The tensor product of two monoidal categoriesWhat structure on a monoidal category would make its 2-category of module categories monoidal and braided?What is known about module categories over general monoidal categories?Which monoidal categories are equivalent to their centers?Enriched monoidal categoriesModule categories over symmetric/braided monoidal categoriesWhen is an Eilenberg-Moore category or Kleisli category braided monoidal? When semisimple?In a closed monoidal abelian category, are the compact projectives a monoidal subcategory?Balanced Tensor Product of Module CategoriesDo the modules over a Hopf algebra in a braided monoidal category form a monoidal category?Tensor product of modules over a monoid in a monoidal category
The tensor product of two monoidal categories
What structure on a monoidal category would make its 2-category of module categories monoidal and braided?What is known about module categories over general monoidal categories?Which monoidal categories are equivalent to their centers?Enriched monoidal categoriesModule categories over symmetric/braided monoidal categoriesWhen is an Eilenberg-Moore category or Kleisli category braided monoidal? When semisimple?In a closed monoidal abelian category, are the compact projectives a monoidal subcategory?Balanced Tensor Product of Module CategoriesDo the modules over a Hopf algebra in a braided monoidal category form a monoidal category?Tensor product of modules over a monoid in a monoidal category
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Given two monoidal categories $mathcalM$ and $mathcalN$, can one form their tensor product in a canonical way?
The motivation I am thinking of is two categories that are representation categories of two algebras $R$ and $S$, and the module category of the tensor product algebra $R otimes_mathbbC S$.
Also, if $mathcalM$ and $mathcalN$ are assumed to be braided monoidal, can we tensor the braidings?
rt.representation-theory ct.category-theory hopf-algebras monoidal-categories braided-tensor-categories
asked 8 hours ago
Nadia SUSYNadia SUSY
1541 silver badge15 bronze badges
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add a comment |
$begingroup$
Given two monoidal categories $mathcalM$ and $mathcalN$, can one form their tensor product in a canonical way?
The motivation I am thinking of is two categories that are representation categories of two algebras $R$ and $S$, and the module category of the tensor product algebra $R otimes_mathbbC S$.
Also, if $mathcalM$ and $mathcalN$ are assumed to be braided monoidal, can we tensor the braidings?
rt.representation-theory ct.category-theory hopf-algebras monoidal-categories braided-tensor-categories
asked 8 hours ago
Nadia SUSYNadia SUSY
1541 silver badge15 bronze badges
$endgroup$
add a comment |
$begingroup$
Given two monoidal categories $mathcalM$ and $mathcalN$, can one form their tensor product in a canonical way?
The motivation I am thinking of is two categories that are representation categories of two algebras $R$ and $S$, and the module category of the tensor product algebra $R otimes_mathbbC S$.
Also, if $mathcalM$ and $mathcalN$ are assumed to be braided monoidal, can we tensor the braidings?
rt.representation-theory ct.category-theory hopf-algebras monoidal-categories braided-tensor-categories
asked 8 hours ago
Nadia SUSYNadia SUSY
1541 silver badge15 bronze badges
$endgroup$
Given two monoidal categories $mathcalM$ and $mathcalN$, can one form their tensor product in a canonical way?
The motivation I am thinking of is two categories that are representation categories of two algebras $R$ and $S$, and the module category of the tensor product algebra $R otimes_mathbbC S$.
Also, if $mathcalM$ and $mathcalN$ are assumed to be braided monoidal, can we tensor the braidings?
rt.representation-theory ct.category-theory hopf-algebras monoidal-categories braided-tensor-categories
rt.representation-theory ct.category-theory hopf-algebras monoidal-categories braided-tensor-categories
asked 8 hours ago
Nadia SUSYNadia SUSY
1541 silver badge15 bronze badges
asked 8 hours ago
Nadia SUSYNadia SUSY
1541 silver badge15 bronze badges
asked 8 hours ago
Nadia SUSYNadia SUSY
1541 silver badge15 bronze badges
asked 8 hours ago
Nadia SUSYNadia SUSY
1541 silver badge15 bronze badges
asked 8 hours ago
Nadia SUSYNadia SUSY
1541 silver badge15 bronze badges
1541 silver badge15 bronze badges
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
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The book Tensor Categories discusses, with many variations, the details of Robert McRae's answer. Just like for vector spaces, there are a number of related but inequivalent "tensor products" of linear categories, with the choice dependent on the types of linear categories considered: sufficiently finite dimensional (vector spaces / categories) have only one reasonable tensor product, but the "algebraic" tensor product of infinite-dimensional objects can be completed in various ways. Locally finite abelian categories is a particularly good choice.
