. Because of the special nature of the domain of your $\theta$, namely it is a Lie groupoid with one object (call it $\mathbf{B}G$; note the boldface B!), then in fact $$ LieGroupoids[W^{-1}](\mathbf{B}G,\mathbf{B}H) \simeq LieGroupoids(\mathbf{B}G,\mathbf{B}H) $$ is an equivalence of categories. The latter category is isomorphic to the category whose objects are homomorphisms $G\to H$ and whose 2-arrows are elements of $H$, acting by conjugation on homomorphisms. Tracing what happens to the quasi-inverse $\phi\colon BH\to BG$ through these equivalences of categories one gets a homomorphism $\psi\colon H\to G$ such that $\psi\circ \theta$ is conjugate to the identity map on $G$, and $\theta\circ \psi$ is conjugate to the identity map on $H$. This is enough to know that $\theta$ is a diffeomorphism, since we can pre- and post-compose $\psi$ with the inner automorphisms to get an inverse for $\theta$.

It is a diffeomorphism, since there is an equivalence of bicategories $DifferentiableStacks \simeq LieGroupoids[W^{-1}]$ where the RHS is the bicategorical localisation of the usual 2-category of Lie groupoids a la Pronk. Because of the special nature of the domain of your $\theta$, namely it is a Lie groupoid with one object (call it $\mathbf{B}G$; note the boldface B!), then in fact $$ LieGroupoids[W^{-1}](\mathbf{B}G,\mathbf{B}H) \simeq LieGroupoids(\mathbf{B}G,\mathbf{B}H) $$ is an equivalence of categories. The latter category is isomorphic to the category whose objects are homomorphisms $G\to H$ and whose 2-arrows are elements of $H$, acting by conjugation on homomorphisms. Tracing what happens to the quasi-inverse $\phi\colon BH\to BG$ to $B\theta$ through these equivalences of categories one gets a homomorphism $\psi\colon H\to G$ such that $\psi\circ \theta$ is conjugate to the identity map on $G$, and $\theta\circ \psi$ is conjugate to the identity map on $H$. This is enough to know that $\theta$ is a diffeomorphism, since we can pre- and post-compose $\psi$ with the inner automorphisms to get an inverse for $\theta$.

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David Roberts ♦
David Roberts ♦
  • 35.5k
  • 11
  • 124
  • 350

It is a diffeomorphism, since there is an equivalence of bicategories $DifferentiableStacks \simeq LieGroupoids[W^{-1}]$ where the RHS is the bicategorical localisation of the usual 2-category of Lie groupoids a la Pronk. Because of the special nature of the domain of your $\theta$, namely it is a Lie groupoid with one object (call it $\mathbf{B}G$; note the boldface B!), then in fact $$ LieGroupoids[W^{-1}](\mathbf{B}G,\mathbf{B}H) \simeq LieGroupoids(\mathbf{B}G,\mathbf{B}H) $$ is an equivalence of categories. The latter category is isomorphic to the category whose objects are homomorphisms $G\to H$ and whose 2-arrows are elements of $H$, acting by conjugation on homomorphisms. Tracing what happens to the quasi-inverse $\phi\colon BH\to BG$ through these equivalences of categories one gets a homomorphism $\psi\colon H\to G$ such that $\psi\circ \theta$ is conjugate to the identity map on $G$, and $\theta\circ \psi$ is conjugate to the identity map on $H$. This is enough to know that $\theta$ is a diffeomorphism, since we can pre- and post-compose $\psi$ with the inner automorphisms to get an inverse for $\theta$.