A pedantic question about defining new structures in a path-independent way.

Question

Sometimes there are multiple equivalent ways of defining the same structure; for example, topological spaces are determined by their open sets, but also by their closed sets. I'm looking for a way of defining new structures that avoids committing to any one particular definition.

Let me elaborate a bit.

Sometimes "non-commitment" occurs without needing special attention. For instance, suppose we've already defined the terms antisymmetric and irreflexive. Now consider two cases. We could go on to define asymmetric as "antisymmetric+irreflexive." Alternatively we could go on to define asymmetric as "Not both $x<y$ and $y<x$." But whichever approach we choose, the same endpoint is attained. For example, in both approaches it holds that:

1. A relation is asymmetric iff it is both antisymmetric and irreflexive, and
2. A relation $<$ is asymmetric iff it never holds that both $x<y$ and $y<x$.

More generally, the sentences that are "true" (also known as theorems) are identical between the different approaches. So we eventually arrive at the same place, albeit via a different routes. If we think of each approach as a path, then the end result is "path-independent."

Unfortunately, definitions of structures (e.g. groups, ordered sets, topological spaces) tend to be path-dependent.

For instance, we could define a topological space as an ordered pair $(X,\mathcal{O})$, where $\mathcal{O}$ is viewed as the collection of open sets. On the other hand, we could define a topological space as an ordered pair $(X,\mathcal{K})$, where $\mathcal{K}$ is viewed as the collection of closed sets.

Notice that, unlike in the previous example, there exist sentences whose truthvalues are dependent on the particular approach taken. For instance, the sentence, "For all topological spaces $(X,\mathcal{O})$, it holds that $\mathcal{O}$ is closed with respect to arbitrary unions," is true in the first approach but false in the second. In some sense, we've "committed" to a particular definition of a topological space; and, in doing so, made a decision that actually effects the end result.

This is bad, because we tend to write/talk about math in approach-independent ways. For instance, we're more likely to write, "Let $X$ denote a topological space and $\mathcal{K}$ denote the set of all closed sets of $X$," as compared to, "Let $(X,\mathcal{K})$ denote a topological space." Even though the latter is more terse, nonetheless the former is clearly better, because its meaning is independent of the details of how a topological space was defined.

So here's what I'm looking for.

Ideally, a full solution would consist of a better way of defining structures, such that we can avoid "committing" to any particular definition of a topological space, ordered set, group etc. We would continue making statements like, "A topological space is determined by its open sets," but would never make a statement like, "A topological space is an ordered pair $(X,\mathcal{O})$ such that..." Thus, the path wherein you define the term "topological space" via open sets would eventually meet up with the path wherein you define the term "topological space" via closed sets; and all sentences that are true in the open-set approach would be true in the closed-set approach, and vice versa.

Less ambitiously, a partial solution would show that, "It doesn't matter which way you do it, the resulting theory is essentially the same." I'll happily accept a category-theoretic solution to the problem, as long as I'm still able to do math in the classical way. For instance, concepts like "the underlying set of a topological space" and "the set of all clopen sets" need to continue making sense. In particular, defining things up to isomorphism (i.e. renaming of the underlying set) is not an option.

Answer 1

Notice that what you are really asking for is comparing two classes of structures and not two particular instances of these structures (judging from the example you gave regarding different definitions of topological spaces).

If the two structures are given by theories $T_1,T_2$ over the same language then you can say that the theories are essentially the same if $T_i$ implies each sentence in $T_j$, for $i,j\in \{1,2\}$. If the two theories are over different languages, say $L_1, L_2$ respectively, then it's a bit more difficult. You might try to first find a common refinement of the two languages and proceed as above. For instance, if you axiomatize groups using the language $\{\cdot, e\}$ by letting $T_1$ be any usual axiomatization and by letting $T_2$ be that usual axiomatization augmented with infinitely many new sentences describing generalized associativity then you can quite easily prove the two theories are essentially the same.

You will run into a bit more difficulty if you try to compare heaps and groups and you will run into a hell of a lot of difficulty if you try to compare the theory of the homotopy of topological spaces and the theory of simplicial sets. These are all cases where the two structures are essentially the same but where working with a theory describing them is quite hopeless (especially the second example).

You seem not to want a categorical solution to the problem, though it is the most natural one. Quite often any definition of structure comes with a definition of morphism. Then collecting these together almost always results in a category (or a weak $\infty $-category). Now, you can say that the two structures are essentially the same if the categories are essentially the same. The latter can be taken to mean anything you want, depending on how much flexibility you want to allow. For instance, requiring the resulting categories to be isomorphic means that the structures are essentially the same in a much stronger sense then requiring the categories to just be equivalent.

In mathematics the relevant notion of categories being essentially the same is categorical equivalence. Given the (huge) class of all categories you can define an equivalence relation on categories by categorical equivalence. Now, you can consider each equivalence class to be the purest form of the definition of a mathematical concept. For instance, the equivalence class of the category $Grp$ of groups (defined in the usual way) includes all equivalent categories, no matter what axiomatization is used to give rise to them. To use the words of your own question, you can think of the particular representatives of an equivalence class to be the different paths that lead to the pure abstract concept, and the equivalence class as being the actual concept being defined. Then independence of paths holds and any particular axiomatization is just a way to describe (indirectly) an equivalence class of categories. A direct, constructive characterization of the equivalence of the resulting categories just in terms of the defining theories is quite hopeless.