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Stream: theory: category theory

Topic: Hilbert spaces and cospans

Amar Hadzihasanovic (Apr 10 2020 at 10:26):

This discussion on adjoint operators made me think of a little result that I proved a while ago, of which I'm fond, but I don't know what use to make of it. I'm sharing it here in hope that someone may have ideas about it.

Amar Hadzihasanovic (Apr 10 2020 at 10:26):

When people talk of $\mathrm{Hilb}$, the category of Hilbert spaces, usually there's two possible choices of morphisms:

• bounded linear maps, or
• short linear maps, aka bounded maps with norm $\leq 1$, aka norm-non-increasing maps.
  The second one has some nice properties, such as the fact that the only isomorphisms are the unitary isomorphisms; let's focus on that one.

Amar Hadzihasanovic (Apr 10 2020 at 10:26):

From the point of view of universal algebra, these morphisms are still a bit odd, because they do not preserve the structure on vector spaces that makes them (pre-)Hilbert spaces, namely, the inner product. The morphisms that are “natural” from this point of view are only the isometries. Let's call $\mathrm{Hilb}_1$ the category of Hilbert spaces and isometries.

Amar Hadzihasanovic (Apr 10 2020 at 10:27):

Now $\mathrm{Hilb}_1$ has pushouts, so we can enough to form the bicategory of cospans $\textit{Cosp}(\mathrm{Hilb}_1)$. This has as morphisms $H \to H'$ cospans $H \hookrightarrow K \hookleftarrow H'$ of isometries, and as 2-morphisms from $H \hookrightarrow K \hookleftarrow H'$ to $H \hookrightarrow K' \hookleftarrow H'$ isometries $K \hookrightarrow K'$ that make the two triangles commute.

Amar Hadzihasanovic (Apr 10 2020 at 10:27):

Then, we can obtain canonically a category by “1-truncating”, that is, identifying any two morphisms which are connected by a 2-morphism. This is similar to how, for example, the category of sets and relations is the 1-truncation of the bicategory of spans of sets.
The result is:

Proposition: The 1-truncation of $\textit{Cosp}(\mathrm{Hilb}_1)$ is equivalent to $\mathrm{Hilb}$.

Amar Hadzihasanovic (Apr 10 2020 at 10:27):

Moreover, bicategories of spans and cospans have a canonical “dagger”, that is a direction-reversing involution on morphisms between two objects, which simply exchanges the two sides of the (co)span.
Denoting by $\tau$ the truncation, if $f$ is a cospan of isometries and $f^\dag$ its dual cospan, we have that $\tau(f^\dag) = \tau(f)^\dag$, the adjoint of $\tau(f)$. Thus we recover adjoint maps from the natural dagger of cospans.

Amar Hadzihasanovic (Apr 10 2020 at 10:28):

The reason I like this result is that it reconstructs $\mathrm{Hilb}$ and its dagger structure, which are both somewhat odd from the point of view of categorical algebra, from

• $\mathrm{Hilb}_1$, which is a full subcategory of a category of models of an algebraic theory,
• the “cospan bicategory” and “truncation” operations, which are natural categorical constructions.

Amar Hadzihasanovic (Apr 10 2020 at 10:28):

But I don't know in what direction this could be taken! Is there anything interesting about the 2-morphisms of $\textit{Cosp}(\mathrm{Hilb}_1)$?

Amar Hadzihasanovic (Apr 10 2020 at 10:28):

One thing that seemed “suggestive” to me is the following: TQFTs are formalised as monoidal functors from a category of cobordisms to the category of Hilbert spaces. But a cobordism is a cospan of embeddings of manifolds, modulo some equivalence.

• Question: * Are there “nice” TQFTs that lift to a functor of bicategories of cospans, or maybe come from just an assignments of isometries to embeddings, via the cospan construction?

My knowledge of TQFTs is not nearly enough to approach this!

Tim Hosgood (Apr 10 2020 at 13:26):

is the $\mathrm{Hilb}$ in your proposition the category with bounded maps or short ones? such a nice result though! is this studied anywhere?

