Any criteria for a category to have all connected limits?

Question

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)?

Answer 1

A category has all connected limits iff it has all pullbacks, equalizers, and filtered limits. First any limit can be built as a filtered limit of finite limits, and if the original limit was connected, then the finite limits can be taken to be connected. (See for instance my answer here; if $I$ is connected then it is easy to see the set of $j\in J$ such that $F_j$ is connected is cofinal in $J$.) Then, any finite connected limit can be built out of pullbacks and equalizers. This is very similar to the construction of general limits from products and equalizers, except that in place of the product you iteratively take pullbacks using the fact that your diagram is connected to be sure you eventually reach every object. (And, instead of taking a single equalizer between big products afterwards, you iteratively take equalizers to one-by-one handle all the morphisms in your diagram.) You can find a detailed construction on nlab.