When does the 2-category V-Cat have pseudo-pullbacks?

Question

Edit: rewritten the question (edit:again), as I realised I wanted weak pseudo-pullbacks, not comma objects.

Consider the 2-category $V$-$Cat$ of $V$-enriched categories. An example is $Cat$ itself, where we take $V=Cat$. I'm interested in the more general setting where $V$ is at least finitely complete and complete, if not possessing small limits and colimits outright. I also know that $V$ is such that small $V$-categories can be interpreted as categories internal to $V$ with object of objects given by a coproduct of the tensor unit.

Pseudo-pullbacks exist in $Cat$, and can be calculated as a strict finite limit $A\times_C C^I \times_C B$ where $C^I$ is the isomorphism category of $C$. Actually I may not be interested in the pseudo-pullbacks, but strict 2-pullbacks, depending on what this example in $Cat$ is.

Based on Finn's answer, I guess this may exist when cotensors by $I$ exist ($I$ is the groupoid with two objects $a,b$ with a unique isomorphism $a \stackrel{\sim}{\to} b$). So this then is my question:

When do pseudo/strict 2-pullbacks (as appropriate) exist in $V$-$Cat$?

Answer 1

As I mentioned in the comment above, "weak limit" is normally defined as for limit, but with the universal property modified to ask only for existence not uniqueness. The 2-dimensional limit notion in which all equations between 1-cells are replaced by suitably coherent invertible 2-cells is usually given the prefix "bi".

Any 2-category with finite limits (in the strict 2-categorical sense) also has isocomma objects (defined like comma objects but with an invertible 2-cell), pseudopullbacks, and bipullbacks.

If V has finite limits (in the ordinary sense) then V-Cat has finite limits (in the strict 2-categorical sense).

(The parts of this that relate to the original version of your question are dealt with in Finn's answer.)

Answer 2

I'm presuming that you want to treat V-Cat as a 2-category rather than as a V-category. There might be a slick change-of-base argument that applies here, but I can't think of one. I think the answer is 'whenever V has finite limits'. Be warned that I haven't checked any of this fully, so it's just a sketch of what I would try.

A 2-category has comma objects if it has pullbacks and cotensors with the arrow category $\mathbf{2}$. Pullbacks in V-Cat should be straightforward. You can define $A^{\mathbf{2}}$ to have objects maps $f \colon I \to A(a,b)$ and say that the hom object $A^{\mathbf{2}}(f,g)$ is the pullback of $A(f,b') \colon A(b,b') \to A(a,b')$ along $A(a,g) \colon A(a,a') \to A(a,b')$. (Check that this does actually give a V-category, because I haven't.)

You have to show that $V\mathrm{-Cat}(X,A^{\mathbf{2}}) \cong \mathrm{Cat}(\mathbf{2},V\mathrm{-Cat}(X,A))$. A functor $F \colon X \to A^{\mathbf{2}}$ sends an object x to $Fx \colon I \to A(Gx,G'x)$ in A and comes with maps $X(x,y) \to A^{\mathbf{2}}(Fx,Fy)$. The latter hom object is a pullback, so you get a commuting square of the sort satisfied by a V-natural transformation (and you can use the projections to define the functors G and G'). Then you have the ingredients of a transformation $\alpha \colon G \to G'$, i.e. an object of $\mathrm{Cat}(\mathbf{2},V\mathrm{-Cat}(X,A))$.

Hope this helps.