Mixing
Inequivalent compact closed symmetric monoidal structures on the same category
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I am looking for interesting examples of categories admitting multiple monoidally inequivalent closed (or compact closed) symmetric monoidal structures.
We know how to construct disconnected toy models [1], but none of direct practical interest. I also seem to recall from [2] that the category of $G$-sets admits two inequivalent such structures, but I am curious to know whether this is a sporadic occurrence or whether there is a large family of interesting examples lurking somewhere out there.
[1]: see e.g. https://arxiv.org/abs/1803.00708
[2]: Borceux, Francis. Handbook of Categorical Algebra
ct.category-theory symmetric-monoidal-categories
asked Sep 1 at 15:35
Stefano Gogioso
38510
add a comment |Â
up vote
11
down vote
favorite
I am looking for interesting examples of categories admitting multiple monoidally inequivalent closed (or compact closed) symmetric monoidal structures.
We know how to construct disconnected toy models [1], but none of direct practical interest. I also seem to recall from [2] that the category of $G$-sets admits two inequivalent such structures, but I am curious to know whether this is a sporadic occurrence or whether there is a large family of interesting examples lurking somewhere out there.
[1]: see e.g. https://arxiv.org/abs/1803.00708
[2]: Borceux, Francis. Handbook of Categorical Algebra
ct.category-theory symmetric-monoidal-categories
asked Sep 1 at 15:35
Stefano Gogioso
38510
add a comment |Â
up vote
11
down vote
favorite
up vote
11
down vote
favorite
I am looking for interesting examples of categories admitting multiple monoidally inequivalent closed (or compact closed) symmetric monoidal structures.
We know how to construct disconnected toy models [1], but none of direct practical interest. I also seem to recall from [2] that the category of $G$-sets admits two inequivalent such structures, but I am curious to know whether this is a sporadic occurrence or whether there is a large family of interesting examples lurking somewhere out there.
[1]: see e.g. https://arxiv.org/abs/1803.00708
[2]: Borceux, Francis. Handbook of Categorical Algebra
ct.category-theory symmetric-monoidal-categories
asked Sep 1 at 15:35
Stefano Gogioso
38510
I am looking for interesting examples of categories admitting multiple monoidally inequivalent closed (or compact closed) symmetric monoidal structures.
We know how to construct disconnected toy models [1], but none of direct practical interest. I also seem to recall from [2] that the category of $G$-sets admits two inequivalent such structures, but I am curious to know whether this is a sporadic occurrence or whether there is a large family of interesting examples lurking somewhere out there.
[1]: see e.g. https://arxiv.org/abs/1803.00708
[2]: Borceux, Francis. Handbook of Categorical Algebra
ct.category-theory symmetric-monoidal-categories
asked Sep 1 at 15:35
Stefano Gogioso
38510
edited Sep 1 at 18:20
asked Sep 1 at 15:35
Stefano Gogioso
38510
asked Sep 1 at 15:35
Stefano Gogioso
38510
asked Sep 1 at 15:35
Stefano Gogioso
38510
38510
add a comment |Â
add a comment |Â
3 Answers
3
active
oldest
votes
up vote
15
down vote
accepted
A pretty interesting class of examples comes about by classifying compact monoidal groupoids. Given a group $G$, a $G$-module $M$, and a (normalized) 3-cocycle $a: G times G times G to M$, one can manufacture a compact monoidal groupoid whose category of objects is $G$, whose morphisms are ordered pairs $(g, m) in G times M$ where we define $textdom(g, m) = textcod(g, m) = g$ and where endomorphism composition is defined by addition in $M$, and where we define the tensor product by $(g, m) otimes (h, n) = (g h, m + g n)$. It's the 3-cocycle that furnishes the associativity data, and we get monoidally inequivalent groupoids whenever the 3-cocycles are not cohomologous. This was observed by Joyal and Street in their paper Braided Monoidal Categories.
