elif: There are longer versions, but in any case the examples are in section 5.5 of the version that you have.

My advice to you is not to ask us the question, but to take apart what $I$-adic completion does in the commutative algebra case, and, importantly, what is it used for. Once you understand that well, you can try to see if an $I$-adic completion for crossed modules of commutative algebras/ cat^1-algebras is going to do something interesting. Again you need to find some examples of these things, say in polynomial rings and to produce some calculations (for yourself).

When the work on pro-C completions of crossed modules was done, all the links between that stuff and higher dimensional algebra were less clear than they are now, so you need to look at that old stuff from the perspective of todays viewpoint, so as to make sure you are approaching it in a sufficiently general categorical way.

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A good question. I thought the example on p.281 (in that version) might help, but I’m not sure it does. That section then goes on to look at Friedlander’s fibration example and that may also be helpful. Have a look at the Sl(2,5) example in more detail.

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