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Abstract
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We study affine structures on a Lie groupoid, including affine
-vector
fields,
-forms
and
-tensors.
We show that the space of affine structures is a
-vector
space over the space of multiplicative structures. Moreover, the space of
affine multivector fields with the Schouten bracket and the space of affine
vector-valued forms with the Frölicher–Nijenhuis bracket are graded strict Lie
-algebras, and
affine
-tensors
constitute a strict monoidal category. Such higher structures can be seen as the
categorification of multiplicative structures on a Lie groupoid.
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Keywords
affine structure, multiplicative structure, $2$-vector
space, strict monoidal category, Lie 2-algebra
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Mathematical Subject Classification 2010
Primary: 53D17, 53D18
Secondary: 22A22, 70G45
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Milestones
Received: 18 March 2019
Revised: 21 February 2020
Accepted: 1 April 2020
Published: 4 September 2020
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