Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: Categorical adjoints as dagger category

John van de Wetering (Feb 07 2021 at 11:38):

Following the advice of @John Baez to ask many --potentially stupid-- questions: Is there some sense in which we can view a category of functors that have adjoints as a dagger category wherein the dagger corresponds to taking the left/right adjoint of the functor? If so, does this mean that a dagger category is a generalisation of adjointed functors or that adjoint functors are some sort of 'categorization' of dagger categories?

John van de Wetering (Feb 07 2021 at 11:39):

To muse a bit about this, if it is true, then monads are precisely the $\dagger$-positive maps, and the unitaries are precisely the adjoint equivalences

Amar Hadzihasanovic (Feb 07 2021 at 15:18):

A compact closed category, seen as a bicategory with a single 0-cell, is a bicategory whose morphisms all have adjoints (the duals) and left&right adjoints are isomorphic, so that would be the "best case" where the operation of taking adjoints is most similar to a dagger. You can start by asking yourself whether you are happy with this being a kind of dagger-bicategory.

Amar Hadzihasanovic (Feb 07 2021 at 15:20):

In fact I think that historically the opposite has happened: dagger categories were invented as a framework to talk about dualities that are not expressible by properties in the language of "bare" category theory, such as the duality induced by a choice of adjoints.

Amar Hadzihasanovic (Feb 07 2021 at 15:23):

So of your two options I think "dagger categories are a generalisation" is closer to the truth.

John Baez (Feb 07 2021 at 17:55):

@John van de Wetering wrote:

If so, does this mean that a dagger category is a generalisation of adjointed functors or that adjoint functors are some sort of 'categorization' of dagger categories?

I've only seen this in special case. The main problem is that taking the dagger of a morphism twice in a dagger category gets you back to the same morphism, whereas taking the left adjoint of a functor twice (if you can) may not, and ditto for the right adjoint.

However, there are some functors whose left adjoint is also their right adjoint:

• Ambidextrous adjunction.

These tend to show up a lot in representation theory: for example if G and H are finite groups and Rep(G) and Rep(H) are their categories of representations, any homomorphism f : G $\to$ H gives a functor $f_\ast:$ Rep(H) $\to$ Rep(G) in an obvious way, and this always has an ambidextrous adjoint. In the literature on group representation theory this fact is called Frobenius reciprocity.

This is a special case of a more general thing: any morphism between Kapranov--Voevodsky 2-vector spaces has an ambidextrous adjoint.

The name sounds scary, but a KV 2-vector space is just a category equivalent to $\mathsf{Vect}^n$ for some $n$. A morphism between them is an exact $\mathsf{Vect}$-enriched functor, and a 2-morphism is a natural transformation.

So, the 2-category of KV 2-vector spaces is a kind of categorification of a dagger category, where taking the adjoint of a 1-morphism acts like a dagger.

John Baez (Feb 07 2021 at 18:08):

Returning to the larger question, I think you have to choose what you're interested in:

• 2-categories where every morphism has a left adjoint
• 2-categories where every morphism has a right adjoint
• 2-categories where every morphism has a left adjoint, and, separately, a right adjoint
• 2-categories where every morphism has an ambidextrous adjoint: a morphism that's both a left adjoint and right adjoint
• dagger 2-categories.

John Baez (Feb 07 2021 at 18:08):

The last may not have been defined yet - but the second to last has been, and it should be a special case of the last.

John Baez (Feb 07 2021 at 18:10):

A "dagger 2-category" would be a 2-category with some extra structure specifying the dagger of every 1-morphism.

A 2-category where every morphism has an ambidextrous adjoint should be a special case, but this is a 2-category with an extra property, so it's nicer.

John van de Wetering (Feb 07 2021 at 18:49):

Amar Hadzihasanovic said:

In fact I think that historically the opposite has happened: dagger categories were invented as a framework to talk about dualities that are not expressible by properties in the language of "bare" category theory, such as the duality induced by a choice of adjoints.

That makes a lot of sense, thanks!

John van de Wetering (Feb 07 2021 at 18:50):

John Baez said:

The name sounds scary, but a KV 2-vector space is just a category equivalent to $\mathsf{Vect}^n$ for some $n$. A morphism between them is an exact $\mathsf{Vect}$-enriched functor, and a 2-morphism is a natural transformation.

So, the 2-category of KV 2-vector spaces is a kind of categorification of a dagger category, where taking the adjoint of a 1-morphism acts like a dagger

With $\mathsf{Vect}^n$ do you mean $n$ copies of $\mathsf{Vect}$?

Martti Karvonen (Feb 07 2021 at 19:32):

The phrase "Dagger 2-category" could be taken to mean two different things, depending on whether you want there to be a dagger on 1-cells or on 2-cells. An example of the former should be given by bicategories of spans (and presumably by 2-categories where every 1-cell has an ambidextrous adjoint etc), whereas an example of the latter is given by DagCat, the 2-category of (small) dagger categories, dagger functors and natural transformations: it turns out that that taking the dagger pointwise of a natural transformation between dagger functors results in a natural transformation. I've usually seen the phrase dagger 2-category to mean the latter notion (so having a dagger on 2-cells): this appears not just in papers I'm involved in, but also in some further works building on these, such as in the recent https://arxiv.org/abs/2101.05210 .

Martti Karvonen (Feb 07 2021 at 19:38):

As far as the original question goes, I have nothing to add to what other said, but I'll turn the question on its head: it's also been asked if the daggers of linear maps (i.e. adjoints) are adjunctions in a suitable category. A positive answer of sorts is given in https://www.jstor.org/stable/2040415?seq=1 : roughly speaking, a bounded linear map between Hilbert spaces and its adjoint induce a (contravariant) adjunction between the lattices of closed subspaces, and up to scalars all adjunctions between the subspace lattices are of this form.

John Baez (Feb 07 2021 at 21:17):

John van de Wetering said:

With $\mathsf{Vect}^n$ do you mean $n$ copies of $\mathsf{Vect}$?

I mean the n-fold product of categories $\mathsf{Vect} \times \cdots \times \mathsf{Vect}$. Just as every finite-dimensional vector space is isomorphic to $\mathbb{F}^n$ where $\mathbb{F}$ is your chosen field, every KV 2-vector space is equivalent to $\mathsf{Vect}^n$. Just as morphisms between finite-dimensional vector spaces can be described using matrices, morphisms between KV 2-vector spaces can be described using matrices of vector spaces. So it's all a straightforward categorification.

John van de Wetering (Feb 08 2021 at 13:44):

Martti Karvonen said:

As far as the original question goes, I have nothing to add to what other said, but I'll turn the question on its head: it's also been asked if the daggers of linear maps (i.e. adjoints) are adjunctions in a suitable category. A positive answer of sorts is given in https://www.jstor.org/stable/2040415?seq=1 : roughly speaking, a bounded linear map between Hilbert spaces and its adjoint induce a (contravariant) adjunction between the lattices of closed subspaces, and up to scalars all adjunctions between the subspace lattices are of this form.

Hmm, This actually feels quite close to some stuff we have done in effectus theory, where for every map $f$ we can define 'possibilistic' maps $f_\diamond$ and $f^\diamond$ between the posets of sharp predicates (in opposite directions) that are Galois connections. I'll need to see if this stuff is indeed related somehow.

John van de Wetering (Feb 08 2021 at 14:01):

Indeed, looking at this more closely, it looks like a special case of Corollary 208VII in Bas Westerbaan's thesis, and his statement that the adjunction characterises a linear map up to scalar is a property called 'rigidity' in the context of von Neumann algebras in Bram Westerbaan's thesis, Section 4.1.5.