Now you were asking about the symmetric monoidal case (where we now assume $G$ is abelian). These are also known as Picard groupoids. In the simplified scenario where we demand strict associativies and consider only the case where $G$ acts trivially on $M$, in which case the underlying category becomes the product $K G times B M$ of the evident discrete monoidal category $K G$ with the evident one-object category $B M$, any symmetric bilinear pairing $G otimes G to M$ can be used to manufacture a symmetry isomorphism for a symmetric monoidal structure on $K G times B M$, and these examples are generally symmetric-monoidally inequivalent. I can't tell how far away such examples are from the "toy"examples" you have in mind, but Picard groupoids are surely of interest -- see for example applications to 2-stage Postnikov systems of spectra here.
answered Sep 1 at 18:22
Todd Trimble♦
42.4k5152252
add a comment |Â
up vote
7
down vote
The category $Cat$ of small categories admits two inequivalent tensor products: the Cartesian product and the funny product. This generalizes up to 2-cat, 3-cat, etc. By the way, a word of advise to those like me with a murky memory: googling "cat funny product" is going to just give you pictures of cats.
answered Sep 1 at 17:48
David White
10.5k45997
But he asked for compactness.
– Todd Trimble♦
Sep 1 at 18:01
2
Omega-cat admits the Crans-Gray-Steiner lax tensor product. I dunno if this meets the compactness requirement, but it is in the same vein.
– Harry Gindi
Sep 1 at 18:10
2
At present, the question asks for compact or compact closed, but anyway, I think all the answers are pointing the OP towards examples from category theory, and that's the important thing.
– David White
Sep 1 at 18:39
add a comment |Â
up vote
4
down vote
The large class of examples I have in mind, though I am not sure if it meets your compactness requirement (definition?) are the Lax tensor products on $n$-Cat. It has been constructed in the cases $n=2$ and $n=omega$ by Gray in the case $n=2$ and Crans, Steiner, and Verity independently in the case $n=omega$.
Edit: To clarify, these are all biclosed, and their right adjoints are the $n$-categories whose objects are strict $n$-functors and whose higher cells are (op)lax natural transformations and (op)lax modifications between them.
answered Sep 1 at 18:14
Harry Gindi
8,306674160
I have edited my question to clarify what I mean by closed and by compact closed.
– Stefano Gogioso
Sep 1 at 18:20
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
15
down vote
accepted
A pretty interesting class of examples comes about by classifying compact monoidal groupoids. Given a group $G$, a $G$-module $M$, and a (normalized) 3-cocycle $a: G times G times G to M$, one can manufacture a compact monoidal groupoid whose category of objects is $G$, whose morphisms are ordered pairs $(g, m) in G times M$ where we define $textdom(g, m) = textcod(g, m) = g$ and where endomorphism composition is defined by addition in $M$, and where we define the tensor product by $(g, m) otimes (h, n) = (g h, m + g n)$. It's the 3-cocycle that furnishes the associativity data, and we get monoidally inequivalent groupoids whenever the 3-cocycles are not cohomologous. This was observed by Joyal and Street in their paper Braided Monoidal Categories.
Now you were asking about the symmetric monoidal case (where we now assume $G$ is abelian). These are also known as Picard groupoids. In the simplified scenario where we demand strict associativies and consider only the case where $G$ acts trivially on $M$, in which case the underlying category becomes the product $K G times B M$ of the evident discrete monoidal category $K G$ with the evident one-object category $B M$, any symmetric bilinear pairing $G otimes G to M$ can be used to manufacture a symmetry isomorphism for a symmetric monoidal structure on $K G times B M$, and these examples are generally symmetric-monoidally inequivalent. I can't tell how far away such examples are from the "toy"examples" you have in mind, but Picard groupoids are surely of interest -- see for example applications to 2-stage Postnikov systems of spectra here.
answered Sep 1 at 18:22
Todd Trimble♦
42.4k5152252
add a comment |Â
up vote
15
down vote
accepted
A pretty interesting class of examples comes about by classifying compact monoidal groupoids. Given a group $G$, a $G$-module $M$, and a (normalized) 3-cocycle $a: G times G times G to M$, one can manufacture a compact monoidal groupoid whose category of objects is $G$, whose morphisms are ordered pairs $(g, m) in G times M$ where we define $textdom(g, m) = textcod(g, m) = g$ and where endomorphism composition is defined by addition in $M$, and where we define the tensor product by $(g, m) otimes (h, n) = (g h, m + g n)$. It's the 3-cocycle that furnishes the associativity data, and we get monoidally inequivalent groupoids whenever the 3-cocycles are not cohomologous. This was observed by Joyal and Street in their paper Braided Monoidal Categories.
