connection on a double category

Let D=(D 1,D 0) be a double category. A connection on D is given by a pair of functions Γ,Γ:arrD 0arrD 1, that assign to each vertical morphism f:YX cells of the form

Y f *X X f Γf 1 X ι X XY ι Y Y 1 Γf f Y f *X X\begin{matrix} Y & \overset{f^*X}{\to} & X \\ \mathllap{f} \downarrow & \mathllap{\Gamma f} \Downarrow & \downarrow \mathrlap{1} \\ X & \underset{\iota_X}{\to} & X \end{matrix} \qquad \qquad \begin{matrix} Y & \overset{\iota_Y}{\to} & Y \\ \mathllap{1} \downarrow & \Downarrow \mathrlap{\Gamma' f} & \downarrow \mathrlap{f} \\ Y & \underset{f^*X}{\to} & X \end{matrix}

(where ι X is the horizontal identity on X) such that Γ and Γ behave suitably with respect to composition and identities in D 0:

(that is, they are identity-on-objects functors) and such that the vertical composite Γ(f)Γ(f)=ι f and the horizontal composite Γ(f)Γ(f)=1 f *X, (that is, f and f *X are companions).

Created on April 18, 2011 19:03:28 by Finn Lawler (