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Stream: deprecated: our papers

Topic: Smash product of monoidal theories

Amar Hadzihasanovic (Jan 27 2021 at 09:56):

@Jade Master was faster than me and started this thread, but I'm opening another one so it doesn't look like I'm (co)appreciating my own work...

I have a new paper on the arxiv: The smash product of monoidal theories.

Abstract. The tensor product of props was defined by Hackney and Robertson as an extension of the Boardman-Vogt product of operads to more general monoidal theories. Theories that factor as tensor products include the theory of commutative monoids and the theory of bialgebras. We give a topological interpretation (and vast generalisation) of this construction as a low-dimensional projection of a "smash product of pointed directed spaces". Here directed spaces are embodied by combinatorial structures called diagrammatic sets, while Gray products replace cartesian products. The correspondence is mediated by a web of adjunctions relating diagrammatic sets, pros, probs, props, and Gray-categories. The smash product applies to presentations of higher-dimensional theories and systematically produces higher-dimensional coherence cells.

In a nutshell --
There is a known way to compose together symmetric monoidal theories $T, S$ in the form of symmetric operads or props. The idea is the following: the category of models of $T$ in a symmetric monoidal category $\mathbf{M}$ has a symmetric monoidal structure. So you can take models of $S$ in the SMC of models of $T$ in $\mathbf{M}$. That's the same as models of $T \otimes S$ in $\mathbf{M}$.
For example, this gives you the theory of commutative monoids as the tensor of the theory of monoids with itself; and the theory of bialgebras as the tensor of the theory of monoids with the theory of comonoids.

There is a classical way of composing together pointed topological spaces, ie spaces with a choice of basepoint $*$. That's the smash product $X \land Y$: you take the cartesian product of spaces $X \times Y$, then you identify all pairs of points of the form $(*,y)$ and $(x,*)$ to a single point.
For example, the smash product $S^n \land S^m$ of the $n$- sphere with the $m$-sphere is $S^{m+n}$.

So one is a very algebraic thing and the other is a very topological thing. But in this article I show that they are two facets of the same construction!

Amar Hadzihasanovic (Jan 27 2021 at 09:59):

The first step is that the tensor product of props has a “more basic” underlying construction, which however does not stay within the same category -- it's a kind of “external” product. It takes two pros (monoidal categories) and returns a prob (braided monoidal category). You can obtain the tensor product of two props universally from the “external” product of their underlying pros, where you forget the symmetric structure.

Amar Hadzihasanovic (Jan 27 2021 at 10:04):

The link is provided by diagrammatic sets, the subject of my previous paper. Diagrammatic sets are combinatorial structures that you can think of as “spaces whose cells have an orientation”.

Diagrammatic sets have

• a (non-symmetric) monoidal structure $- \otimes -$ called the Gray product;
• a geometric realisation $|-|$, that is, a functor to spaces, which turns Gray products into cartesian products.

You can define a smash product $- \land -$ of pointed diagrammatic sets which has the same definition as the smash product of spaces, except you replace cartesian products with Gray products.

(In the paper I use a different symbol, \owedge, which doesn't seem available here...)

Then the geometric realisation takes smash products to smash products: $|X \land Y| \simeq |X| \land |Y|$.

Amar Hadzihasanovic (Jan 27 2021 at 10:08):

So smash products of pointed diagrammatic sets functorially give you smash products of spaces.
In the article, I show that they also functorially give you the (external) tensor product of pros!

A pointed diagrammatic set $X$ with a single 0-cell presents a pro through a left adjoint functor $P$. This functor has a full and faithful right adjoint, which embeds the category of pros into the category of pointed diagrammatic sets.

It follows that every pro is presented by a pointed diagrammatic set.

Amar Hadzihasanovic (Jan 27 2021 at 10:14):

Suppose $X$ and $Y$ are presentations of the pros $PX, PY$. Then take their smash product. But first reverse the orientation of all cells in $Y$ (I call this $Y^\circ$).