Now you were asking about the symmetric monoidal case (where we now assume $G$ is abelian). These are also known as Picard groupoids. In the simplified scenario where we demand strict associativies and consider only the case where $G$ acts trivially on $M$, in which case the underlying category becomes the product $K G times B M$ of the evident discrete monoidal category $K G$ with the evident one-object category $B M$, any symmetric bilinear pairing $G otimes G to M$ can be used to manufacture a symmetry isomorphism for a symmetric monoidal structure on $K G times B M$, and these examples are generally symmetric-monoidally inequivalent. I can't tell how far away such examples are from the "toy"examples" you have in mind, but Picard groupoids are surely of interest -- see for example applications to 2-stage Postnikov systems of spectra here.
answered Sep 1 at 18:22
Todd Trimble♦
42.4k5152252
add a comment |Â
up vote
15
down vote
accepted
up vote
15
down vote
accepted
A pretty interesting class of examples comes about by classifying compact monoidal groupoids. Given a group $G$, a $G$-module $M$, and a (normalized) 3-cocycle $a: G times G times G to M$, one can manufacture a compact monoidal groupoid whose category of objects is $G$, whose morphisms are ordered pairs $(g, m) in G times M$ where we define $textdom(g, m) = textcod(g, m) = g$ and where endomorphism composition is defined by addition in $M$, and where we define the tensor product by $(g, m) otimes (h, n) = (g h, m + g n)$. It's the 3-cocycle that furnishes the associativity data, and we get monoidally inequivalent groupoids whenever the 3-cocycles are not cohomologous. This was observed by Joyal and Street in their paper Braided Monoidal Categories.
Now you were asking about the symmetric monoidal case (where we now assume $G$ is abelian). These are also known as Picard groupoids. In the simplified scenario where we demand strict associativies and consider only the case where $G$ acts trivially on $M$, in which case the underlying category becomes the product $K G times B M$ of the evident discrete monoidal category $K G$ with the evident one-object category $B M$, any symmetric bilinear pairing $G otimes G to M$ can be used to manufacture a symmetry isomorphism for a symmetric monoidal structure on $K G times B M$, and these examples are generally symmetric-monoidally inequivalent. I can't tell how far away such examples are from the "toy"examples" you have in mind, but Picard groupoids are surely of interest -- see for example applications to 2-stage Postnikov systems of spectra here.
answered Sep 1 at 18:22
Todd Trimble♦
42.4k5152252
A pretty interesting class of examples comes about by classifying compact monoidal groupoids. Given a group $G$, a $G$-module $M$, and a (normalized) 3-cocycle $a: G times G times G to M$, one can manufacture a compact monoidal groupoid whose category of objects is $G$, whose morphisms are ordered pairs $(g, m) in G times M$ where we define $textdom(g, m) = textcod(g, m) = g$ and where endomorphism composition is defined by addition in $M$, and where we define the tensor product by $(g, m) otimes (h, n) = (g h, m + g n)$. It's the 3-cocycle that furnishes the associativity data, and we get monoidally inequivalent groupoids whenever the 3-cocycles are not cohomologous. This was observed by Joyal and Street in their paper Braided Monoidal Categories.
Now you were asking about the symmetric monoidal case (where we now assume $G$ is abelian). These are also known as Picard groupoids. In the simplified scenario where we demand strict associativies and consider only the case where $G$ acts trivially on $M$, in which case the underlying category becomes the product $K G times B M$ of the evident discrete monoidal category $K G$ with the evident one-object category $B M$, any symmetric bilinear pairing $G otimes G to M$ can be used to manufacture a symmetry isomorphism for a symmetric monoidal structure on $K G times B M$, and these examples are generally symmetric-monoidally inequivalent. I can't tell how far away such examples are from the "toy"examples" you have in mind, but Picard groupoids are surely of interest -- see for example applications to 2-stage Postnikov systems of spectra here.
