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Stream: deprecated: mathematics

Topic: almost linear map

Matteo Capucci (he/him) (Apr 14 2022 at 10:07):

Suppose $R$ is a 'ripped monoid' (better terminology welcome), i.e. a set with associative and unital operations in every arity, not just finite ones, hence equipped with operations $\sum_{i \in I}$ for every set $I$, that are compatible etc.
Examples of these monoids: the positive real numbers (including $+\infty$) with sums, any cocomplete lattice, and I guess compact spaces with the operator that takes accumulation points of sets.
A map of ripped monoids then commutes with all the operations, included the nullary ones.
Now consider the ripped monoid $(2^X, \bigcup)$, and a map of ripped monoids $f:2^X \to \R^+$. This satisfies $f(\bigcup_{i \in I} X_i) = \sum_{i \in I} f(X_i)$, so it is sensible to say $f$ is determined by its values on singletons. Yet one also has $f(X_1 \cup X_1) = f(X_1)+f(X_1)$ so it seems every such map has to map everything to $0$.
Can you think of a sensible restriction on the notion of map such those maps $2^X \to R$ are in bijection with maps of sets $X \to R$? Yes I'm looking for an adjunction :P

Matteo Capucci (he/him) (Apr 14 2022 at 10:10):

I guess I'm trying to understand how much powerset is the 'free ripped monoid' monad. It seems to be the 'free idempotent ripped monoid' monad, i.e. the free lattice (?) monad.

Nathanael Arkor (Apr 14 2022 at 10:24):

i.e. a set with associative and unital operations in every arity, not just finite ones

This sounds like a an infinitary operad.

Matteo Capucci (he/him) (Apr 14 2022 at 10:40):

It very likely is

Matteo Capucci (he/him) (Apr 14 2022 at 10:40):

But iirc a map of operads is exactly what I described commuting on the nose with the operations

Matteo Capucci (he/him) (Apr 14 2022 at 10:42):

It occurred to me now that I might simply be looking for lax/oplax maps between these things, with the understanding that they are actually 2-dimensional objects, like preorders

Oscar Cunningham (Apr 14 2022 at 10:53):

Ripped monoids have to be commutative, right? Because we're not assuming any ordering on the index sets.

Oscar Cunningham (Apr 14 2022 at 10:56):

I think you can equip any ripped monoid with a partial order given by $x\geq y$ iff the exists $a$ with $y+a = x$. Note that if $x = y+a$ and $y = x+b$ then we can evaluate $x + a + b + a + b+\dots$ in two ways to prove $x=y$. So this ordering doesn't make anything equivalent that wasn't already equal.

Then instead of 'almost linear' we can say 'sublinear'.

Matteo Capucci (he/him) (Apr 14 2022 at 13:13):

Great Oscar!

Matteo Capucci (he/him) (Apr 14 2022 at 13:14):

Yes, subliner is what I called 'oplax maps' above. I think that's spot on.

Oscar Cunningham (Apr 14 2022 at 13:26):

then we can evaluate $x + a + b + a + b+\dots$ in two ways to prove $x=y$

Actually, I can't quite convince myself of this. Getting from $x + (a+b)=x$ to $x + (a+b)+ (a+b)+\dots=x$ seems to involve escaping from infinitely many brackets.

John Baez (Apr 14 2022 at 15:28):

I think there's a significant difference between an algebraic structure with operations of arity $\alpha$ for all cardinals less than some cardinal $\kappa$, and algebraic structures with operations of every arity. The former can be handled by an bounded [[infinitary Lawvere theory]].

John Baez (Apr 14 2022 at 15:30):

The powerset monad corresponds to an infinitary Lawvere theory that has operations of every arity, so it's more scary.

John Baez (Apr 14 2022 at 15:31):

But it's the monad for [[suplattices]]. A "ripped idempotent commutative monoid" is usually called a [[suplattice]], so look there for more information, @Matteo Capucci (he/him).

John Baez (Apr 14 2022 at 15:35):

I got interested in suplattices and the power set monad a while back, and I contributed to this page. I think @Todd Trimble has too, and he probably understands everything about infinitary Lawvere theories and their corresponding monads better than I do.

Matteo Capucci (he/him) (Apr 14 2022 at 15:37):

thanks John, I tend to get lost in the sea of flavours of latt(ic)es out there, and I wasn't sure anymore which corresponded to $P$

Matteo Capucci (he/him) (Apr 14 2022 at 15:38):

are you aware of people studying these structures with sublinear maps instead of linear ones? by that I mean maps such that $f(\bigvee_i a_i) \leq \bigvee_i f(a_i)$

John Baez (Apr 14 2022 at 15:38):

I don't know about that. But I know lots of fun stuff about suplattices!

Matteo Capucci (he/him) (Apr 14 2022 at 15:39):

if you'd like to tell me, and if's not what you already kindly wrote down on the nLab, i'm happy to listen! :D

John Baez (Apr 14 2022 at 15:40):

It's a subset of what's on the nLab.

John Baez (Apr 14 2022 at 15:40):

Most interestingly, the powerset of a set is an Bool-enriched category version of the category of presheaves on a set.

John Baez (Apr 14 2022 at 15:41):

The map $X \to PX$ is thus a version of the Yoneda embedding.

John Baez (Apr 14 2022 at 15:41):

Also, a monoid in SupLat is called a "unital quantale".

John Baez (Apr 14 2022 at 15:42):

This is mostly interesting if you have already bumped into quantales, and want a high-level view of them.

Reid Barton (Apr 14 2022 at 15:42):

Matteo Capucci (he/him) said:

are you aware of people studying these structures with sublinear maps instead of linear ones? by that I mean maps such that $f(\bigvee_i a_i) \leq \bigvee_i f(a_i)$

If $f$ is also meant to be order-preserving then this implies "linearity".

John Baez (Apr 14 2022 at 15:43):

You mean that inequality plus monotonicity implies the corresponding equality?

Reid Barton (Apr 14 2022 at 15:44):

Yes

Reid Barton (Apr 14 2022 at 15:44):

As there is always a map colim F -> F(colim)

Reid Barton (Apr 14 2022 at 15:45):

i.e., $f(a_j) \le f(\bigvee_i a_i)$ for each $j$ automatically.

John Baez (Apr 14 2022 at 15:53):

Thanks. Duh. I knew that principle: every functor "laxly" preserves colimits.

John Baez (Apr 14 2022 at 15:54):

This explains why nobody talks about Matteo's "sublinear" maps between suplattices. :upside_down: