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Monoidal centres and groupoid-graded categories

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Branko Nikolić and Ross Street

We denote the monoidal bicategory of two-sided modules (also called
profunctors, bimodules and distributors) between categories by Mod;
the tensor product is cartesian product of categories.
For a groupoid G, we study the monoidal centre ZPs(G,Mod^op)
of the monoidal bicategory Ps(G, Mod^op)
of pseudofunctors and pseudonatural transformations; the tensor product is pointwise.
Alexei Davydov defined the full centre of a monoid in a monoidal category.
We define a higher dimensional version: the full monoidal centre of a monoidale (= pseudomonoid) in a monoidal bicategory M, and it is a braided monoidale in the monoidal centre ZM of M.
Each fibration pi : M -> G between groupoids provides an example of a full monoidal centre of a monoidale in
Ps(G, Mod^op).
For a group G, we explain how the G-graded categorical structures, as considered by Turaev and Virelizier in order to construct topological invariants, fit into this monoidal bicategory context.
We see that their structures are monoidales in the monoidal centre of the monoidal bicategory of k-linear
categories on which G acts.

Keywords:
monoidal centre; graded center; graded category; Day convolution; bidual

2020 MSC:
18M15; 18N10; 18D15; 18D20; 18D60; 57K16; 57K31

*Theory and Applications of Categories,*
Vol. 40, 2024,
No. 1, pp 3-31.

Published 2024-03-21.

http://www.tac.mta.ca/tac/volumes/40/1/40-01.pdf

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