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Stream: deprecated: id my structure

Topic: Noncommutative stable ordered valuation algebra?

Naso (May 08 2023 at 01:28):

I've been investigating some instances of a certain algebraic structure whose definition I'm trying to pin down.

I am hoping to get some feedback on this definition from people with better intuitions and also a suggestion on what to call it.
(Previously someone suggested it may be related to monoidal Grothendieck fibrations, but I haven't been able to figure that out.)

This is essentially a variation of something called a (noncommutative, labelled, ordered, stable) valuation algebra, but I've used categorical notions to try reduce the number of axioms.

Let $(X, \mathcal{T})$ be a topological space.

Definition:
A [Name?] on $(X, \mathcal{T})$ is a triple $(\Phi, \otimes, \epsilon)$ where:

i. $\Phi : \mathcal{T}^{op} \to \mathcal{Pos}$ is a poset-valued presheaf,
ii. $\otimes : \int \Phi \times \int \Phi \to \int \Phi$ is a binary operation on the covariant Grothendieck poset of $\Phi$,
iii. $\epsilon : 1 \to \Phi$ is a global element of $\Phi$,

such that the following axioms are satisfied for all $A, B \in \mathcal{T}$:

i. (Ordered semigroup.) The operation $\otimes$ is associative, and monotone: for all $a_1,a_2,b_1,b_2$ with $a_1 \preceq a_2$ and $b_1 \preceq b_2$, we have $a_1 \otimes b_1 \preceq a_2 \otimes b_2$.

ii. (Labelling.) For all $a \in \Phi A, b \in \Phi B$, we have $a \otimes b \in \Phi (A \cup B)$.

iii. (Neutrality.) For all $a \in \Phi A$, we have $\epsilon_A \otimes a = a \otimes \epsilon_A = a$, and also, $\epsilon_A \otimes \epsilon_B = \epsilon_{A \cup B}$.

iv. (Combination.) For all $a \in \Phi A, b \in \Phi B$, we have $(a \otimes b) |_A = a \otimes b |_{A \cap B}$ and $(a \otimes b) |_B = a |_{A \cap B} \otimes B$.

Note that the covariant Grothendieck ordering ($\leq_G$) is defined as follows: for $a \in \Phi A$ and $b \in \Phi B$,

$a \leq_G b \iff B \subseteq A \land a |_B \leq_{\Phi B} b$.

(This ordering has semantic significance for our application domain.)

In addition, we usually want to consider valuation algebras such that the all the projection maps have a right adjoint that we call extension but I wasn't sure whether to make this part of the definition or not?

After that, we define a (lax) morphism of VA's $f : (\Phi,\otimes,\epsilon) \to (\Phi',\otimes',\epsilon')$ to simply be a monotone function $f : \int \Phi \to \int \Phi'$ that weakly preserves the neutral elements and product, i.e. $\epsilon_A' \leq f ( \epsilon_A )$ and $f(a) \otimes' f(b) \leq f(a \otimes b)$.

(Maybe we should also ask that $f : \Phi \to \Phi'$ is a lax natural transformation?)

Richard Samuelson (May 08 2023 at 05:40):

There is some relationship between hypergraph PROPs and stable valuation algebras.

In particular, every hypergraph PROP $\mathcal{H}$ determines a stable valuation algebra $(\Psi, 2^D)$. Valuations are pairs $(s, f) \in \Psi$, where $s: 0 \to n$ is a state in $\mathcal{H}$, and $f: [n] \to D$ is a function assigning labels to the numbers $\{1, ..., n\}$.