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Disquisitions on Monoidal Categories and Operads
Disquisitions on Monoidal Categories and Operads
Geometric construction of quiver tensor products
Geometric construction of quiver tensor products
Organizers
Speaker
Daigo Ito
Time
Monday, March 2, 2026 4:00 PM - 5:00 PM
Venue
A3-1-301
Online
Zoom 559 700 6085
(BIMSA)
Abstract
By a classical theorem of Beilinson, the perfect derived category $\operatorname{Perf}(\mathbb{P}^n)$ of projective space is equivalent to the category of derived representations of a certain quiver with relations. The vertex-wise tensor product of quiver representations corresponds to a symmetric monoidal structure $\otimes_{\mathrm{quiv}}$ on $\operatorname{Perf}(\mathbb{P}^n)$. We demonstrate that this symmetric monoidal structure can be geometrically described as an “extended convolution product.” Specifically, the Fourier--Mukai kernel is given by the closure of the torus multiplication map in $(\mathbb{P}^n)^3$. We further extend the result to any smooth proper toric variety of Bondal--Ruan type. Under toric mirror symmetry, this extended convolution product corresponds to the tensor product of constructible sheaves on a real torus.