Abstract
In recent years, philosophers of science have explored categorical equivalence as a promising criterion for when two (physical) theories are equivalent. On the one hand, philosophers have presented several examples of theories whose relationships seem to be clarified using these categorical methods. On the other hand, philosophers and logicians have studied the relationships, particularly in the first order case, between categorical equivalence and other notions of equivalence of theories, including definitional equivalence and generalized definitional (aka Morita) equivalence. In this article, I will express some skepticism about categorical equivalence as a criterion of physical equivalence, both on technical grounds and conceptual ones. I will argue that “category structure” (alone) likely does not capture the structure of a theory, and discuss some recent work in light of this claim.
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Notes
- 1.
This proposal was inspired and strongly influenced by Halvorson [28], who had previously argued that category theory would likely be useful for representing theories, particularly for settling questions of (in)equivalence, but who did not make a concrete proposal for how to do this in practice. Since then, there has been a large literature on this subject in philosophy of physics. (Of course, mathematicians have long used category theory to explore related issues!) For a detailed review of the literature on theoretical equivalence in physics, see Weatherall [59, 60].
- 2.
- 3.
In many physical theories, there are natural candidates for the “models of the theory”; the arrows require more careful attention, though in practice, ambiguities concerning what one should take as arrows of the category of models reflect real interpretational differences. For instance, the models of general relativity are relativistic spacetimes, which are four-dimensional manifolds, satisfying certain topological conditions, endowed with a smooth metric of Lorentz signature. One natural choice of arrows for this category are the isometries, which are diffeomorphisms between spacetime manifolds that preserve the metric.
- 4.
One might be skeptical that there is any single criterion—that is, any necessary or mutually sufficient conditions—for equivalence of theories; instead, one might think that there are many criteria out there that each capture different senses of equivalence, and that the most fruitful approach is to develop a bestiary of such criteria and to ask, in particular cases, in which senses theories are and are not equivalent. From this perspective, the worry is that categorical equivalence, as described in the literature, may not adequately capture the sense of equivalence that it is intended to capture—or, perhaps, any interesting sense at all.
- 5.
Note, here, that the ambiguity alluded to above matters: at issue is whether the categorical equivalence of two categories of models adequately captures the required relationship between the mathematical structures invoked in physical theory; if not, then categorical equivalence, as a criterion of theoretical equivalence that includes an additional condition regarding empirical significance, arguably fails.
- 6.
A third direction—pursued, for instance, by Nguyen [41] and by Butterfield [18]—has been more critical. These authors argue that there are further considerations, related to the interpretation of theories that are necessary to establish equivalence. (See also Coffey [19] and Sklar [50].) In a sense, this is uncontroversial: after all, categorical equivalence, as described above, requires theories to be empirically equivalent, in a way that is compatible with their categorical equivalence, and empirical equivalence depends on interpretation. But there is still a matter of controversy over whether there are further senses in which interpretation should matter. I set this cluster of issues aside in what follows, as my goal is to raise a different set of concerns about categorical equivalence.
- 7.
- 8.
Specifically, they conjecture that categorical equivalence implies Morita equivalence for theories with finite signatures.
- 9.
See Weatherall [58] for a more detailed discussion of the ways in which categorical equivalence has been applied to better understand classical field theories.
- 10.
- 11.
- 12.
There were hints in the philosophical literature that something like this should hold. In the first instance, Earman [22, 23] suggested that Einstein algebras might resolve the apparent indeterminism in general relativity revealed by the hole argument [24]. But as Rynasiewicz [49] argued, in related cases in algebraic topology the sorts of maps that are used to run the hole argument also arise between algebras of functions, suggesting that it is hopeless to expect the hole argument to go away if one moves to Einstein algebras. (See also [6].) This relationship between the maps between Einstein algebras and relativistic spacetimes is at the heart of the equivalence result. For a different view of the hole argument, compatible with the attitudes adopted in the present paper, see Weatherall [56].
- 13.
I do not take the list here to be exhaustive! For instance, I do not (otherwise) raise any concerns about using groupoids—categories where every arrow is invertible—as opposed to categories with richer arrow structure, to represent theories. I also note that Barth [10] has also criticized categorical equivalence—or rather, what he has called the solution-category approach—along lines that are related to some of the points below, though I do not reproduce all of his concerns here and I think some of my concerns are different. Still, my thinking has certainly been influenced by his.
- 14.
How different really is this situation from the structure one considers in general relativity? It would take us too far afield to evaluate this question in detail, though Weatherall [55] argues that the structures are much more similar in character than they initially appear.
- 15.
On the other hand, once one has used categories to build models of a theory, it is often natural to construct categories of such models. But when one does so for theories whose constructions are “native” to category theory one tends to get much richer structures.
- 16.
