A cogroupoid object in $\mathsf{CAlg}_R$ is called a Hopf algebroid over $R$. How are cocategory objects in $\mathsf{CAlg}_R$ called? (Unfortunately, bialgebroid is already taken, which seems to mean something else.) We can define comodules as usual; the notion of a comodule over a Hopf algebroid does not use the antipode. Is there some literature about these comodules? If we allow stacks to be fibered in non-groupoids, these comodules can probably be seen as quasi-coherent modules on the corresponding "stack".
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1$\begingroup$ A "cocategory of object in $\mathsf{CAlg}_R$" is the "commutative case of bialgebroid" (as in the linked nlab page). In more recent literature a "cogroupoid object in $\mathsf{CAlg}_R$" is the "commutative case of Hopf algebroid", while Hopf algeboid itself is a bialgebroid with an antipode ncatlab.org/nlab/show/Hopf+algebroid. $\endgroup$– Dimitri ChikhladzeCommented Apr 4, 2017 at 1:12
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$\begingroup$ @DimitriChikhladze: Thank you, this is useful to know.. $\endgroup$– HeinrichDCommented Apr 4, 2017 at 5:38
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Converting Dimitri Chikhladze's comment to an answer:
A "cocategory of object in $\mathsf{CAlg}_R$" is the "commutative case of bialgebroid" (as in the linked nlab page). In more recent literature a "cogroupoid object in $\mathsf{CAlg}_R$" is the "commutative case of Hopf algebroid", while Hopf algeboid itself is a bialgebroid with an antipode https://ncatlab.org/nlab/show/Hopf+algebroid.
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$\begingroup$ Can you explain more why this is the commutative case of a bialgebroid? The nlab definition at ncatlab.org/nlab/show/bialgebroid is quite complicated and I don't see the connection. $\endgroup$ Commented Jan 22, 2020 at 22:27