Title: On the geometric notion of connection and its expression in tangent categories,
Abstract: 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 (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.