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Stream: deprecated: mathematics

Topic: parametrised operations

Matteo Capucci (he/him) (Sep 11 2021 at 09:44):

I stumbled upon a structure I'd call a 'parametrised monoidal operation' and I was wondering if someone has ever thought about it before (googling is not satisfying)
In its most simplified form, it's an operation $\mu : P \times Q \times Q \to Q$ on a set $Q$, together with a map $\eta : P \to Q$, that satisfies parametrised associativity and unitality:

$$\mu(p, \mu(p, q_1,q_2),q_3) = \mu(p,q_1,\mu(p,q_2,q_3))\\ \mu(p,\eta(p),q)=\mu(p,q,\eta(p))=q$$

The more astute among you might have noticed this is secretly assuming a comonoidal structure on $P$, which comes for free in a Cartesian setting. Otherwise, to generalize this structure to an arbitrary monoidal category one has to esplicitly ask for it (but notice: this pair of comonoidal-monoidal structure, even assuming $P=Q$, is weaker than an Hopf algebra structure, since no compatibility between the operations is asked)
The even more astute among you might have noticed this is a comonoid in lenses, and its monoidal generalization is a comonoid in optics.

Examples of such structures:

1. Any monoid, with $P=1$, or simply with $\mu$ ignoring $P$ altogether
2. Any ring, with $P=2=\{+,×\}$ selecting either of the two monoidal structures and $\eta$ mapping each operation to its identity
3. I suspect that Lie groups might provide interesting examples? I picture a group having a continuous family of group structures indexed by $P$. Something like that is going on in deformation theory but iirc the deformed operations are not associative and unital throughout

John Baez (Sep 11 2021 at 12:39):

Neat! I don't know how to get a parametrized bunch of monoids from a Lie group....

except that any monoid in a symmetric monoidal category has two monoidal structures, the usual one and the "opposite" one where we switch before we multiply, so a monoid in Set or Top or various other categories has a monoid structure parametrized by $2$.

John Baez (Sep 11 2021 at 12:41):

In the deformation theory of associative algebras we do ask that the multiplication be associative and unital (and give us an algebra) throughout the deformation process. So there are lots of $\mathbb{R}$-parametrized monoids running around in algebra, e.g. in "deformation quantization" where we deform a commutative product into a noncommutative one.

Matteo Capucci (he/him) (Sep 11 2021 at 12:43):

oh so i didn't recall correctly

Matteo Capucci (he/him) (Sep 11 2021 at 12:43):

this sounds potentially interesting then

John Baez (Sep 11 2021 at 12:43):

For example, Weyl algebras and Clifford algebras give deformation quantizations of polynomial algebras, where we get an associative multiplication depending on a parameter often called $\hbar \in \mathbb{R}$: Planck's constant.

Matteo Capucci (he/him) (Sep 11 2021 at 12:51):

also it occurred me now: elliptic curves? this is delicate since I'm not very familiar with them but I take that $P$ could be $\mathbb C^2$ (or probably some more complex variety, I guess it should be the 'moduli space of elliptic curves over $\mathbb C$'), then $Q$ is a torus and $p=(\omega_1, \omega_2) \in P$ are used by $\mu$ to select the group law corresponding to the elliptic curve generated by the choice of $p$

John Baez (Sep 11 2021 at 12:58):

As an abstract group all elliptic curves are the same; what you're changing is the complex structure.

Matteo Capucci (he/him) (Sep 11 2021 at 12:58):

ha, fair

John Baez (Sep 11 2021 at 12:59):

That is, if you treat the underlying set of a family of elliptic curves as the same (which you're doing in your setup), it's not the group law that's changing, it's the complex structure.

John Baez (Sep 11 2021 at 13:01):

You could consider the set as variable, which you're doing when your changing your guys $(\omega_1, \omega_2)$ and defining the elliptic curve to be $\mathbb{C}$ module the lattice generated by $(\omega_1, \omega_2)$. But that's a different kind of "parametrized monoid", where the underlying object changes (to something isomorphic).

Matteo Capucci (he/him) (Sep 11 2021 at 13:06):

Indeed. This is not unheard of, in fact I suspect this (a law like $(p :P) \to Q_p \times Q_p \to Q_p$) would be what you get if instead of lenses you were to consider dependent lenses, in which $Q$ can be a dependent type over $P$.

Matteo Capucci (he/him) (Sep 11 2021 at 13:06):

That's maybe more interesting