Dear John, Thank you for your mail and the precisions you give in it, but I'm not interested, for the time being, in general questions about 2-categories. Let me repeat precisely my question: If S is a category with finite limits and Cat(S) is the 2-category of internal categories of S, under which condition does Cat(S) have strict inverters? Can you, or anybody give a precise answer? (Of course I know that Cat(S) is cotensored with 2) . David Roberts says that finite limits in S suffice. As I I said I don't believe that. I'm perhaps wrong. In that case, could he, you, or anybody tell me how to construct strict inverters when all I assume is that S has finite limits? It is always a pleasure to hear from you. All the best , Jean Le 15 févr. 17 à 08:41, John Power a écrit :

Dear Jean,

Max wrote an expository paper which I believe was called "Elementary Observations on 2-Categorical Limits" and was published in the Bulletin of the Australian Mathematical Society I think around 1990. He would have had a discussion of inverters there.

Strict inverters are a kind of strict weighted limit (see, for instance, https://golem.ph.utexas.edu/category/2014/04/elementary_observations_on_2ca....) and a 2-category has all strict weighted limits if it has all strict conical limits and all strict cotensors, as a 2-category is a Cat-enriched category. So if one can prove that Cat(S) has strict conical limits and strict cotensors, one can construct strict inverters by following the procedure in the link above.

For strict cotensors, it suffices to prove that a 2-category has strict cotensors with the arrow category. I believe that is straightforward for Cat(S) if you follow the case of S = Set.

Once again, it is always lovely to hear from you.

All the best,

John.

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