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 $V$-$Cat$?

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    $\begingroup$ David, it might be useful to the reader if you spent a line or two explaining the notions that you use in the formulation of your question. Your question is only one line long: it certainly wouldn't hurt to expand a bit. $\endgroup$ – André Henriques Dec 5 '10 at 13:28
  • $\begingroup$ I guess by $V$-Cat you mean $V$-enriched categories!? I do not see where the construction of lax-pullbacks of categories fails here... $\endgroup$ – Thomas Nikolaus Dec 5 '10 at 13:28
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    $\begingroup$ You remember me a old (unsolved) question I think when I was a student. ANyway if V-Cat has 2-Pullback's you only need the commma about elementary situation $1: A\to A\leftarrow A: 1$ (See "Fibrations and Yoneda's lemma" by ROss Street Lecture Notes in Mathematics, 1974, Volume 420/1974). $\endgroup$ – Buschi Sergio Dec 5 '10 at 16:07
  • $\begingroup$ Rewritten the question - I realise I didn't want comma objects, but weak/pseudo pullbacks $\endgroup$ – David Roberts Dec 5 '10 at 23:05
  • $\begingroup$ What do you mean by weak pullback, David? This is usually defined to be like a pullback except that you do not demand uniqueness in the universal property, only existence. I am guessing that you may mean bipullback, which is like a pullback except that one only ever asks for commutativity of diagrams of 1-cells to hold "up to isomorphism". Any 2-category with finite limits has comma objects, pseudopullbacks, and bipullbacks, and the assumptions on V you list are more than enough to ensure that V-Cat has finite limits. (Or all limits if V is complete.) $\endgroup$ – Steve Lack Dec 6 '10 at 3:12

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.)


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.

  • $\begingroup$ This may be enough to get me over the line. Thanks! I'll wait, though, and see if anyone points out something like 'this is an easy consequence of such and such a result about completeness of V-categories'. Then I'll know my question is really naive :-) $\endgroup$ – David Roberts Dec 6 '10 at 1:30

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