Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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
   • PermaLink
   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
   • PermaLink
   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