Once you have made such a choice, tensoring (braided) monoidal structures is typically easy. You probably will need to require that the monoidal structure is "continuous" for however you chose to complete your tensor products. Again, see the Tensor Categories book for details.
answered 5 hours ago
Theo Johnson-FreydTheo Johnson-Freyd
30.6k8 gold badges83 silver badges257 bronze badges
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add a comment |
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If your categories are locally finite abelian, I think you are looking for the Deligne tensor product of $mathcalM$ and $mathcalN$. The Deligne tensor product $mathcalMboxtimesmathcalN$ does inherit braided monoidal structure from $mathcalM$ and $mathcalN$ if these are braided monoidal.
answered 6 hours ago
Robert McRaeRobert McRae
2162 silver badges4 bronze badges
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$begingroup$
What happens if the categories are not locally finite?
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– Nadia SUSY
5 hours ago
$begingroup$
For example, $k$-linear and finite dimensional.
$endgroup$
– Nadia SUSY
5 hours ago
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The book Tensor Categories discusses, with many variations, the details of Robert McRae's answer. Just like for vector spaces, there are a number of related but inequivalent "tensor products" of linear categories, with the choice dependent on the types of linear categories considered: sufficiently finite dimensional (vector spaces / categories) have only one reasonable tensor product, but the "algebraic" tensor product of infinite-dimensional objects can be completed in various ways. Locally finite abelian categories is a particularly good choice.
Once you have made such a choice, tensoring (braided) monoidal structures is typically easy. You probably will need to require that the monoidal structure is "continuous" for however you chose to complete your tensor products. Again, see the Tensor Categories book for details.
answered 5 hours ago
Theo Johnson-FreydTheo Johnson-Freyd
30.6k8 gold badges83 silver badges257 bronze badges
$endgroup$
add a comment |
$begingroup$
The book Tensor Categories discusses, with many variations, the details of Robert McRae's answer. Just like for vector spaces, there are a number of related but inequivalent "tensor products" of linear categories, with the choice dependent on the types of linear categories considered: sufficiently finite dimensional (vector spaces / categories) have only one reasonable tensor product, but the "algebraic" tensor product of infinite-dimensional objects can be completed in various ways. Locally finite abelian categories is a particularly good choice.
Once you have made such a choice, tensoring (braided) monoidal structures is typically easy. You probably will need to require that the monoidal structure is "continuous" for however you chose to complete your tensor products. Again, see the Tensor Categories book for details.
answered 5 hours ago
Theo Johnson-FreydTheo Johnson-Freyd
30.6k8 gold badges83 silver badges257 bronze badges
$endgroup$
add a comment |
$begingroup$
The book Tensor Categories discusses, with many variations, the details of Robert McRae's answer. Just like for vector spaces, there are a number of related but inequivalent "tensor products" of linear categories, with the choice dependent on the types of linear categories considered: sufficiently finite dimensional (vector spaces / categories) have only one reasonable tensor product, but the "algebraic" tensor product of infinite-dimensional objects can be completed in various ways. Locally finite abelian categories is a particularly good choice.
Once you have made such a choice, tensoring (braided) monoidal structures is typically easy. You probably will need to require that the monoidal structure is "continuous" for however you chose to complete your tensor products. Again, see the Tensor Categories book for details.
answered 5 hours ago
Theo Johnson-FreydTheo Johnson-Freyd
30.6k8 gold badges83 silver badges257 bronze badges
$endgroup$
The book Tensor Categories discusses, with many variations, the details of Robert McRae's answer. Just like for vector spaces, there are a number of related but inequivalent "tensor products" of linear categories, with the choice dependent on the types of linear categories considered: sufficiently finite dimensional (vector spaces / categories) have only one reasonable tensor product, but the "algebraic" tensor product of infinite-dimensional objects can be completed in various ways. Locally finite abelian categories is a particularly good choice.
