It's just a Kan fibration with all fibres principal homogeneous $G$-spaces. Take an $\infty$-category $C$ and a functor $C\to BG,$ where BG is the classifying groupoid of an $\infty$-group (a grouplike $E_1$-space). Pulling back the overcategory projection $EG=BG_{/\ast}\to BG,$ where $\ast$ is the unique object of $BG$, gets you the Kan fibration you wanted.

A Kan fibration over a 1-category is a biCartesian fibration in groupoids, and being G-principal is a condition on the fibres.

To see this is the right condition, let $c$ be an object of $C$. Then $c$ is classified by a functor $\ast\to C,$ and the fibre over $c$ must be the pullback of $EG$ to a point, which is now just $\Omega BG\simeq G$ by recognizing $BG$ as the delooping of $G$.

It's just a Kan fibration with all fibres principal homogeneous $G$-spaces. Take an $\infty$-category $C$ and a functor $C\to BG,$ where BG is the classifying groupoid of an $\infty$-group (a grouplike $E_1$-space). Pulling back the overcategory projection $EG=BG_{/\ast}\to BG,$ where $\ast$ is the unique object of $BG$, gets you the Kan fibration you wanted.

A Kan fibration over a 1-category is a biCartesian fibration in groupoids, and being $G$-principal is a condition on the fibres.

To see this is the right condition, let $c$ be an object of $C$. Then $c$ is classified by a functor $\ast\to C,$ and the fibre over $c$ must be the pullback of $EG$ to a point, which is now just $\Omega BG\simeq G$ by recognizing $BG$ as the delooping of $G$.

It's just a Kan fibration with all fibres principal homogeneous $G$-spaces. Take an $\infty$-category $C\to BG,$ where BG is the classifying groupoid of an $\infty$-group (a grouplike $E_1$-space). Pulling back $EG=BG_{/\ast}$ gets you the Kan fibration you wanted.

A Kan fibration over a 1-category is a biCartesian fibration in groupoids, and being G-principal is a condition on the fibres.

It's just a Kan fibration with all fibres principal homogeneous $G$-spaces. Take an $\infty$-category $C$ and a functor $C\to BG,$ where BG is the classifying groupoid of an $\infty$-group (a grouplike $E_1$-space). Pulling back the overcategory projection $EG=BG_{/\ast}\to BG,$ where $\ast$ is the unique object of $BG$, gets you the Kan fibration you wanted.

A Kan fibration over a 1-category is a biCartesian fibration in groupoids, and being G-principal is a condition on the fibres.

To see this is the right condition, let $c$ be an object of $C$. Then $c$ is classified by a functor $\ast\to C,$ and the fibre over $c$ must be the pullback of $EG$ to a point, which is now just $\Omega BG\simeq G$ by recognizing $BG$ as the delooping of $G$.