Oscar Cunningham (Apr 10 2020 at 13:35):

This is nice!
Is there a classical analogue? I'd say the analogue of isometries would be injective functions. So I think the cospans would be partial injections. That makes sense as a classical analogue of norm-decreasing maps.

Oscar Cunningham (Apr 10 2020 at 13:37):

@Tim Hosgood It's the short ones. The cospan $H\hookrightarrow K \hookleftarrow H'$ corresponds to the map that sends $H$ to $H'$ by projecting the image of $H$ in $K$ onto the image of $H'$. This is always short.

Amar Hadzihasanovic (Apr 10 2020 at 13:56):

Oscar Cunningham said:

This is nice!
Is there a classical analogue? I'd say the analogue of isometries would be injective functions. So I think the cospans would be partial injections. That makes sense as a classical analogue of norm-decreasing maps.

Yes, exactly: one gets the category of partial injections. In fact the $\ell^2$ functor defined by your supervisor Chris Heunen arises from this construction applied to its restriction from (sets, injections) to (Hilbert spaces, isometries).

Amar Hadzihasanovic (Apr 10 2020 at 13:58):

Of course, partial injections are more easily obtained as spans of injections. But spans of isometries only give you partial isometries.

Amar Hadzihasanovic (Apr 10 2020 at 14:51):

So, since operators on a single Hilbert space are so important in the theory, something we can do is look at the hom-category $[H,H]$ in the bicategory of cospans, above the hom-set of short operators on $H$ -- which is of course a monoidal category...

Amar Hadzihasanovic (Apr 10 2020 at 14:52):

In the simplest case of $0$ the trivial Hilbert space, there is a unique map $0 \to 0$ in $\mathrm{Hilb}$. But there is a (unique) cospan of isometries $0 \to 0$ for each Hilbert space $H$...

Amar Hadzihasanovic (Apr 10 2020 at 14:54):

Morphisms between these cospans are exactly injections of Hilbert spaces, and their composition is a pushout of morphisms from the initial object, ie a coproduct, ie a direct sum of Hilbert spaces.

Amar Hadzihasanovic (Apr 10 2020 at 14:55):

So the monoidal category $[0,0]$ is equivalent to $(\mathrm{Hilb}_1, \oplus, 0)$.

Amar Hadzihasanovic (Apr 10 2020 at 14:56):

(I guess that will be true of the cospan bicategory on any category with pushouts and an initial object -- the hom-category on the initial object is the original category with the monoidal structure given by coproducts).

Amar Hadzihasanovic (Apr 10 2020 at 15:00):

What is a monoid in $[0,0]$, ie a monad on $0$ in $\textit{Cosp}(\mathrm{Hilb}_1)$? Let $H$ be a Hilbert space, corresponding to a morphism $0 \to 0$. There is always a unique 2-morphism $\mathrm{id}_0 \Rightarrow H$, so that will have to be the unit.

Amar Hadzihasanovic (Apr 10 2020 at 15:03):

The multiplication is an isometry $H \oplus H \hookrightarrow H$. So there can be one only if $H$ is infinite-dimensional.

Amar Hadzihasanovic (Apr 10 2020 at 15:04):

(Or $0$)

Amar Hadzihasanovic (Apr 10 2020 at 15:07):

But unitality would force both the components to be the identity, but that's not an isometry unless $H = 0$... so I don't think there's any interesting monoid.

Amar Hadzihasanovic (Apr 10 2020 at 15:10):

I vaguely recall isometries $H \oplus H \hookrightarrow H$ being relevant to Geometry of Interaction, though.

Amar Hadzihasanovic (Apr 10 2020 at 15:43):

(Now that I think about it, I haven't told the story exactly right. It's not true that direct sum is a coproduct in $\mathrm{Hilb}_1$ -- in fact it only satisfies a universal property wrt pullback squares... but that's enough to still define a composition of cospans, and a bicategory of them...)

Amar Hadzihasanovic (Apr 10 2020 at 15:47):

(Which is why instead of getting “a unique monoid on every object” as we should in a cocartesian category, we don't really get any nontrivial ones :grimacing:)

(=_=) (Apr 10 2020 at 16:03):

Amar Hadzihasanovic said:

This discussion on adjoint operators made me think of a little result that I proved a while ago, of which I'm fond, but I don't know what use to make of it. I'm sharing it here in hope that someone may have ideas about it.