Once you have made such a choice, tensoring (braided) monoidal structures is typically easy. You probably will need to require that the monoidal structure is "continuous" for however you chose to complete your tensor products. Again, see the Tensor Categories book for details.
answered 5 hours ago
Theo Johnson-FreydTheo Johnson-Freyd
30.6k8 gold badges83 silver badges257 bronze badges
answered 5 hours ago
Theo Johnson-FreydTheo Johnson-Freyd
30.6k8 gold badges83 silver badges257 bronze badges
answered 5 hours ago
Theo Johnson-FreydTheo Johnson-Freyd
30.6k8 gold badges83 silver badges257 bronze badges
answered 5 hours ago
Theo Johnson-FreydTheo Johnson-Freyd
30.6k8 gold badges83 silver badges257 bronze badges
30.6k8 gold badges83 silver badges257 bronze badges
add a comment |
add a comment |
$begingroup$
If your categories are locally finite abelian, I think you are looking for the Deligne tensor product of $mathcalM$ and $mathcalN$. The Deligne tensor product $mathcalMboxtimesmathcalN$ does inherit braided monoidal structure from $mathcalM$ and $mathcalN$ if these are braided monoidal.
answered 6 hours ago
Robert McRaeRobert McRae
2162 silver badges4 bronze badges
$endgroup$
$begingroup$
What happens if the categories are not locally finite?
$endgroup$
– Nadia SUSY
5 hours ago
$begingroup$
For example, $k$-linear and finite dimensional.
$endgroup$
– Nadia SUSY
5 hours ago
add a comment |
$begingroup$
If your categories are locally finite abelian, I think you are looking for the Deligne tensor product of $mathcalM$ and $mathcalN$. The Deligne tensor product $mathcalMboxtimesmathcalN$ does inherit braided monoidal structure from $mathcalM$ and $mathcalN$ if these are braided monoidal.
answered 6 hours ago
Robert McRaeRobert McRae
2162 silver badges4 bronze badges
$endgroup$
$begingroup$
What happens if the categories are not locally finite?
$endgroup$
– Nadia SUSY
5 hours ago
$begingroup$
For example, $k$-linear and finite dimensional.
$endgroup$
– Nadia SUSY
5 hours ago
add a comment |
$begingroup$
If your categories are locally finite abelian, I think you are looking for the Deligne tensor product of $mathcalM$ and $mathcalN$. The Deligne tensor product $mathcalMboxtimesmathcalN$ does inherit braided monoidal structure from $mathcalM$ and $mathcalN$ if these are braided monoidal.
answered 6 hours ago
Robert McRaeRobert McRae
2162 silver badges4 bronze badges
$endgroup$
If your categories are locally finite abelian, I think you are looking for the Deligne tensor product of $mathcalM$ and $mathcalN$. The Deligne tensor product $mathcalMboxtimesmathcalN$ does inherit braided monoidal structure from $mathcalM$ and $mathcalN$ if these are braided monoidal.
answered 6 hours ago
Robert McRaeRobert McRae
2162 silver badges4 bronze badges
answered 6 hours ago
Robert McRaeRobert McRae
2162 silver badges4 bronze badges
answered 6 hours ago
Robert McRaeRobert McRae
2162 silver badges4 bronze badges
answered 6 hours ago
Robert McRaeRobert McRae
2162 silver badges4 bronze badges
2162 silver badges4 bronze badges
$begingroup$
What happens if the categories are not locally finite?
$endgroup$
– Nadia SUSY
5 hours ago
$begingroup$
For example, $k$-linear and finite dimensional.
$endgroup$
– Nadia SUSY
5 hours ago
add a comment |
$begingroup$
What happens if the categories are not locally finite?
$endgroup$
– Nadia SUSY
5 hours ago
$begingroup$
For example, $k$-linear and finite dimensional.
$endgroup$
– Nadia SUSY
5 hours ago
$begingroup$
What happens if the categories are not locally finite?
$endgroup$
– Nadia SUSY
5 hours ago
$begingroup$
What happens if the categories are not locally finite?
$endgroup$
– Nadia SUSY
5 hours ago
$begingroup$
For example, $k$-linear and finite dimensional.
$endgroup$
– Nadia SUSY
5 hours ago
$begingroup$
For example, $k$-linear and finite dimensional.
$endgroup$
– Nadia SUSY
5 hours ago
add a comment |
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