An anonymous referee notes that attempts to make “empirical equivalence” precise exist in the literature—as, for instance, in van Fraassen [51]. Fair enough. But as van Fraassen himself would surely acknowledge, especially given his more recent work, how our scientific theories come to represent worldly situations involves a great deal of interpretation, intention, practice, and context [52]. One can develop formal tools for expressing these intentions, etc., at which point precise standards of equivalence may be employed. But this hardly yields a formal test of empirical equivalence analogous to the way in which definitional equivalence, say, is a formal test of equivalence of first order theories. The worry, which I articulate presently, is that the vast bulk of the work of establishing equivalence occurs when one tries to take the messy details of how a scientific theory is used to represent the world and distill them into the formal apparatus—and not when one checks the properties of a certain functor.
- 17.
- 18.
The significance of this example for our understanding of classical electromagnetic theory is discussed by Belot [11].
- 19.
Note that this concern is a bit different from that raised by the authors noted in footnote 6, because it is not specifically about formal representations of a theory. Rather, the concern is that theories are not to be pinned down at all, whether formally or not. Indeed, one might take the remarks here to be a kind of skepticism about any account of the “structure of theories” or even the “semantics of theories”. For an interesting discussion of these issues, see [25], who argue that physical theories should be associated with a network of different formal axiomatic theories, rather than a single theory. Thank you to Hajnal Andréka and István Németi for drawing my attention to this work.
- 20.
The term “gadget”, here, comes from John Baez, who introduced it (in conversation) because “object” has a technical meaning in category theory and “structure” seems to carry too much baggage; basically, a mathematical “gadget” is any sort of thing that mathematicians define and study. I adopt and discuss this ideology in Weatherall [56]; I think it is particularly clearly expressed in Burgess [17], though he does not go on to emphasize the relationship to “structure-preserving maps” that I introduce presently. Although the basic point is very closely related to famous arguments by Benacerraf [12], it is important to distinguish the ideology about mathematical gadgets that I am adopting from what is sometimes called “mathematical structuralism”. Structuralism, as I understand it, is a view about the ontology of mathematical objects. I do not mean to make any particular claims about ontology here, and I take the claims I make here to be compatible with many different philosophies of mathematics.
- 21.
Why not the categorical isomorphisms—that is, the invertible functors? One certainly could take these to be the relevant standard of equivalence, but this is rarely done in category theory. One reason is that categorical isomorphisms preserve “too much”, in the sense that the intended structure of a category does not include a determinate number of objects (only a determinate number of non-isomorphic objects). Another, related, reason is that it is often natural to think of functors as themselves having structure-preserving maps, known as natural isomorphisms, between them. Categorical equivalences are functors that have “almost” inverses—that is, inverses up to natural isomorphism. In this sense, categorical equivalence might be understood as isomorphism up to isomorphism.
- 22.
These remarks are not meant to be surprising, particularly to anyone who is accustomed to working with categories. In fact, category theorists often emphasize that it is the arrows that do all the work, and there are even single-sort axiomatizations of category theory in which only arrows appear.
- 23.
I am not worrying, here, about size considerations. But for someone who is worried about whether the category of groups is well-defined, we can consider all groups of cardinality less than some inaccessible cardinal, \(\kappa \).
- 24.
Hudetz makes this condition precise, but for present purposes an informal description suffices.
- 25.
There is a connection here to classic work by Makkai [38], on a duality between syntax and semantic in first order theories; see also [3, 36]. At least in some cases, one can reconstruct a theory, uniquely up to a suitable notion of equivalence, from its category of models. But the relationship between such results and the categories encountered in the philosophy of physics literature is not clear.
- 26.
In fact, the category of sets can be defined “directly”, as the category satisfying certain axioms, as opposed to by beginning with a prior definition of sets and functions. See Lawvere [33].
- 27.
I call this the ‘G’ property because it was proposed by Bob Geroch during a conversation with Hans Halvorson at a meeting in Pittsburgh in April, 2013. Essentially the same condition was also discussed, apparently independently, by Dewar and Eva [21], though their motivation for considering the condition was different: they suggested that violating it would indicate that a theory has “excess structure”. I do not engage further with their proposal here. It is also considered by mathematicians, under a different name: the ‘G’ property is precisely the condition that the automorphism class group of a category be trivial (cf. [26], Problem 1.B). (I am grateful to Hans Halvorson for bringing Freyd’s work to my attention back in 2013.).
- 28.
Here we make use of the fact that every full, faithful, and essentially surjective functor as an almost-inverse, i.e., is an equivalence of categories. For definitions of full, faithful, and essentially surjective, see Leinster [35].
- 29.
Below I discuss concerns about whether this category has the “right” arrows. But one might also object that it is not clear what objects the category should have, on the grounds that it is not clear if we should limit attention to spacetimes that are connected, maximal, etc. [39]. I set this issue aside here, but note that the two questions may interact in interesting ways.
- 30.
