There are many theorems in category theory that give criteria for the existence of a large class of limits in terms of the existence of a few special, more manageable types of limits. For instance:
- A category has all (small) limits if it has finite limits and (small) cofiltered limits.
- A category has all (small) limits if it has (small) products and equalizers.
- A category has all finite limits if it has a terminal object and pullbacks.
Are there any such criteria for a category to admit all (small) connected limits (that is, limits of functors whose domain is a connected category)?