There is a different left adjoint functor $G$ from diagrammatic sets to Gray-categories. Now, the category of probs can be included (pseudo-injectively) into the category of Gray-categories. That's an instance of @John Baez and Dolan's periodic table!

In general $GX$ is not a prob. But when $X$ and $Y$ are presentations of pros, then $G(X \land Y^\circ)$ is always a prob.

The theorem is that $G(X \land Y^\circ)$ is naturally isomorphic to the tensor product of pros $PX \otimes PY$.

Amar Hadzihasanovic (Jan 27 2021 at 10:15):

So through one functor, $X \land Y^\circ$ becomes the smash product of spaces $|X| \land |Y|$. Through another functor, it becomes the tensor product of pros $PX \otimes PY$.

Amar Hadzihasanovic (Jan 27 2021 at 10:16):

This has a really pretty topological interpretation in string diagrams, but for that you have to read the paper. :)

Amar Hadzihasanovic (Jan 27 2021 at 10:23):

A simple example:

There is a diagrammatic set $X$ generated by a single 0-cell and a single 1-cell. Then:

• $PX$ is the monoid of natural numbers $\mathbb{N}$, ie the free strict monoidal category on one object.
• $|X|$ is the 1-sphere $S^1$.

In this case $X = X^\circ$. You can take the smash product $X \land X$. Then:

• $G(X \land X)$ is the free braided strict monoidal category on one object, that is, the theory of braids;
• $|X \land X|$ is the 2-sphere $S^2$.

Amar Hadzihasanovic (Jan 27 2021 at 10:30):

Why should this be interesting to theoretical computer scientists?

If you start from a presentation with oriented equations then the smash products gives you another presentation with oriented equations, aka a rewrite system.
And not only that -- it also produces some higher-dimensional "stuff". If you know a bit of rewriting theory, that higher-dimensional stuff tells you that certain new critical branchings that are created in the smash products can be joined, ie, are confluent. So it's computationally interesting stuff.

So my hope is that you can use this to automatically get “nice” presentations (convergent, coherent...) from nice presentations.

I'm done talking.

Matteo Capucci (he/him) (Jan 27 2021 at 13:56):

Wow, Amar, that's very beautiful. Kudos!

Nathanael Arkor (Jan 27 2021 at 14:05):

I've just read the introduction and skimmed the rest of the paper (I hope to read it more thoroughly later) and the central idea seems very elegant and conceptually illuminating. I really like the higher-dimensional string diagrams too: they're very clear.

Nathanael Arkor (Jan 27 2021 at 14:06):

Does the tensor product of algebraic theories follow from this development too, or does that require some adaptation to preserve the cartesian structure?

Amar Hadzihasanovic (Jan 27 2021 at 14:23):

Algebraic theories (as cartesian props) are a reflective subcategory of the category of props, which forms a kind of ideal for the tensor product, in the sense that $T \otimes S$ is always cartesian when either $T$ or $S$ is. That's because the reflector is in fact “tensoring with the theory of commutative comonoids”. So you don't need to do anything special -- if you start with cartesian props, you'll end up with cartesian props.

That said, as I mentioned you still need to do a little work to get the tensor product of props from the “external” product of their underlying pros, but it's really not much (basically you reflect the prob onto a prop, then ask that the inclusions of the factors into the composite are maps of props).

I think the “correct” way to think of this (but for now it was overkill to develop formally, it's only briefly mentioned in the paper) is this: the smash product of an $n$-tuply monoidal theory and an $m$-tuply monoidal theory (in the Baez-Dolan periodic table sense) is an $(n+m)$-tuply monoidal theory.
A symmetric monoidal theory is one that is stable -- it is $n$-tuply monoidal for all $n$. That's why the smash of symmetric with symmetric should give you symmetric, but for example (monoidal $\land$ monoidal) gives you (braided monoidal).

Of course if you think of “1-theories” then you stabilise quickly, but you should get interesting phenomena with higher-dim theories. :)