answered Sep 1 at 18:22
Todd Trimble♦
42.4k5152252
answered Sep 1 at 18:22
Todd Trimble♦
42.4k5152252
answered Sep 1 at 18:22
Todd Trimble♦
42.4k5152252
answered Sep 1 at 18:22
Todd Trimble♦
42.4k5152252
42.4k5152252
add a comment |Â
add a comment |Â
up vote
7
down vote
The category $Cat$ of small categories admits two inequivalent tensor products: the Cartesian product and the funny product. This generalizes up to 2-cat, 3-cat, etc. By the way, a word of advise to those like me with a murky memory: googling "cat funny product" is going to just give you pictures of cats.
answered Sep 1 at 17:48
David White
10.5k45997
But he asked for compactness.
– Todd Trimble♦
Sep 1 at 18:01
2
Omega-cat admits the Crans-Gray-Steiner lax tensor product. I dunno if this meets the compactness requirement, but it is in the same vein.
– Harry Gindi
Sep 1 at 18:10
2
At present, the question asks for compact or compact closed, but anyway, I think all the answers are pointing the OP towards examples from category theory, and that's the important thing.
– David White
Sep 1 at 18:39
add a comment |Â
up vote
7
down vote
The category $Cat$ of small categories admits two inequivalent tensor products: the Cartesian product and the funny product. This generalizes up to 2-cat, 3-cat, etc. By the way, a word of advise to those like me with a murky memory: googling "cat funny product" is going to just give you pictures of cats.
answered Sep 1 at 17:48
David White
10.5k45997
But he asked for compactness.
– Todd Trimble♦
Sep 1 at 18:01
2
Omega-cat admits the Crans-Gray-Steiner lax tensor product. I dunno if this meets the compactness requirement, but it is in the same vein.
– Harry Gindi
Sep 1 at 18:10
2
At present, the question asks for compact or compact closed, but anyway, I think all the answers are pointing the OP towards examples from category theory, and that's the important thing.
– David White
Sep 1 at 18:39
add a comment |Â
up vote
7
down vote
up vote
7
down vote
The category $Cat$ of small categories admits two inequivalent tensor products: the Cartesian product and the funny product. This generalizes up to 2-cat, 3-cat, etc. By the way, a word of advise to those like me with a murky memory: googling "cat funny product" is going to just give you pictures of cats.
answered Sep 1 at 17:48
David White
10.5k45997
The category $Cat$ of small categories admits two inequivalent tensor products: the Cartesian product and the funny product. This generalizes up to 2-cat, 3-cat, etc. By the way, a word of advise to those like me with a murky memory: googling "cat funny product" is going to just give you pictures of cats.
answered Sep 1 at 17:48
David White
10.5k45997
answered Sep 1 at 17:48
David White
10.5k45997
answered Sep 1 at 17:48
David White
10.5k45997
answered Sep 1 at 17:48
David White
10.5k45997
10.5k45997
But he asked for compactness.
– Todd Trimble♦
Sep 1 at 18:01
2
Omega-cat admits the Crans-Gray-Steiner lax tensor product. I dunno if this meets the compactness requirement, but it is in the same vein.
– Harry Gindi
Sep 1 at 18:10
2
At present, the question asks for compact or compact closed, but anyway, I think all the answers are pointing the OP towards examples from category theory, and that's the important thing.
– David White
Sep 1 at 18:39
add a comment |Â
But he asked for compactness.
– Todd Trimble♦
Sep 1 at 18:01
2
Omega-cat admits the Crans-Gray-Steiner lax tensor product. I dunno if this meets the compactness requirement, but it is in the same vein.
– Harry Gindi
Sep 1 at 18:10
2
At present, the question asks for compact or compact closed, but anyway, I think all the answers are pointing the OP towards examples from category theory, and that's the important thing.
– David White
Sep 1 at 18:39
But he asked for compactness.
– Todd Trimble♦
Sep 1 at 18:01
But he asked for compactness.
