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1. • CommentRowNumber1.
   • CommentAuthordomenico_fiorenza
   • CommentTimeJun 26th 2011
   • (edited Jun 26th 2011)
   • PermaLink
   Author: domenico_fiorenza
   Format: MarkdownItexIn one of its incarnations 2-Vect should be the 2-category whose objects are (commutative, with unit) algebras in Vect (to be thought of as placeholders for their categories of modules), whose 1-morphisms are bimodules (an $A$-$B$-bimodule is to be thought as a "linear" map from $Mod_A$ to $Mod_B$), and whose 2-morphisms are morphisms of bimodules. There should be a natural 2-monoidal structure on 2-Vect, for which Vect is the unit object (I guess the tensor product uses the span $A\leftarrow A\otimes B\to B$ to define $Mod_A\times Mod_B\to Mod_{A\otimes B}\times Mod_{A\otimes B}\stackrel{\otimes}{\to} Mod_{A\otimes B}$ which geometrically corresponds to pulling back quasicoherent sheaves from the factors to a product and taking their tensor product there). From this point of view, the dual of $Mod_A$ (if it exists) should be a category of modules $Mod_{A^\vee}$, where $A^\vee$ is an algebra endowed with an $A^\vee\otimes A$-$\mathbb{K}$ bimodule (the pairing) and a $\mathbb{K}$-$A^\vee\otimes A$-bimodule (the copairing) satisfying a few compatibility axioms. In particular, the composition of the pairing and the copairing should give a $\mathbb{K}$-$\mathbb{K}$-bimodule (i.e. an element of Vect), to be thought as the dimension of $Mod_A$. Is this correct?

   In one of its incarnations 2-Vect should be the 2-category whose objects are (commutative, with unit) algebras in Vect (to be thought of as placeholders for their categories of modules), whose 1-morphisms are bimodules (an $A$-$B$-bimodule is to be thought as a “linear” map from $Mod_A$ to $Mod_B$), and whose 2-morphisms are morphisms of bimodules. There should be a natural 2-monoidal structure on 2-Vect, for which Vect is the unit object (I guess the tensor product uses the span $A\leftarrow A\otimes B\to B$ to define $Mod_A\times Mod_B\to Mod_{A\otimes B}\times Mod_{A\otimes B}\stackrel{\otimes}{\to} Mod_{A\otimes B}$ which geometrically corresponds to pulling back quasicoherent sheaves from the factors to a product and taking their tensor product there).

   From this point of view, the dual of $Mod_A$ (if it exists) should be a category of modules $Mod_{A^\vee}$, where $A^\vee$ is an algebra endowed with an $A^\vee\otimes A$-$\mathbb{K}$ bimodule (the pairing) and a $\mathbb{K}$-$A^\vee\otimes A$-bimodule (the copairing) satisfying a few compatibility axioms.

   In particular, the composition of the pairing and the copairing should give a $\mathbb{K}$-$\mathbb{K}$-bimodule (i.e. an element of Vect), to be thought as the dimension of $Mod_A$.

   Is this correct?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 26th 2011
   • (edited Jun 26th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> There should be a natural 2-monoidal structure on 2-Vect Yes, the 2-category of bimodules over algebras over a commutative ring is braided monoidal, with tensor product induced from tensoring of algebras. I think this was one of the examples that motivated Mike's work on pro-arrow equipments (because that makes it easier to exhibit that braided monoidal structure by allowing one to use algebra homomorphisms as structure morphisms, instead of just the bimodules which they induce). > [...] > Is this correct? Yes, that looks correct. I seem to remember that this and examples are also discussed in FHLT, but I can't check right now.

   There should be a natural 2-monoidal structure on 2-Vect

   Yes, the 2-category of bimodules over algebras over a commutative ring is braided monoidal, with tensor product induced from tensoring of algebras. I think this was one of the examples that motivated Mike’s work on pro-arrow equipments (because that makes it easier to exhibit that braided monoidal structure by allowing one to use algebra homomorphisms as structure morphisms, instead of just the bimodules which they induce).

   […] Is this correct?

   Yes, that looks correct. I seem to remember that this and examples are also discussed in FHLT, but I can’t check right now.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJun 26th 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItex> I think this was one of the examples that motivated Mike's work on pro-arrow equipments Yes, that's right. And this example is not just braided monoidal, it's symmetric monoidal (because the ground ring is *commutative*). In order to get a braided monoidal example you have to do fancier things that I don't understand. And yes, what you describe about duality is also true. In fact, every object of this 2-category is dualizable; the dual of $A$ is $A^{op}$, the algebra with the opposite multiplication. The pairing and copairing bimodules are $A$ itself, regarded as a $A$-$A$-bimodule and thereby as a $\mathbb{K}$-$A\otimes A^{op}$-bimodule and dually. And the "dimension" or "Euler characteristic" of an algebra is its $HH_0$. (If you work with derived categories of bimodules instead, you can get all of Hochschild homology.)

   I think this was one of the examples that motivated Mike’s work on pro-arrow equipments

   Yes, that’s right. And this example is not just braided monoidal, it’s symmetric monoidal (because the ground ring is commutative). In order to get a braided monoidal example you have to do fancier things that I don’t understand.

   And yes, what you describe about duality is also true. In fact, every object of this 2-category is dualizable; the dual of $A$ is $A^{op}$, the algebra with the opposite multiplication. The pairing and copairing bimodules are $A$ itself, regarded as a $A$-$A$-bimodule and thereby as a $\mathbb{K}$-$A\otimes A^{op}$-bimodule and dually. And the “dimension” or “Euler characteristic” of an algebra is its $HH_0$. (If you work with derived categories of bimodules instead, you can get all of Hochschild homology.)

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeJun 27th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexThere are similar issues with related truncation, the [[Loday-Pirashvili tensor category]]. Unfortunately, most objects there are not projective so things like dual basis do not seem to work, and I do not know a 2-categorical rescue.

   There are similar issues with related truncation, the Loday-Pirashvili tensor category. Unfortunately, most objects there are not projective so things like dual basis do not seem to work, and I do not know a 2-categorical rescue.