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

## 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.

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