– Todd Trimble♦
Sep 1 at 18:01
2
2
Omega-cat admits the Crans-Gray-Steiner lax tensor product. I dunno if this meets the compactness requirement, but it is in the same vein.
– Harry Gindi
Sep 1 at 18:10
Omega-cat admits the Crans-Gray-Steiner lax tensor product. I dunno if this meets the compactness requirement, but it is in the same vein.
– Harry Gindi
Sep 1 at 18:10
2
2
At present, the question asks for compact or compact closed, but anyway, I think all the answers are pointing the OP towards examples from category theory, and that's the important thing.
– David White
Sep 1 at 18:39
At present, the question asks for compact or compact closed, but anyway, I think all the answers are pointing the OP towards examples from category theory, and that's the important thing.
– David White
Sep 1 at 18:39
add a comment |Â
up vote
4
down vote
The large class of examples I have in mind, though I am not sure if it meets your compactness requirement (definition?) are the Lax tensor products on $n$-Cat. It has been constructed in the cases $n=2$ and $n=omega$ by Gray in the case $n=2$ and Crans, Steiner, and Verity independently in the case $n=omega$.
Edit: To clarify, these are all biclosed, and their right adjoints are the $n$-categories whose objects are strict $n$-functors and whose higher cells are (op)lax natural transformations and (op)lax modifications between them.
answered Sep 1 at 18:14
Harry Gindi
8,306674160
I have edited my question to clarify what I mean by closed and by compact closed.
– Stefano Gogioso
Sep 1 at 18:20
add a comment |Â
up vote
4
down vote
The large class of examples I have in mind, though I am not sure if it meets your compactness requirement (definition?) are the Lax tensor products on $n$-Cat. It has been constructed in the cases $n=2$ and $n=omega$ by Gray in the case $n=2$ and Crans, Steiner, and Verity independently in the case $n=omega$.
Edit: To clarify, these are all biclosed, and their right adjoints are the $n$-categories whose objects are strict $n$-functors and whose higher cells are (op)lax natural transformations and (op)lax modifications between them.
answered Sep 1 at 18:14
Harry Gindi
8,306674160
I have edited my question to clarify what I mean by closed and by compact closed.
– Stefano Gogioso
Sep 1 at 18:20
add a comment |Â
up vote
4
down vote
up vote
4
down vote
The large class of examples I have in mind, though I am not sure if it meets your compactness requirement (definition?) are the Lax tensor products on $n$-Cat. It has been constructed in the cases $n=2$ and $n=omega$ by Gray in the case $n=2$ and Crans, Steiner, and Verity independently in the case $n=omega$.
Edit: To clarify, these are all biclosed, and their right adjoints are the $n$-categories whose objects are strict $n$-functors and whose higher cells are (op)lax natural transformations and (op)lax modifications between them.
answered Sep 1 at 18:14
Harry Gindi
8,306674160
The large class of examples I have in mind, though I am not sure if it meets your compactness requirement (definition?) are the Lax tensor products on $n$-Cat. It has been constructed in the cases $n=2$ and $n=omega$ by Gray in the case $n=2$ and Crans, Steiner, and Verity independently in the case $n=omega$.
Edit: To clarify, these are all biclosed, and their right adjoints are the $n$-categories whose objects are strict $n$-functors and whose higher cells are (op)lax natural transformations and (op)lax modifications between them.
answered Sep 1 at 18:14
Harry Gindi
8,306674160
edited Sep 1 at 19:51
answered Sep 1 at 18:14
Harry Gindi
8,306674160
answered Sep 1 at 18:14
Harry Gindi
8,306674160
answered Sep 1 at 18:14
Harry Gindi
8,306674160
8,306674160
I have edited my question to clarify what I mean by closed and by compact closed.
– Stefano Gogioso
Sep 1 at 18:20
add a comment |Â
I have edited my question to clarify what I mean by closed and by compact closed.
– Stefano Gogioso
Sep 1 at 18:20
I have edited my question to clarify what I mean by closed and by compact closed.
– Stefano Gogioso
Sep 1 at 18:20
I have edited my question to clarify what I mean by closed and by compact closed.
– Stefano Gogioso
Sep 1 at 18:20
add a comment |Â
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