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I’m a bit unhappy about the fact that the definition of creation of limits on the nlab is different from the definition of MacLane and on Wikipedia. I understand that they both use ‘strict’ definitions and the nlab wants a non-strict one, but the nlab definition doesn’t even generalize the standard one. For the benefit of people who might be confused about this in the future, let me pinpoint the difference:
The nlab says:
Let $F:C\to D$ be a functor and $J:I\to C$ a diagram. We say that $F$ creates limits for $J$ if $J$ has a limit whenever the composite $F\circ J$ has a limit, and $F$ both preserves and reflects limits of $J$.
… whereas a non-strict version of the standard (MacLane/Wikipedia) definition would be
Let $F:C\to D$ be a functor and $J:I\to C$ a diagram. We say that $F$ creates limits for $J$ if $J$ has a limit whenever the composite $F\circ J$ has a limit, and in this case $F$ both preserves and reflects limits of $J$.
Thus, the nlab definition forbids the existence of limits in $C$ if the limit in $D$ doesn’t exist, whereas the standard definition doesn’t say anything in this case. Right?
I’m not necessarily asking to change this, but I think that at least the difference should be emphasized more - I have to admit that I only noticed the difference after looking at the page for the third time :-)
Are there other sources that use the “nlab version” of the definition? (strict or non-strict)
There’s a related thread here but I thought I start a new one with the correct title.
Does the difference really matter that much though? My experience has been that one only ever really talks about creation of limits that exist, which is why the nLab definition makes more sense to me. (In what sense can something that doesn’t exist be “created”?)
In any case it might be good to add a comment to the page that highlights the subtlety
Maybe what we should do is first give the definition assuming, as a precondition, that the limit exists in the base, and then remark that there are two ways one might choose to extend this to the case when the limit doesn’t exist.
I just made an edit emphasizing the non-equivalence more, and stating MacLane’s definition explicitly instead of the slightly hazy “sometimes the definition is given differently” (before seeing Mike’s last comment). Feel free to change or rollback.
Nobody objected to #4, so I went ahead and tried it.
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