Construction of 2-limits in 2-categories

Question

Limits in a category can be built as a combination of the basic limits that are products and equalizers. Is there a similar construction for 2-limits in a 2-category? If yes, is it from products and equalizers, or some other basic limits? How do you, for example, built a comma object (a special case of 2-limit) using this construction? How does it instantiate in the 2-category of 1-categories?

Answer 1

You need more than products and equalisers, but only a bit more: it is enough to have products, equalisers, and cotensors with $\mathbf{2}$ (the free arrow, i.e. the category $\{ 0 \to 1 \}$). First of all, because every category is obtained as an iterated conical colimit of diagrams starting from copies of $\mathbf{1}$ and $\mathbf{2}$, it follows that you have cotensors with any category. Then, by enriched category theory, we get all 2-limits.

For example, given $f : X \to Z$ and $g : Y \to Z$ in a 2-category, the comma object $(f \downarrow g)$ can be constructed as the conical limit of the following diagram, $$\require{AMScd} \begin{CD} @. @. Z \\ @. @. @VV{g}V \\ @. \mathbf{2} \pitchfork Z @>>> Z \\ @. @VVV \\ X @>>{f}> Z \end{CD}$$ where the two arrows $\mathbf{2} \pitchfork Z \to Z$ are induced by the two functors $\mathbf{1} \to \mathbf{2}$.

All that said, a word of caution: what I said applies only to strict 2-limits, which can be quite different from bilimits. To construct bilimits one needs instead (bicategorical) products, inserters, and equifiers.