Is there a generic definition of "strongly indistinguishable"?

Question

This is related to a previous question.

Consider the quasiordered set $Q = \{\bot, q,q', \top\}$ with $q \lesssim q'$ and $q \lesssim q',$ such that $\bot$ is the unique least element and $\top$ the unique greatest element. In this structure, there is clearly a strong sense in which $q$ and $q'$ are indistinguishable. For example, we have that:

$$(\forall x \in Q) \;\; x \lesssim q \iff x \lesssim q'$$

Now consider, on the other hand, the diamond poset $D = \{\bot,d,d',\top\}.$ Then okay, $d$ and $d'$ are "weakly" indistinguishable, for example no first-order formula in the language of posets can distinguish them, since after all, we can find an poset automorphism $\alpha : D \rightarrow D$ that permutes $d$ and $d'$. Which means, that for example, we can uniformly substitute $d$ with $d'$ without changing anything. For example, $d \leq d$, so therefore $d' \leq d'$. However, notice that we cannot just substitute $d$ with $d'$ willy-nilly, for example just because $d \leq d$ is true, does not mean $d \leq d'$ is true. So we might say that $d$ and $d'$ fail to be "strongly indistinguishable."

Along a similar vein, let $\alpha : D \rightarrow D$ denote the unique non-trivial such automorphism (the one that permutes $d$ and $d'$). Then although it holds universally that $$x \leq y \iff \alpha(x) \leq \alpha(y)$$

nonetheless it does not hold universally that

$$x \leq y \iff x \leq \alpha(y).$$

So in some sense, $\alpha$ is not a "strong automorphism" of $D$.

What I want to know is, what is the "correct" definition of "strongly indistinguishable" and/or "strong automorphism"?

This seems to be a very general phenomenon, and we can often get the "correct" definitions in an ad hoc manner. This allows us to take a quotient that identifies strongly indistinguishable elements. For example, we can define:

1. In a quasiordered set... two points $x$ and $y$ are strongly indistinguishable iff $x \lesssim y$ and $y \lesssim x.$

2. In a pseudometric space... iff $d(x,y)=0.$

3. In a topological space... iff their neighbourhood filters are equal.

4. In a measure space... two measurable sets $A$ and $B$ are strongly indistinguishable iff $\mu(A\, \Delta\, B) = 0.$

So, I am wondering, is there is a generic definition of "strongly indistinguishable" that yields the correct specific definition when applied to any given structure? We would like to be able to take a quotient that identifies indistinguishable elements while preserving basically everything that is interesting about the structure of interest.

Edit. I'm also interested to know whether there is an abstract category-theoretic definition of "strong automorphism."

Answer 1

The only idea that springs to mind that is the same for all four is category-theoretic in nature.

If $Q$ is a preorder (I'm not familiar with the name quasi-order, but I think that's what you mean), then the poset $TQ$ you get by identifying equivalent elements, along with the quotient map $Q \to TQ$, has the following universal property: if $P$ is any poset, and $Q \to P$ is an order homomorphism, then there is a unique way to factor it as $Q \to TQ \to P$.

Put differently, there is a bijection between the set of homomorphisms $TQ \to P$ and the set of homomorphisms $Q \to P$, and the bijection is given by composing with the map $Q \to TQ$.

The above property uniquely determines (up to isomorphism) the poset $TQ$ and the map $Q \to TQ$. The only 'choice' we made was that we decided to single out the class of posets from the class of all preorders.

The general notion is that of a reflective subcategory: $\mathscr{C}$ is a full subcategory of $\mathscr{D}$, and the inclusion functor $i : \mathscr{C} \to \mathscr{D}$ has a left adjoint $T$. We have the property that

$$ \hom(TD, C) \cong \hom(D, C) \quad (= \hom(D, iC))$$

for $D \in \mathscr{D}$ and $C \in \mathscr{C}$. And as in the special case above, we have a natural transformation $\mathbf{1}_{\mathscr{D}} \to i \circ T$. e.g. we get maps $D \to TD$.

Your first example comes from $\mathbf{Poset} \subseteq \mathbf{Preorder}$.

The second example comes from the category of metric spaces as a full subcategory of the category of quasi-metric spaces.

The third example is, I think, the category of $T_0$ spaces as a subcategory of the category of all topological spaces.

The fourth one... well, I'm not versed enough in measure theory to know if there is a standard name for the two categories involved.

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Of course, this is somewhat tangential to the question you actually asked: I've answered "what is the 'quotient' map that identifies 'indistinguishable' elements?" (I put 'quotient' in quotes, since it might not actually be a quotient). But from the category theoretic point of view, it's the maps that matter moreso than the elements anyways, so it's probably the question you should have asked.