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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 4th 2013
   • (edited Feb 4th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am constructing a table _[[structure on algebras and their module categories - table]]_ and am including it into the relevant entries. This is a bit experimental for the moment. More details and variants should be added and maybe some of the relations stated in a better way. Help is appreciated.

   I am constructing a table

   structure on algebras and their module categories - table

   and am including it into the relevant entries. This is a bit experimental for the moment. More details and variants should be added and maybe some of the relations stated in a better way. Help is appreciated.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 4th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexSo far I have the following: **[[Tannaka duality]] for associative algebras** | [[associative algebra]] | [[category of modules]] | |--|--| | $A$ | $Mod_A$ | | $R$-[[associative algebra]] | $Mod_R$-[[2-module]] | | [[sesquialgebra]] | [[2-ring]] = [[monoidal category|monoidal]] [[presentable category]] with [[colimit]]-preserving [[tensor product]] | | [[bialgebra]] | strict [[2-ring]]: [[monoidal category]] with [[fiber functor]] | | [[Hopf algebra]] | [[rigid monoidal category]] with [[fiber functor]] | | [[quasitriangular bialgebra]] | [[braided monoidal category]] with [[fiber functor]] | | [[triangular bialgebra]] | [[symmetric monoidal category]] with [[fiber functor]] | | [[quasitriangular Hopf algebra]] ([[quantum group]]) | [[rigid monoidal category|rigid]] [[braided monoidal category]] with [[fiber functor]] | | [[triangular Hopf algebra]] | [[rigid monoidal category|rigid]] [[symmetric monoidal category]] with [[fiber functor]] |

   So far I have the following:

   Tannaka duality for associative algebras

   associative algebra	category of modules
   $A$	$Mod_A$
   $R$-associative algebra	$Mod_R$-2-module
   sesquialgebra	2-ring = monoidal presentable category with colimit-preserving tensor product
   bialgebra	strict 2-ring: monoidal category with fiber functor
   Hopf algebra	rigid monoidal category with fiber functor
   quasitriangular bialgebra	braided monoidal category with fiber functor
   triangular bialgebra	symmetric monoidal category with fiber functor
   quasitriangular Hopf algebra (quantum group)	rigid braided monoidal category with fiber functor
   triangular Hopf algebra	rigid symmetric monoidal category with fiber functor