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On the geometric notion of connection and its expression in tangent categories

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Rory B. B. Lucyshyn-Wright

*Tangent categories* provide an axiomatic approach to key
structural aspects of differential geometry that exist not only in the
classical category of smooth manifolds but also in algebraic geometry,
homological algebra, computer science, and combinatorics. Generalizing
the notion of \textit{(linear) connection} on a smooth vector bundle,
Cockett and Cruttwell have defined a notion of connection on a
differential bundle in an arbitrary tangent category. Herein, we
establish equivalent formulations of this notion of connection that reduce
the amount of specified structure required. Further, one of our
equivalent formulations substantially reduces the number of axioms
imposed, and others provide useful abstract conceptualizations of
connections. In particular, we show that a connection on a differential
bundle $E$ over $M$ is equivalently given by a single morphism $K$ that
induces a suitable decomposition of $TE$ as a biproduct. We also show
that a connection is equivalently given by a vertical connection $K$ for
which a certain associated diagram is a limit diagram.

Keywords:
connection; tangent category; linear connection; affine connection; vector
bundle; differential bundle

2010 MSC:
18D99, 53C05, 53B05, 18F15

*Theory and Applications of Categories,*
Vol. 33, 2018,
No. 28, pp 832-866.

Published 2018-08-25.

http://www.tac.mta.ca/tac/volumes/33/28/33-28.pdf

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