Explicit description of a pullback of $(2,1)$-categories

Question

In the 1-category of 2-categories, with objects being categories enriched over Cat, and morphisms being 2-functors, is there an explicit way to describe a pullback of two functors $G:E\to D$ and $F:C\to D$? In particular I am interested in the case of $(2,1)$-categories, i.e. categories enriched over groupoids, but the general case would be great too.

I have been looking for resources, but I could only find literature on higher limits in higher categories, instead of 'ordinary' limits in categories with higher categories as objects.

Ideally, the answer would contain of an explicit description of the 0, 1 and 2 cells of the category. It seems reasonable that the 0-cells are just the objects of the ordinary underlying category, and perhaps the 1 and 2 cells given by the pullback of the relevant categories in $\mathcal{Cat}$ or $\mathcal{Gpd}$, but I can also imagine that there is a larger choice of 1-cells; those which are connected by a 2-cell in $D$. These two possible definitions of $C\times_D E$ are, most probably, not equivalent.

Any advice, ideas, or references are more than welcome.

Answer 1

For a strict pullback of strict 2-functors, the 0-, 1-, and 2-cells of the pullback are precisely the pullbacks of the underlying sets of cells.

Strict pullbacks of pseudofunctors do not generally exist. For instance, if $C$ and $E$ are the ordered set with 3 elements $x\xrightarrow{f} y\xrightarrow{g} z$, and $D$ is this category with an extra morphism $x\xrightarrow{h} z$ that is isomorphic to $g f$, while $F$ and $G$ act as the identity on objects but $F(gf) = gf$ while $G(gf) =h$, then in the strict pullback there would be no morphisms $x\to z$, hence nothing for the composite of $f$ and $g$ to be.

If in addition to the 1-cells involving a 2-cell in $D$, the 0-cells involve a 1-cell in $D$, then you have a comma category or pseudo-pullback. This is usually the sensible thing to look at when working with pseudofunctors (and pseudonatural transformations), which is why there's more literature on it; the 1-category of pseudofunctors is not very well behaved. There are various choices depending on what you take to be invertible. For instance, if everything is invertible then an object will consist of objects of $C$ and $E$ and an isomorphism or equivalence between their images in $D$, a 1-cell will consist of 1-cells in $C$ and $E$ and a 2-isomorphism filling the appropriate square in $D$, and a 2-cell will consist of 2-cells in $C$ and $E$ making a cylinder commute in $D$.

If you want the 0-cells to be a strict pullback but the 1-cells to involve a 2-cell in $D$, then you are looking at comma objects or pseudo-pullbacks in the 2-category of 2-categories, functors, and icons. These always exist, even if the functors are only pseudo: the 0-cells are as in the strict case, the 1-cells are a pair of 1-cells in $C$ and $E$ with a (globular, not square!) 2-isomorphism between their images in $D$, and the 2-cells are a pair of 2-cells in $C$ and $E$ making a square of 2-cells commute in $D$.

The various kinds of pullback are, indeed, not equivalent in general. But they are equivalent (though not isomorphic) if one of the functors $C\to D$ or $E\to D$ is a "fibration" in a suitable sense, i.e. admits lifting of isomorphisms/equivalences/2-isomorphisms (depending on the kind of pullback you're looking at).