# nLab connection on a double category

Let $D=\left({D}_{1},{D}_{0}\right)$ be a double category. A connection on $D$ is given by a pair of functions $\Gamma ,\Gamma \prime :\mathrm{arr}{D}_{0}\to \mathrm{arr}{D}_{1}$, that assign to each vertical morphism $f:Y\to X$ cells of the form

$\begin{array}{ccc}Y& \stackrel{{f}^{*}X}{\to }& X\\ f↓& \Gamma f⇓& ↓1\\ X& \underset{{\iota }_{X}}{\to }& X\end{array}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\begin{array}{ccc}Y& \stackrel{{\iota }_{Y}}{\to }& Y\\ 1↓& ⇓\Gamma \prime f& ↓f\\ Y& \underset{{f}^{*}X}{\to }& X\end{array}$\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 ${\iota }_{X}$ is the horizontal identity on $X$) such that $\Gamma$ and $\Gamma \prime$ 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 $\Gamma \prime \left(f\right)\cdot \Gamma \left(f\right)={\iota }_{f}$ and the horizontal composite $\Gamma \left(f\right)\circ \Gamma \prime \left(f\right)={1}_{{f}^{*}X}$, (that is, $f$ and ${f}^{*}X$ are companions).