Pseudolimits equivalent to limits

Question

In Proposition 4.1 of this article is proved that if $F: I\rightarrow\mathbf{Cat}$ is a pseudofunctor and $I$ is a filtered poset, then the pseudo colimit of $F$ is equivalent to its strict colimit. Is there any similar result for pseudo limits, maybe even in the case in which the target of $F$ is a general strict 2-category and $I$ is something more general than a filtered poset? If so, can anybody suggest a reference in which these properties of pseudo limits are proved, and maybe even other basic results on pseudo limits?

Answer 1

This is hard to generalize or dualize and actually depends very heavily on special properties of filtered categories and of $\mathbf{Cat}$-specifically, that $\mathbf{Cat}$ contains a small set of objects (namely $\bullet$ and $\bullet\to \bullet$), maps out of which commute with filtered colimits, and which detect equivalences. This implies that the strict colimit functor $\mathbf{Cat}^I\to \mathbf{Cat}$ preserves equivalences of diagrams and so, by general nonsense, must coincide with the pseudocolimit functor.

But it's quite rare for strict limits and colimits to preserve equivalences/to coincide with the pseudo notions. Even in $\mathbf{Cat}$, equalizers and pseudo-equalizers don't coincide in general; nor do pullbacks and pseudo-pullbacks; nor do cofiltered limits and pseudo-cofiltered limits. There are a few kinds of limits that coincide with pseudolimits in any 2-category, most notably, products and splitting of idempotents. In fact, it's a theorem that the class of diagrams indexed by a 1-category whose limits have this property in every 2-category is generated in an appropriate sense by products and idempotents. If you allow $I$ to be a 2-category and consider weighted limits, then you get some more important classes of such so-called "flexible" limits, called inserters and equifiers.

A good general starting point for all things 2-categorical is Steve Lack's "2-Categories Companion." Two particular references to start with on the topic of strict versus pseudo limits are Bourke-Garner, "On semiflexible, flexible, and pie algebras" and Bird-Kelly-Power-Street, "Flexible limits for 2-categories."