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Stream: learning: questions

Topic: Nucleus of a Profunctor with value in a Boolean algebra

Peva Blanchard (Feb 15 2024 at 22:38):

Hi,

I stumbled upon this article on "concepts and profunctor nuclei", with a focus on the enriched aspect of it. And also Tai-Danae Bradley's thesis.

I am considering the case of the $hom$ profunctor $L^{op} \times L \rightarrow V$ where $V$ is the enriching category. Most of the examples mentioned in the articles above involve an enriching category $V$ that is either $Set$, or $2 = \{0,1\}$ (preorders), or the unit interval $[0,1]$ (fuzzy stuff).

In this analogy of considering $L^{op} \times L \rightarrow V$ as a matrix whose entries are "truth-values", I was wondering if anyone has studied the case where $V$ is the boolean algebra $\mathcal{P}N$, power set of some finite set $N$. In particular, the nucleus of $hom$.

Peva Blanchard (Feb 15 2024 at 22:39):

More precisely, the adjoint pairs $L^* \dashv L_*$ are given by the following. For any presheaf $\phi : L^{op} \rightarrow \mathcal{P}N$, and co-presheaf $\psi : L \rightarrow \mathcal{P}N$:

$$\begin{align*} L^* \phi &= \bigcap_c ( L(c, \_) \cup \neg \phi c ) \\ L_* \psi &= \bigcap_d ( L(\_, d) \cup \neg \psi d ) \end{align*}$$

Thus the nucleus is described by the pairs $(\phi, \psi)$ such that

$$\begin{align*} \psi d &= \bigcap_c ( L(c, d) \cup \neg \phi c ) \\ \phi c &= \bigcap_d ( L(c, d) \cup \neg \psi d ) \end{align*}$$

I don't know how to approach these equations. Maybe I could use characteristic functions and get a system of polynomial equations, and so on. But before going that road, I was wondering if someone had already done the work.

Peva Blanchard (Feb 15 2024 at 22:40):

ps: My motivation behind is the following. In social choice theory, Arrow's theorem states that, given some "nice-to-assume-to-have" properties, the only way to aggregate preferences of (a finite set of) individuals is to select the preference of a single individual, aka the dictator. This comes down to the fact that, when $N$ is finite, the only boolean algebra homomorphisms $2^N \rightarrow 2$ are the projections.

More formally, each individual $i$ defines a total order $\leq_i$ over a common set of objects. Arrow studies the solutions to aggregating those orders into one order. An order being a category enriched over $2$, a way to relax the problem is to consider a larger boolean algebra. The "free-est" aggregation should look like that: for each pair of objects $a$ and $b$, we define $hom(a,b)$ as the subset of individuals $i$ who prefers $b$ over $a$, i.e., $a \leq_i b$. In other words, we are enriching over $\mathcal{P}N$.