Limits in the 2-Category of 2-Hilbert Spaces
Posted by John Baez
guest post by Jamie Vicary
I’m interested in understanding the limits that exist in 2Hilb, the 2-category of finite-dimensional 2-Hilbert spaces. (The category 2Vect of finite-dimensional 2-vector spaces over Vect${}_{\mathbb{C}}$ would be just as good.)
One sort of limit that people study in 2-categories is the ‘iso-inserter’, which is like a weakened version of an equaliser.
Definition. In a 2-category, given parallel 1-cells $f,g:A \to B$, their iso-inserter is a 0-cell $S$ equipped with a 1-cell $s: S \to A$ and an invertible 2-cell $\sigma : f s \Rightarrow g s$, satisfying these conditions:
- For any 1-cell $z: Z \to A$ and invertible 2-cell $\zeta: f z \Rightarrow g z$, there is a unique 1-cell $c:Z \to S$ such that $s c=z$ and $\sigma c = \zeta$.
- Given 1-cells $c,d:Z \to S$ and a 2-cell $\nu:s c \Rightarrow s d$ such that $(\sigma d) (f \nu) = (g \nu) (\sigma c)$, there is a unique 2-cell $\mu:c \Rightarrow d$ with $s \mu = \nu$.
Unfortunately, it seems to me that 2Hilb doesn’t have all iso-inserters; just try calculating the iso-inserter of $\mathrm{id} _{\mathrm{Hilb}}$ with itself! Just from the first condition, you can sort of tell that the 0-cell $S$ should be a 2-Hilbert space having objects which are Hilbert spaces equipped with an automorphism, and 1-cells which are linear maps that ‘get along’ with the choices of automorphism. Unfortunately, this requirement of ‘getting along’ is quite stringent, and $S$ winds up being infinite-dimensional. Since we defined 2Hilb to only have finite-dimensional objects, this object doesn’t exist.
This doesn’t mean we should give up, because iso-inserters are a very constrained type of limit, and they won’t be the right type to consider in some 2-categories. The most powerful class of limits are called ‘bilimits’, but you have to be quite brave to wheel them onto the battlefield; because of the way they’re defined, they can be quite difficult to get to grips with in a particular 2-category, and as far as I’m aware, no direct definition (along the lines of the one given above for an iso-inserter) has been published for any nontrivial sort of bilimit. For this reason people often make do with simpler classes of limits, such as the ‘pie-limits’, of which the iso-inserter is an example. These are easier to define explicitly, but are quite ‘brittle’: the comparison 1-cells are unique up to isomorphism, rather than up to equivalence.
For some 2-categories this ‘brittleness’ is too much to take, and it’s possible for a 2-category to have bilimits but not pie-limits. Hopefully, this is what’s going on with 2Hilb. Fortunately, for these categories, there is a powerful coherence theorem we can use: every 2-category with bilimits is biequivalent to a 2-category with pie-limits. So, here’s my question: is there a 2-category, biequivalent to 2Hilb, that has all (finite) pie-limits?
Perhaps such a 2-category doesn’t exist. This would be a real shame, because 2Hilb is a categorification of Hilb, which has excellent completeness properties, and which itself is a categorification of $\mathbb{C}$, which also has excellent completeness properties. The formal connection between these different types of ‘completeness’ is strengthened by Bruce Bartlett’s recent observation, described in Proposition 3.5 of his thesis, that 2Hilb is the 2-category of Cauchy-complete H*-categories, which is a nice categorification of the statement that Hilb is the category of Cauchy-complete inner-product spaces over $\mathbb{C}$.
All the facts about 2-categorical limits that I’ve mentioned here are discussed in these two papers:
- G. M. Kelly, Elementary observations on 2-categorical limits, Bull. Austral. Math. Soc. 39 (1989), 301–317.
- J. Power, 2-categories, BRICS Notes Series, NS-98-7.
Re: Limits in the 2-Category of 2-Hilbert Spaces
Have you thought about whether $\mathbf{2Hilb}$ is Cauchy-complete in some appropriate 2-categorical sense?
Indeed, what is the appropriate 2-categorical sense?
I ask this because, given that $\mathbf{2Hilb}$ doesn’t seem to have all the limits you want, you might start by asking whether it satisfies even very weak completeness conditions; and Cauchy-completeness can be regarded as a very weak (co)completeness condition.