I have some problems to understand a notation introduced in Ronald Brown's Crossed complexes and higher homotopy groupoids as non commutative tools for higher dimensional local-to-global problems (p 10) on double categories:
A double category, $K$, consists of a triple of category structures
$$ (K_2,K_1, \partial_1^{-}, \partial_1^{+}, \circ_1, \epsilon_1), \ \ \ \ \ \ (K_2,K_1, \partial_2^{-}, \partial_2^{+}, \circ_2, \epsilon_2) $$
$$ (K_1,K_0, \partial^{-}, \partial^{+}, \circ, \epsilon) $$
as partly shown in the diagram
$$ \array{ K_2 & \stackrel{\overset{\partial_2^{-}}{\to}}{\underset{\partial_2^{+}}{\to}} & K_1 \\ ^{\partial_1^{+}}\downarrow \downarrow^{\partial_1^{-}} && ^{\partial^{+}}\downarrow \downarrow^{\partial^{-}} \\ K_1 & \stackrel{\overset{\partial^{-}}{\to}}{\underset{\partial^{+}}{\to}} & K_0 } $$
The elements of $K_0,K_1,K_2$ will be called respectively points or objects, edges, squares. The maps $\partial^{\pm}, \partial_i^{\pm}, i=1,2$ , will be called face map, the maps $\epsilon_i : K_1 \to K_2, i = 1, 2$ , resp. $\epsilon : K_0 \to K_1$ will be called degeneracies. The boundaries of an edge and of a square are given by the diagrams
$$ \partial^{-} \longrightarrow \partial^{+} $$
and
$$ \begin{matrix} . &\stackrel{\partial_1^{-}}{\to} & . \\ {}^{\partial_2^{-}}\downarrow & & \downarrow^{\partial_2^{+}} \\ . &\underset{\partial_1^{+}}{\to} & . \end{matrix} $$
The last notation I not understand: what is the meaning of the diagram $ \partial^{-} \longrightarrow \partial^{+} $ and which sense it can be regarded as a "boundary" of an edge? Is it a map between images of $K_1$ in $K_0$ under $\partial^{-}$ and $\partial^{-}$? Same question about the boundary diagram of an edge? What about the choosen directions of the arrows there? According to which rule / logic are these formed? What else could this notation mean?
It seems to be somehow a non-standard terminology. If we try to think about these in most simple minded way in terms of simplicial sets or chain complexes, then the $K_i$ should be regarded as $i$-simplices and boundaries of $i$-simplices are usually formal sums (iterated by $(\pm)$-sign of $(i-1)$-simplices. Maybe the direction of the arrows above inducates if a summand of the formal sum is multiplied by a plus or minus sign (I'm not sure, that's just a naive guess of mine). But on the other hand the face maps $\partial^{\pm}, \partial_i^{\pm} $ involved in the diagrams build in the in the setting of simlicial sets are not the members of some simplices, but a kind of "meta data", which gouverns the glueing relations between the simplices themselves. So the concept of boundaries in the definition above not really allow this interpretation of boundaries in the standard setting of simplicial sets. That's exactly the main point of my confusion.
(by the way: does anybody know how to render the pairs of horizontal arrows in first diagram to the same length? I tried my best but I recommend to take a look in the linked paper where the correct diagrams are depicted on page 10, because it may happen that my attempts to draw them here not fully capture an important point behind their meaning)