# On the definition of double categories?

I'm trying to understand double categories but I'm having a hard time.

A preliminary definition is:

Definition. Let $\mathscr{C}$ be a category. We say $\mathscr{I}$ is an internal category to $\mathscr{C}$ if $\mathscr{I}=(\mathscr{I}_0, \mathscr{I}_1, s, t, u, \circ)$ where:

$(i)$ $\mathscr{I}_0$ is an object of $\mathscr{C}$;

$(ii)$ $\mathscr{I}_1$ is an object of $\mathscr{C}$:

$(iii)$ $s, t:\mathscr{I}_1\longrightarrow \mathscr{I}_0$ are morphisms of $\mathscr{C}$;

$(iv)$ $u:\mathscr{I}_0\longrightarrow \mathscr{I}_1$ is a morphism of $\mathscr{C}$;

$(v)$ $\circ:\mathscr{I}_1\times_{\mathscr{I}_0}\mathscr{I}_1\longrightarrow \mathscr{I}_1$ is a morphism of $\mathscr{C}$.

This data are subject to the properties which define a category whose class of objects is $\mathscr{I}_0$, whose set of morphisms is $\mathscr{I}_1$, and where $s$ is the source map, $t$ is the target map, $u$ is the identity assigning map and where $\circ$ is the composition.

In other words, we're just interpreting the definition of a category $\mathscr{I}$ inside the category $\mathscr{C}$.

Definition. A double category is a category internal to $\mathbf{Cat}$.

Above $\mathbf{Cat}$ stands for the category of all categories (maybe small??).

I'm having some trouble to understand this definition for I can't think about some example to have in mind.

Can anyone provide me some down to earth examples? Furtheremore, is there some modern reference which deals with double categories?

Thanks.

Think of a double category as having horizontal arrows as the arrows of $J_0$, vertical arrows as the objects of $J_1$, and squares between them as the arrows of $J_1$. Most of the examples I know are formal, i.e. I don't think double categories are used as much as 2-categories and bicategories outside of category theory proper, but you can find several on the ncatlab, which is also the modern reference id recommend. But the reference I'd recommend without that restriction is Kelly's Elements of 2-Categories, which begins with double categories. All of category theory is pretty modern, after all!

I see that this question is quite old never the less maybe this answer could be of help to someone, if not the OP.

Double categories are not as rare as one would expect.

For instance there is a double category of adjunctions as shown in MacLane's Category theory for the Working Mathematicians.

Another double category is the one having sets and functions forming the object-category and binary relations and relative morphisms forming the arrow-category.

Then there families of double categories that can be build for other categories. For instance for every category $\mathbf C$ you have a double category whose object-category is $\mathbf C$ itself and whose arrow-category is...well the arrow category of $\mathbf C$ (that is the category having arrows of $\mathbf C$ as objects and commutative squares as morphisms).

Of course the examples become even more if you relax the strictness of the composition functor and enter in the world of weak double categories: these are basically for bicategories are for strict 2-categories. Many of these weak double categories arise from a general construction, the module category construction.

If you need additional details feel free to ask.