Cospans come up in C* -algebras as well, in the form of linking algebras. Spans in that context are called C* -correspondences, which are used in KK-theory, and these C* -correspondences are implemented using Hilbert bimodules. Sorry, I'm hazy about the details here.

John Baez (Apr 10 2020 at 18:21):

Usually "nice" TQFTs lift in various ways. The most productive way to go up one notch is to consider symmetric monoidal 2-functors from

John Baez (Apr 10 2020 at 18:22):

[(n-2)-manifolds, cobordisms, cobordisms between cobordisms]

John Baez (Apr 10 2020 at 18:22):

to

John Baez (Apr 10 2020 at 18:23):

[2-Hilbert spaces, linear functors, natural transformations]

Amar Hadzihasanovic (Apr 10 2020 at 21:41):

I think this is a description of a bicategory equivalent to $\textit{Cosp}(\mathrm{Hilb}_1)$, which may be easier to work with to understand what is going on.

Objects are still Hilbert spaces.
Now a morphism $H \to H'$ is given by a pair of Hilbert spaces $K_0, K_1$, together with a unitary isomorphism $u: H \oplus K_0 \to H' \oplus K_1$.

Amar Hadzihasanovic (Apr 10 2020 at 21:42):

I will denote such a morphism just by $u: H \oplus K_0 \to H' \oplus K_1$, since it contains all the information...

Amar Hadzihasanovic (Apr 10 2020 at 21:45):

A 2-morphism from $u: H \oplus K_0 \to H' \oplus K_1$ to $u': H \oplus K'_0 \to H' \oplus K'_1$ is a pair of isometries $i_0: K_0 \hookrightarrow K'_0, i_1: K_1 \hookrightarrow K'_1$ with the property that

$(\mathrm{id}_H \oplus i_0) ; u' = u ; (\mathrm{id}_{H'} \oplus i_1)$.

Amar Hadzihasanovic (Apr 10 2020 at 21:47):

Vertical composition of 2-morphisms is just composing the isometries pairwise.

Amar Hadzihasanovic (Apr 10 2020 at 21:55):

The horizontal composition of $u: H \oplus K_0 \to H' \oplus K_1$ and of $v: H' \oplus K'_0 \to H'' \oplus K'_1$ is

... well, the unitary isomorphism $H \oplus (K_0 \oplus K'_0) \to H'' \oplus (K_1 \oplus K'_1)$ that you get by composing $u$ and $v$ along $H'$ and permuting inputs and outputs when necessary (writing it down explicitly is not very illuminating).

Amar Hadzihasanovic (Apr 10 2020 at 21:57):

And the horizontal composite of two 2-morphisms should be clear: just stick the isometries “one next to another” in the direct sums...

(=_=) (Apr 11 2020 at 00:39):

John Baez said:

[(n-2)-manifolds, cobordisms, cobordisms between cobordisms]

arXiv:1501.00792 is probably the reference for this.

John Baez (Apr 11 2020 at 00:52):

I'm glad you know that paper by my student Franciscus Rebro, @Rongmin Lu! It was sort of subsumed by this one by two other students:

John Baez (Apr 11 2020 at 00:53):

• Kenny Courser and Daniel Cicala, Bicategories of spans and cospans.

John Baez (Apr 11 2020 at 00:54):

Note: version 1, and only version 1, studies the bicategory of spans-of-spans in a category with finite limits and shows it's symmetric monoidal.

(=_=) (Apr 11 2020 at 07:40):

John Baez said:

I'm glad you know that paper by my student Franciscus Rebro

All credit to Google, though I should probably start working my way through the UCR back catalogue for more of your and your students' greatest hits.

It was sort of subsumed by this one by two other students:

Thanks, good to know!

Notification Bot (Sep 20 2021 at 13:02):

Amar Hadzihasanovic has marked this topic as resolved.

Notification Bot (Sep 20 2021 at 13:02):

Amar Hadzihasanovic has marked this topic as unresolved.