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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeMay 17th 2011
   • (edited May 19th 2011)
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   Author: zskoda
   Format: MarkdownItexThe stub for "associative" [[bialgebroid]]. Bialgebroids are to bialgebras what on dual side groupoids are to groups. More references at [[Hopf algebroids]].

   The stub for “associative” bialgebroid. Bialgebroids are to bialgebras what on dual side groupoids are to groups. More references at Hopf algebroids.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeMay 19th 2011
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   Author: zskoda
   Format: MarkdownItexRelated new stubs [[dynamical extension of a monoidal category]] and [[dynamical Yang-Baxter equation]].

   Related new stubs dynamical extension of a monoidal category and dynamical Yang-Baxter equation.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeOct 2nd 2012
   • (edited Oct 2nd 2012)
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   Author: zskoda
   Format: MarkdownItexSome new material at [[bialgebroid]], following * [[Gabriella Böhm]], _Hopf algebroids_, (a chapter of) Handbook of algebra, [arxiv:math.RA/0805.3806](http://arxiv.org/abs/0805.3806) All modules and morphisms will be over a fixed ground commutative ring $k$. A __left $A$-bialgebroid__ is an $A\otimes_k A^{op}$-[[ring]] $(H,\mu_H,\eta)$, together with the $A$-bimodule map $\Delta : A\to H\otimes_A H$, which is coassociative and counital with counit $\epsilon$, such that (i) the $A$-bimodule structure used on $H$ is $a.h.a':= s(a)t(a')h$, where $s := \eta(-\otimes 1_A):A\to H$ and $t:=\eta(1_A\otimes -):A^{op}\to H$ are the algebra maps induced by the unit $\eta$ of the $A\otimes A^{op}$-ring $H$ (ii) the coproduct $\Delta : H\to H\otimes_A H$ corestricts to the [[Takeuchi product]] and the corestriction $\Delta : H\to H\times_A H$ is a $k$-algebra map, where the Takeuchi product $H\times_A H$ has a multiplication induced factorwise (iii) $\epsilon$ is a left [[character]] on the $A$-ring $(H,\mu_H,s)$ Notice that $H\otimes_A H$ is in general not an algebra, just an $A$-bimodule. The definition of a __right $A$-bialgebroid__ differs by the $A$-bimodule structure on $H$ given instead by $a.h.a':= h s(a')t(a)$ and the counit $\epsilon$ is a _right_ character on the $A$-coring $(H,\mu_H,t)$ ($t$ and $s$ can be interchanged in the last requirement).

   Some new material at bialgebroid, following

   • Gabriella Böhm, Hopf algebroids, (a chapter of) Handbook of algebra, arxiv:math.RA/0805.3806

   All modules and morphisms will be over a fixed ground commutative ring $k$.

   A left $A$-bialgebroid is an $A\otimes_k A^{op}$-ring $(H,\mu_H,\eta)$, together with the $A$-bimodule map $\Delta : A\to H\otimes_A H$, which is coassociative and counital with counit $\epsilon$, such that

   (i) the $A$-bimodule structure used on $H$ is $a.h.a':= s(a)t(a')h$, where $s := \eta(-\otimes 1_A):A\to H$ and $t:=\eta(1_A\otimes -):A^{op}\to H$ are the algebra maps induced by the unit $\eta$ of the $A\otimes A^{op}$-ring $H$

   (ii) the coproduct $\Delta : H\to H\otimes_A H$ corestricts to the Takeuchi product and the corestriction $\Delta : H\to H\times_A H$ is a $k$-algebra map, where the Takeuchi product $H\times_A H$ has a multiplication induced factorwise

   (iii) $\epsilon$ is a left character on the $A$-ring $(H,\mu_H,s)$

   Notice that $H\otimes_A H$ is in general not an algebra, just an $A$-bimodule.

   The definition of a right $A$-bialgebroid differs by the $A$-bimodule structure on $H$ given instead by $a.h.a':= h s(a')t(a)$ and the counit $\epsilon$ is a right character on the $A$-coring $(H,\mu_H,t)$ ($t$ and $s$ can be interchanged in the last requirement).

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeDec 16th 2020
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   Author: zskoda
   Format: MarkdownItexI am writing an explicit definition of a left bialgebroid for the wikipedia in a different way than I previously put into the entry so I am copying a variant here to have yet another equivalent definition, avoiding $A\otimes A^{op}$-rings. <a href="https://ncatlab.org/nlab/revision/diff/bialgebroid/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/bialgebroid/14">v14</a>, <a href="https://ncatlab.org/nlab/show/bialgebroid">current</a>

   I am writing an explicit definition of a left bialgebroid for the wikipedia in a different way than I previously put into the entry so I am copying a variant here to have yet another equivalent definition, avoiding $A\otimes A^{op}$-rings.

   diff, v14, current

5. • CommentRowNumber5.
   • CommentAuthorTim_Porter
   • CommentTimeDec 16th 2020
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   Author: Tim_Porter
   Format: MarkdownItexMinor fixes <a href="https://ncatlab.org/nlab/revision/diff/bialgebroid/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/bialgebroid/15">v15</a>, <a href="https://ncatlab.org/nlab/show/bialgebroid">current</a>

   Minor fixes

   diff, v15, current

6. • CommentRowNumber6.
   • CommentAuthorTim_Porter
   • CommentTimeOct 31st 2021
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   Author: Tim_Porter
   Format: MarkdownItexFormatting <a href="https://ncatlab.org/nlab/revision/diff/bialgebroid/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/bialgebroid/16">v16</a>, <a href="https://ncatlab.org/nlab/show/bialgebroid">current</a>

   Formatting

   diff, v16, current