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Wikipedia begins the list of ingredients in the definition of a 2-category as follows:

  • A class of 0-cells (or objects) $A$, $B$, ....
  • For all objects $A$ and $B$, a category $\mathbf{C}(A,B)$. The objects $f,g:A\to B$ of this category are called 1-cells and its morphisms $\alpha:f\Rightarrow g$ are called 2-cells; the composition in this category is usually written $\circ$ or $\circ_1$ and called vertical composition or composition along a 1-cell.

What is the reason for requiring $f$ and $g$ to be morphisms between the same pair of objects? Would it be possible/meaningful to generalize the definition a little and allow $f$ and $g$ to be between different pairs of objects: $f:A\to B$, $g:C\to D$?

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up vote 3 down vote accepted

Yes, it is possible, and meaningful, and leads to the notion of (pseudo) double category.

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Thanks! Why does one feel the need to specialize to 2-categories? Is is because 2-categories appear more frequently than more general double categories? – merle Aug 20 '13 at 16:11
Yes, $2$-categories (to be precise: bicategories) arise quite naturally ( And of course they are easier to handle than double categories. – Martin Brandenburg Aug 20 '13 at 16:16
Thanks, will read that. – merle Aug 20 '13 at 16:19

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