Different possibilities arise from using goroupoids, since group objects in groupoids, or, equivalently, , groupoid objects groups, are equivalent to crossed modules, which are in some sense “more nonabelian” than groups, and should be thought of as 2-dimensional groups, since they model weak, pointed, homotopy 2-types. There is much more on this in my expository article

R. Brown `Groupoids and crossed objects in algebraic topology’,

Homology, homotopy and applications, 1 (1999) 1-78.(electronic)

which also contains a matrix version of the Eckmann-Hilton argument,

This also explains about the use of certain kinds of double groupoids, which have advantages over the commonly used 2-groupoids. A homotopy strict double groupoid is defined for a pointed pair of spaces, $(X,A,x)$ and is “equivalent” to the second relative homotopy group of the pair, considered as a crossed module. One takes homotopy classes of maps of a square to $X$ which take the edges of the square to $A$ and the vertices to the base point. I find these much easier to use than the globular 2-groupoids, and

I get confused about the use of the term “fundamental 2-groupoid” of a space, since it is not a strict structure, even in dimension 1!

I ended up do the going through a similar route, except I did not use the borders, which is a great idea. Thanks for the feed back.

]]>Nothing fancy. I blew up a .pdf containing the images I wanted (with borders around them) to 200%, print-screened it into mspaint, and saved it as a .png. The borders were so I knew where to cut. There must be a better way to do this, but it works and doesn’t take too much time.

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