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Another comment (but too long to fit): Have you tried making sense out of "Morita equivalence" in this setting? The condition for fully-faithful is clear. For groupoids, $\varphi:H \to G$ is essentially-surjective if $t \circ pr_1:G_1 \times_{G_0} H_0 \to G_0$ is a cover (or if you want something stronger, submersion, in relation to your other question), but this really just says that for every object $x$ of $G_0$ there is an arrow $f(y) \to x$ in $G$- SINCE $G$ is a groupoid, this is the same as essentially surjective, but for categories, it is not. I believe that this problem is related to the non-principality you have been running into.

However, I don't think that this is a real problem. You can generalize my answer to THIS question: Torsors for monoidsTorsors for monoids, to show that torsors for a category object $C$ are the same as torsors for its groupoid of all invertible elements, $\tilde C$ (as objects, but they have different morphisms). But, this means that the stack associated to $C$ should satisfy that any map $T \to C$ with $T$ an object of your site, that the underlying object of its associated $C$-torsor ($\tilde C$-torsor) should be the weak pullback $T \times_{C} C_0$, which is the same as $T \times_{\tilde C} C_0$.

The upshot is, you don't WANT principality, but only principality with respect to the underlying groupoid. But, you do want to allow different morphisms between bibundles than before.

Another comment (but too long to fit): Have you tried making sense out of "Morita equivalence" in this setting? The condition for fully-faithful is clear. For groupoids, $\varphi:H \to G$ is essentially-surjective if $t \circ pr_1:G_1 \times_{G_0} H_0 \to G_0$ is a cover (or if you want something stronger, submersion, in relation to your other question), but this really just says that for every object $x$ of $G_0$ there is an arrow $f(y) \to x$ in $G$- SINCE $G$ is a groupoid, this is the same as essentially surjective, but for categories, it is not. I believe that this problem is related to the non-principality you have been running into.

However, I don't think that this is a real problem. You can generalize my answer to THIS question: Torsors for monoids, to show that torsors for a category object $C$ are the same as torsors for its groupoid of all invertible elements, $\tilde C$ (as objects, but they have different morphisms). But, this means that the stack associated to $C$ should satisfy that any map $T \to C$ with $T$ an object of your site, that the underlying object of its associated $C$-torsor ($\tilde C$-torsor) should be the weak pullback $T \times_{C} C_0$, which is the same as $T \times_{\tilde C} C_0$.

The upshot is, you don't WANT principality, but only principality with respect to the underlying groupoid. But, you do want to allow different morphisms between bibundles than before.

Another comment (but too long to fit): Have you tried making sense out of "Morita equivalence" in this setting? The condition for fully-faithful is clear. For groupoids, $\varphi:H \to G$ is essentially-surjective if $t \circ pr_1:G_1 \times_{G_0} H_0 \to G_0$ is a cover (or if you want something stronger, submersion, in relation to your other question), but this really just says that for every object $x$ of $G_0$ there is an arrow $f(y) \to x$ in $G$- SINCE $G$ is a groupoid, this is the same as essentially surjective, but for categories, it is not. I believe that this problem is related to the non-principality you have been running into.

However, I don't think that this is a real problem. You can generalize my answer to THIS question: Torsors for monoids, to show that torsors for a category object $C$ are the same as torsors for its groupoid of all invertible elements, $\tilde C$ (as objects, but they have different morphisms). But, this means that the stack associated to $C$ should satisfy that any map $T \to C$ with $T$ an object of your site, that the underlying object of its associated $C$-torsor ($\tilde C$-torsor) should be the weak pullback $T \times_{C} C_0$, which is the same as $T \times_{\tilde C} C_0$.

The upshot is, you don't WANT principality, but only principality with respect to the underlying groupoid. But, you do want to allow different morphisms between bibundles than before.

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David Carchedi
David Carchedi
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Another comment (but too long to fit): Have you tried making sense out of "Morita equivalence" in this setting? The condition for fully-faithful is clear. For groupoids, $\varphi:H \to G$ is essentially-surjective if $t \circ pr_1:G_1 \times_{G_0} H_0 \to G_0$ is a cover (or if you want something stronger, submersion, in relation to your other question), but this really just says that for every object $x$ of $G_0$ there is an arrow $f(y) \to x$ in $G$- SINCE $G$ is a groupoid, this is the same as essentially surjective, but for categories, it is not. I believe that this problem is related to the non-principality you have been running into.

However, I don't think that this is a real problem. You can generalize my answer to THIS question: Torsors for monoids, to show that torsors for a category object $C$ are the same as torsors for its groupoid of all invertible elements, $\tilde C$ (as objects, but they have different morphisms). But, this means that the stack associated to $C$ should satisfy that any map $T \to C$ with $T$ an object of your site, that the underlying object of its associated $C$-torsor ($\tilde C$-torsor) should be the weak pullback $T \times_{C} C_0$, which is the same as $T \times_{\tilde C} C_0$.

The upshot is, you don't WANT principality, but only principality with respect to the underlying groupoid. But, you do want to allow different morphisms between bibundles than before.