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Stream: learning: questions

Topic: Tensors and PROPS

Nathaniel Virgo (May 14 2020 at 09:23):

A PROP is a category where the objects are natural numbers and the monoidal product on objects is addition. For example, one can define a PROP of linear transformations, where the morphisms are $n\times m$ matrices, composition is matrix multiplication and the monoidal product is the matrix direct sum.

However, for a category of matrices it would be more usual to define the monoidal product as the matrix tensor product. So the objects are still natural numbers, but the monoidal product on objects is multiplication.

Question 1: is there a name for categories like this, that are like PROPS but where the monoidal product is multiplication instead of addition on objects? I'll call them TEPs for now, for tensor and permutation category.

Question 2: is there a magic wand I can wave, where given a PROP (possibly with additional restrictions) I can obtain a corresponding TEP, preferably in a unique way? That is, I'd like to take the process of going from the PROP of matrices to the usual category of matrices, equivalent to Vect, and generalise in a useful way to a different PROP, if that can be done.

Matteo Capucci (he/him) (May 14 2020 at 09:42):

Since multiplication is repeated addition, you can probably get from a PROP to a TEP in a unique way, right?

Matteo Capucci (he/him) (May 14 2020 at 09:43):

I don't know if that gives you tensor product of matrices, though

Jem (May 14 2020 at 11:18):

Small note - PROPs are symmetric monoidal, not just monoidal. We need to use the symmetry in order for the tensor product in the TEP to be functoral. (I've not verified that it is functoral, but did need to use it to define the map on morphisms.)

Cole Comfort (May 14 2020 at 15:37):

Nathaniel Virgo said:

A PROP is a category where the objects are natural numbers and the monoidal product on objects is addition. For example, one can define a PROP of linear transformations, where the morphisms are $n\times m$ matrices, composition is matrix multiplication and the monoidal product is the matrix direct sum.

However, for a category of matrices it would be more usual to define the monoidal product as the matrix tensor product. So the objects are still natural numbers, but the monoidal product on objects is multiplication.

Question 1: is there a name for categories like this, that are like PROPS but where the monoidal product is multiplication instead of addition on objects? I'll call them TEPs for now, for tensor and permutation category.

Question 2: is there a magic wand I can wave, where given a PROP (possibly with additional restrictions) I can obtain a corresponding TEP, preferably in a unique way? That is, I'd like to take the process of going from the PROP of matrices to the usual category of matrices, equivalent to Vect, and generalise in a useful way to a different PROP, if that can be done.

Borrowing from quantum computing, I call these props qudit models. For every fixed natural number $d$ and every commutative sermring/ring/field $S$, the full subcategory of $({\sf Mat} (S),\otimes)$ with objects $d^n:n \in\mathbb{N}$ is a prop.

Matrices are bimonoidal categories containing the additive prop and all qudit props. I don't think there is any easy way to get from an additive prop to a qubit prop. You might be able to use the bimonoidal structure to try to recover some of it , though.

For example, the prop $({\sf Span}^\sim (\sf FinOrd),+) \cong ({\sf Mat}(\mathbb{\N}),\oplus)$ is generated by a bicommutative hopf algebra.
However, I show that the full qubit model, ZX& of $({\sf Mat}(\mathbb{N}),\otimes)$, contains the image of this bicommutative hopf algebra + their adjoints + nontrivial extra equations involving the not gate and and gate.

However, there is a connection between additive props and qudit props over different base categories. Linear relations over a field is an additive prop; however, $\mathbb{Z}_2$ linear relations is the same as the projective quotient of Hadamard free real stabilizer fragment of the ZX-calculus. I think this correspondence can only go so far, probably no further than stabilizers, because the structure of the qubit fragment becomes too expressive.

John Baez (May 14 2020 at 18:19):

Jem said:

Small note - PROPs are symmetric monoidal, not just monoidal.

Smaller note: a merely monoidal category where the objects are natural numbers and the tensor product of objects is addition is called a PRO. If it's braided monoidal it's called a PROB. If it's symmetric monoidal it's called a PROP. (The last P stands for "permutations".)