# Can comma categories of small categories be understood as limits/colimits in $\textbf{Cat}$?

Let $F: C \to D$ be a functor of small categories. One can form the comma categories $F/$ and $/F$ with objects \begin{align*} (c,d,\phi) && \phi: F(c) \to d \\ (c,d,\psi) && \psi: d \to F(c) \end{align*} where $c\in Ob(C)$, $d \in Ob(D)$, and $\phi,\psi \in Arr(D)$. Can these categories be expressed as a limit or colimit in $\textbf{Cat}$?

## 1 Answer

To be a pedant, I think the term "2-pullback" is ambiguous, and perhaps should simply be defined to be a comma object, which is not quite the same thing as a lax pullback. The objects of the lax pullback category of functors $F: A \to C$ and $G: B \to C$ consist of quintuple $(a \in A, b \in B, c \in C, Fa \overset{f}{\to} b, b \overset{g}{\to} Gc)$, whereas the objects of a comma category consist of just a triple $(a \in A, b \in B, Fa \overset{f}{\to} Gc)$.

There is a variant where you require the maps $f,g$ to be invertible, resulting in the notions of iso-comma category and pseudopullback, which, though not isomorphic, are equivalent. But comma categories and lax pullbacks need not even be equivalent, as for example in the case of slice and co-slice categories.

Note that in order to view a slice or coslice category, or more generally a comma category, as a limit, you need to think in terms of weighted limits in the $\mathsf{Cat}$-enriched sense, as discussed here.

• The term "2-pullback" actually means something quite different from a comma object. To an Australian-style 2-category theorist, it means a strict (Cat-enriched) pullback, while on the nLab it is sometimes used to mean a weak pullback up to isomorphism (what an Australian would call a "bipullback"). Comma objects and lax pullbacks are both something different (and different from each other). Dec 30 '16 at 11:37