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1. • CommentRowNumber1.
   • CommentAuthorJohn Baez
   • CommentTimeAug 2nd 2013
   • (edited Aug 2nd 2013)
   • PermaLink
   Author: John Baez
   Format: MarkdownItexI edited * [[Deligne tensor product of abelian categories]] to make it clear that this tensor product only works for _finite_ abelian categories, which are what we get from looking at _finite-dimensional_ representations of _finite-dimensional_ associative algebras. It's all very finite... and over fields, too, at least in the treatment by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, which is all I have access to now. It would be nice to see this kind of thing done more generally. The slides by Ignacio López Franco, linked on this page, are a hint of how to do it.

   I edited

   • Deligne tensor product of abelian categories

   to make it clear that this tensor product only works for finite abelian categories, which are what we get from looking at finite-dimensional representations of finite-dimensional associative algebras. It’s all very finite… and over fields, too, at least in the treatment by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, which is all I have access to now. It would be nice to see this kind of thing done more generally. The slides by Ignacio López Franco, linked on this page, are a hint of how to do it.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeAug 2nd 2013
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI think Lyubashenko was also writing some things about Deligne's product of abelian categories.

   I think Lyubashenko was also writing some things about Deligne’s product of abelian categories.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeAug 4th 2013
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   Author: Urs
   Format: MarkdownItexI have added a brief clarification to the entry. It seems that with the right generality understood, the statement about the tensor product of module categrories remains true without the finiteness constraints, see example 2.2.7 of [Chirvasitu & Johnson-Freyd 11](http://ncatlab.org/nlab/show/Deligne+tensor+product+of+abelian+categories#ChirvasituJohnson-Freyd11).

   I have added a brief clarification to the entry. It seems that with the right generality understood, the statement about the tensor product of module categrories remains true without the finiteness constraints, see example 2.2.7 of Chirvasitu & Johnson-Freyd 11.

4. • CommentRowNumber4.
   • CommentAuthorJohn Baez
   • CommentTimeMar 21st 2024
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   Author: John Baez
   Format: MarkdownItexAdded reference to paper by Ignacio L&#243;pez Franco: Tensor products of finitely cocomplete and abelian categories. <a href="https://ncatlab.org/nlab/revision/diff/Deligne+tensor+product+of+abelian+categories/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/Deligne+tensor+product+of+abelian+categories/15">v15</a>, <a href="https://ncatlab.org/nlab/show/Deligne+tensor+product+of+abelian+categories">current</a>

   Added reference to paper by Ignacio López Franco: Tensor products of finitely cocomplete and abelian categories.

   diff, v15, current