Examples of (co)complete 2-categories that aren't (co)complete as a 1-category

Question

Title says it all. Are there examples of 2-categories that are (co)complete (with 2-(co)limits) such that their underlying 1-categories aren't (co)complete as a 1-category?

Answer 1

It depends on what you mean by by 2-limits.

If you mean strict 2-limits, then no. In a strict 2-category, any 2-categorical (co)limit of an ordinary diagram on a locally discrete 2-category is also a 1-categorical limit for that diagram in the underlying 1-category, essentially because the functor $\mathrm{ob} : \mathbf{Cat} \to \mathbf{Set}$ preserves all limits. So, in particular, the underlying 1-category of a strictly (co-)complete 2-category is again (co-)complete.

If you mean bi-limits, then yes.$\newcommand{\C}{\mathbf{C}}\newcommand{\D}{\mathbf{D}}$ Consider the 2-category consisting of:

• pointed categories $(\C, c_0)$
• (weakly) pointed functors $(F : \C \to \D,\; \varphi : Fc_0 \to d_0)$
• pointed transformations $(\alpha : F \Rightarrow F',\; \varphi' \alpha_{c_0} = \varphi)$.

This is complete and co-complete (in the sense of bilimits), but its underlying 1-category has no strictly initial object, since if $I$ is the “walking isomorphism” $[ 0 \cong 1 ]$, then any pointed category $(\C,c_0)$ admits at least two distinct pointed functors into $(I,0)$: the two constant functors, each with a unique isomorphism on the point.