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Stream: learning: questions

Topic: Is there a nice 2-category of matrices?

Peva Blanchard (Jun 08 2024 at 10:32):

I'm learning about adjunctions in 2-categories other than the 2-category of (small) categories.

From what I understand, a 1-cell $f : X \rightarrow Y$ is a left adjoint to a 1-cell $g: Y \rightarrow X$ when

• there exists a unit 2-cell $\eta: id_X \rightarrow gf$
• there exists a co-unit 2-cell $\epsilon: fg \rightarrow id_Y$
• and they satisfy the triangle identities.

I've been trying to define a 2-category of matrices so that adjunction would correspond to transposition, in vain. Or more precisely, I can't find a right definition for the 2-cells.

• the 0-cells are the natural numbers $0, 1, 2, \dots$
• a 1-cell $M: n \rightarrow m$ is a (e.g., real-valued) matrices with $m$ rows and $n$ columns
• and 2-cells are ???

If I think of a matrix as a profunctor enriched in $\mathcal{V} = (\mathbb{R}, \le, +, 0)$, then 2-cells would correspond to $\mathcal{V}$ natural transformations. Unfolding the definition, this means that there is a (unique) 2-cell $A \Rightarrow B$ iff $A_{ij} \le B_{ij}$ for all indices $i,j$.

But this definition doesn't seem to yield an adjunction between a matrix $A$ and its transposition $A^T$.

Hence my question: is it possible to define a 2-category of matrices so that adjunction corresponds to transposition?

David Michael Roberts (Jun 08 2024 at 11:08):

The transpose of a matrix is more like the kind of structure a dagger category provides, in particular the dagger category of (finite-dimensional?) Hilbert spaces.
Or else it's more like working with an enriched category that is not a Cat-category. Maybe something like enriching over categories that are themselves enriched in something nearly degenerate, something like metric spaces? Just spitballing. It's late here, and I'm not thinking these suggestions through properly...

David Corfield (Jun 08 2024 at 11:18):

Perhaps not directly answering you, but there are some interesting related thoughts in this MO question.

Simon Burton (Jun 08 2024 at 13:06):

There's some discussion related to this here, section 2.6. Btw, I'll be talking about this paper at the upcoming ACT conference.

Simon Burton (Jun 08 2024 at 13:12):

And this talk by Todd Trimble discusses (at ~8:45) the 2-category of sets & relations, and what adjunctions look like there.

John Baez (Jun 08 2024 at 15:09):

A rather famous 2-category of with matrices as morphisms is the 2-category of [[Kapranov-Voevodsky 2-vector spaces]].

John Baez (Jun 08 2024 at 15:10):

The objects of this 2-category are categories of the form $\mathsf{Vect}^n$, and the morphisms are $m \times n$ matrices of vector spaces, and the 2-morphisms are $m \times n$ matrices of linear maps.

John Baez (Jun 08 2024 at 15:12):

If we loosen up the definition to include are categories equivalent to $\mathsf{Vect}^n$, as we should, then any category of representations of a finite group is a Kapranov-Voevodsky 2-vector space.

John Baez (Jun 08 2024 at 15:14):

We get the math of 2-vector spaces by taking the math of vector spaces replacing $\mathbb{C}$ (or your favorite field) by $\mathbf{Vect}$, replacing $+$ by $\oplus$, and replacing $\times$ by $\otimes$. So we are 'categorifying' linear algebra.

The 2-category of 2-vector spaces, usually called $\mathbf{2Vect}$, has the property you hint at: every 1-morphism has both a left and right adjoint.

Peva Blanchard (Jun 08 2024 at 16:52):

Great thank you all for the links!

Peva Blanchard (Jun 08 2024 at 16:59):

@Simon Burton I just noticed that you already shared your paper with me one or two months ago. But I think I am better equipped now to try reading it.