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Stream: theory: mathematics

Topic: Derived triangular bialgebras

Brendan Murphy (Mar 20 2024 at 00:05):

Let $(\mathcal{C}, \otimes, \mathbb{1})$ be a nice symmeteic monoidal additive (1-)category. For any bialgebra object $A$ in $\mathcal{C}$ the category $\mathsf{Mod}_A$ inherits a monoidal strucure from $\mathcal{C}$, given by tensoring the underlying objects of two modules and then getting an induced action by the comultiplication of $A$. My understanding is that any braided monoidal structure on $\mathsf{Mod}_A$ is determined by the component of the braiding $A \otimes A \to A \otimes A$, and we can write down some simple conditions on such an endomorphism (reminiscent of the axioms of a braided monoidal category) which say that it induces a braided monoidal structure (and this all goes through for symmetric monoidal structures as well if we require the endomorphism to square to the identity).

Has anyone worked out the analogous result when we start with $\mathcal{C}$ an $(\infty, 1)$-category? I guess I'm actually interested in a simpler question, which is if $A$ is a dg-bialgebra over some base commutative ring $R$ then what conditions do we need on a degree zero element of $A \otimes A$ to induce a symmetric monoidal structure on the $(\infty, 1)$-category of $A$ modules? And to what extent can we reverse this process?

Brendan Murphy (Mar 20 2024 at 00:09):

Even more specifically: if $R$ is a commutative noetherian ring and $a_1,\ldots,a_m$ a sequence of elements in its jacobson radical then we can view the theory of local cohomology wrt $V(a_1,\ldots, a_m)$ as coming from a coreflective subcategory of "derived torsion" complexes in $D(R)$. This subcategory is equivalent to the category of modules over the endomorphism ring $\mathcal{E}$ (as a dg algebra) of the koszul complex on $a_1, \ldots, a_m$. The coreflection endofunctor of $D(R)$ is symmetric monoidal and so we can get a symmetric monoidal structure on the category of $\mathcal{E}$ modules. Is this determined by some data internal to $\mathcal{E}$?