nLab
connection on a double category

Let D=(D 1,D 0)D=(D_1, D_0) be a double category. A connection on DD is given by a pair of functions Γ,Γ:arrD 0arrD 1\Gamma, \Gamma' \colon \mathrm{arr} D_0 \to \mathrm{arr} D_1, that assign to each vertical morphism f:YXf \colon Y \to X 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\iota_X is the horizontal identity on XX) such that Γ\Gamma and Γ\Gamma' behave suitably with respect to composition and identities in D 0D_0:

(that is, they are identity-on-objects functors) and such that the vertical composite Γ(f)Γ(f)=ι f\Gamma'(f) \cdot \Gamma(f) = \iota_f and the horizontal composite Γ(f)Γ(f)=1 f *X\Gamma(f) \circ \Gamma'(f) = 1_{f^*X}, (that is, ff and f *Xf^*X are companions).

Created on April 18, 2011 19:03:28 by Finn Lawler (134.226.56.7)