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Stream: learning: questions

Topic: Displayed vs indexed fuzzy relations

Ralph Sarkis (Nov 15 2023 at 14:47):

I am only starting to recognize the equivalence between displayed and indexed stuff shows up in many places, and I am a beginner in the yoga of the Grothendieck construction. I am writing about an equivalence between relations valued in a complete lattice and families of binary relations indexed by that lattice, and I was wondering what is the quickest way to see it as Grothendieck construction. Let me give the gist very briefly, with some more details here.

Ralph Sarkis (Nov 15 2023 at 14:47):

Let $L$ be a complete lattice.

Definition: An $L$-relation on a set $X$ is a function $d: X\times X \to L$ with no additional constraint.
Definition: An $L$-structure on a set $X$ is a continuous functor $R: L \to \mathcal{P}(X\times X)$, we see both $L$ and the powerset $\mathcal{P}(X\times X)$ as posetal categories (the order of the latter is the inclusion).
Proposition: There is a correspondence between $L$-relations and $L$-structures on $X$.

Ralph Sarkis (Nov 15 2023 at 14:47):

Definition: The category $L\mathbf{Rel}_s$ has sets equipped with an $L$-relation as objects, and a morphism $f: (X,d) \to (Y,\Delta)$ is a function $f: X \to Y$ satisfying $d(x,x') = \Delta(f(x),f(x'))$.
Defintion: The category $L\mathbf{Str}_s$ has sets equipped with an $L$-structure as objects, and a morphism $f: (X,R) \to (Y, S)$ is a function $f: X \to Y$ satisfying $f(R_e) = S_e$ for all $e \in L$. You can probably see it as some comma category $L \downarrow \mathcal{P}(-\times -)$.
Proposition: The categories $L\mathbf{Rel}_s$ and $L\mathbf{Str}_s$ are isomorphic.

Ralph Sarkis (Nov 15 2023 at 14:48):

Definition: The category $L\mathbf{Rel}$ has sets equipped with an $L$-relation as objects, and a morphism $f: (X,d) \to (Y,\Delta)$ is a function $f: X \to Y$ satisfying $d(x,x') \geq \Delta(f(x),f(x'))$.
Defintion: The category $L\mathbf{Str}$ has sets equipped with an $L$-structure as objects, and a morphism $f: (X,R) \to (Y, S)$ is a function $f: X \to Y$ satisfying $f(R_e) \subseteq S_e$ for all $e \in L$. You can probably see it as some lax or colax comma category $L \downarrow \mathcal{P}(-\times -)$.
Proposition: The categories $L\mathbf{Rel}$ and $L\mathbf{Str}$ are isomorphic.

Ralph Sarkis (Nov 15 2023 at 14:48):

To reiterate my question: Are these propositions simple corollaries of a known result ?

Matteo Capucci (he/him) (Nov 15 2023 at 18:13):

How are an L-relation and an L-structure equivalent?

Matteo Capucci (he/him) (Nov 15 2023 at 18:15):

They seem different from what you expect a Grothendieck construction to identify. A Grothendieck construction would send $X \times X \to L$ to $E \to X$, where the latter has to be interpreted suitably (i.e. as an L-functor of sorts)

Matteo Capucci (he/him) (Nov 15 2023 at 18:17):

Your correspondence reminds me more of the equivalence between H-sets and H-sheaves for a Heyting algebra H

Ralph Sarkis (Nov 15 2023 at 18:21):

Maybe I got the wrong feeling about it being a Grothendieck construction then. I actually have an isomorphism between $L\mathbf{Rel}$ and $L\mathbf{Str}$ proven in the file above. On objects we have:
image.png
image.png

Matteo Capucci (he/him) (Nov 15 2023 at 18:44):

I see

Matteo Capucci (he/him) (Nov 15 2023 at 18:44):

So this is not an H-sets/H-sheaves correspondence

Matteo Capucci (he/him) (Nov 15 2023 at 18:46):

I'm leaning towards a combination of (1) a general correspondence between functions $A \to PB$ and $B \to PA$ and (2) the fact that a complete lattice has a $P$-algebra structure

Matteo Capucci (he/him) (Nov 15 2023 at 18:49):

Indeed, given a function $L \to P(X \times X)$ one has a corresponding 'transpose' $X \times X \to PL$. Postcomposing the latter with $\inf : PL \to L$ yields the L-relation in the second picture.

Matteo Capucci (he/him) (Nov 15 2023 at 18:52):

I didn't use the fact the L-structure is continuous though

Matteo Capucci (he/him) (Nov 15 2023 at 18:54):

Viceversa, transposing an L-relation gives you $L \to P(X \times X)$ but this sends $\epsilon$ to the subset of $X \times X$ of those pairs related by precisely $\epsilon$. You probably pass to the downsets to get the continuity..?

Ralph Sarkis (Nov 15 2023 at 20:19):

Hmm, that is underwhelming...