A class of limits is saturated if it is closed under the “construction” of other limits out of limits in .
Any such can serve as a weight for defining the weighted limit of a -functor , for any -category . We say that a -category is -complete if it admits all such limits, for all , and thath a -functor between -complete -categories is -continuous if it preserves all such limits.
The saturation of is the class of all weights (with small) such that
It is an open question whether the second condition is implied by the first in general.
Finally, is saturated if it is its own saturation.
In this case, we may consider instead classes of small categories; we write for the class of weights . We say that a category lies in the saturation of if the weight lies in the saturation of , and that is saturated if it is its own saturation.
Note that in practically all cases, the saturation of will contain weights that are not of the form . Moreover, even when there are nontrivial saturated classes of weights that do not contain any nontrivial conical weights, such as the saturation of the weight for “cartesian squares” .
However, for conical weights the answer to the above open question is known to be affirmative. On the one hand, if is a class of -weights such that every -complete category is also -complete, then every -continuous funtor is also -continuous. See AK for a proof of this.
On the other hand, if is a class of -categories and is a -weight such that every -complete category is also -complete, then every -continuous functor is also -continuous. In fact, this is still true if instead of we consider a class of weights all of which take only nonempty sets as values. See KP for a proof of this.
The main theorem of AK (which introduced the notion under the name “closure”) is the following.
lies in the saturation of if and only if it lies in the closure of the representables under -weighted colimits in .
The following examples are all for , restricted to the conical case.
The class of L-finite limits is saturated; it is the saturation of the class of finite limits. It is also the saturation of the finite class containing only terminal objects and pullbacks, and the saturation of the class containing only finite products and equalizers.
The class of connected limits is saturated. It is the saturation of the class consisting of wide pullbacks and equalizers. Similarly, the class of L-finite connected limits is the saturation of the finite class of pullbacks and equalizers. See also pullback and wide pullback for their saturations.
There are also interesting examples for other .
When , the classes of PIE-limits and flexible limits are saturated. The former is, essentially by definition, the saturation of the class containing products, inserters, and equifiers. The latter can be proven to be the saturation of the class containing products, inserters, equifiers, and splitting of idempotents.
It is also worth mentioning some non-examples.
For , the class of strict pseudo-limits is not saturated; it does not even contain the representables. (The same is true for strict lax limits.) It is unclear precisely what its saturation looks like.