One might worry that this argument depends essentially on GR being a groupoid—i.e., that it has no non-invertible maps. This means that there is no information about which spacetimes might be, for instance, embeddable in one another. Perhaps by adding more arrows, such as isometric embeddings, one could produce a category that has the ‘G’ property. But I doubt it, because one could then consider spacetimes that were, roughly speaking, asymmetric at all scales. This would generate more complicated structures, but still no automorphisms. I suspect that a similar functor could be generated under these circumstances, though I do not claim that this is a proof.
- 31.
This sort of situation arises often when one has highly structured (or highly asymmetric) objects in a category. There are very few maps available that preserve all of the relevant structure.
- 32.
There is another response available to the Bau-Di example, which is to say: in fact, the internal structure of the objects in these categories is not so different after all. This response is motivated by the idea that (adopting the terminology of [62]) the objects of these categories are “co-determinate”, in the sense that any two-dimensional vector space, with ordered basis, is determined “freely” by that basis, i.e., by an ordered pair; and every two-dimensional vector space with ordered basis determines, in particular, an ordered pair (consisting of the basis elements). So, perhaps, once we choose an ordered basis for a vector space, the entire vector space structure should be seen as “determined by” (or, roughly, definable from) that basis. From this perspective, that property ‘G’ holds of these categories is not a problem for property ‘G’. But alas, this response is too fast. The reason is that these categories are too rigid, and one can easily come up with other categories, equivalent to both, for which this “co-determination” relationship does not seem to hold. Consider, for instance, the category whose objects are sets with one elements and whose arrows are functions preserving that element. (Or: the category \(\mathbf {1}\), with a single object and a single arrow.) This category is equivalent to both Bau and Di! And yet it is hard to see how a set with one element could determine a two-dimensional vector space in any interesting sense (since the free vector space on one element is one dimensional). I am grateful to Thomas Barrett for pushing me on this point.
- 33.
Rings are also generally taken to have a multiplicative identity.
- 34.
In fact, although many examples of rings that are not isomorphic to their opposites are known, they are not exactly trivial to state. See Jacobson [31, Sect. 2.8] or Lam [32, Sect. 1]. I believe it was Hans Halvorson who first brought this example to my attention. Observe that groups, too, have opposites, defined in a similar way, but in general groups are isomorphic to their opposites, where the isomorphism takes group elements to their inverses.
- 35.
Some readers might be tempted, in light of this, to revise the notion of “isomorphism” associated with rings, so that all rings that are suitably “the same” are isomorphic. But this strikes me as a disastrous proposal. A ring homomorphism that could not distinguish left multiplication from right multiplication would wash out the structure of non-commutative rings!.
- 36.
That said: there remains an interesting question raised by this first approach, which is: if categories C and D are equivalent, with \(F:C\rightarrow D\) realizing that equivalence, then what, if any, structural relationship holds between objects c in C and F(c) in D? I am grateful to Thomas Barrett for emphasizing this point, which I completely endorse.
- 37.
In a recent talk, Thomas Barrett described a similar, but distinct, program, on which it is “well-behaved” functors that realize equivalences. I will not attempt to reconstruct (or scoop) his ideas here, but note only that it is another proposal that falls into this second category—or, perhaps, somewhere in between the first and second approaches, depending on how it is spelled out.
- 38.
One might be tempted by a possible resonance with the previous proposal, and try to modify the ‘G’ property, using the notion of definable functor, as follows: a category satisfies the ‘H’ property if every full, faithful, and essentially surjective definable functor \(F:C\rightarrow C\) is naturally isomorphic to \(1_C\). But this proposal is unlikely to work, since the functor \(Op:\mathbf{Ring} \rightarrow \mathbf{Ring} \) apparently counts as definable.
- 39.
I do not mean to criticize Hudetz here. He is explicit about the assumptions he is making when he defines definable functors, and makes clear why in the cases of interest, definable functors are well-defined. But I read his assumptions as sufficient conditions for making sense of definable functors, which is weaker than a theory of the sorts of structures that definable functors relate.
- 40.
Observe that for some categories—toposes—there is an “internal language” associated with the category. But this notion of internal language is not the same as the notion of “language of objects of a category” associated with definable functors.
- 41.
Hudetz has recently made some progress in this direction: see, for instance, [30].
- 42.
Compare this perspective to classic arguments due to Sklar [50], recently amplified by, for instance, Coffey [19], Nguyen [41], and Butterfield [18], to the effect that a “purely formal” criterion of equivalence could never be adequate. (Recall note 6.) Here it is a semantic relationship—that is, a relationship between the interpreted, applied theories—that is ultimately the starting point, and then the formal methods are a guide to evaluating such relationships. Consider, too, a connection to [44], which argues that empirically equivalent theories are more or less certain to be equivalent in some stronger sense, or else to differ in ways that make more clearly preferable; from the present perspective, categorical (in)equivalence is a way of establishing how much, if anything, is missing from some empirical equivalence.
- 43.
- 44.
Defenders of these approaches might well balk at this point. Do they really need to be committed to the view that categories of models are representations of theories “once and for all”? But if the goal is to determine if two theories are equivalent as theories, then presumably that condition needs to capture everything salient about the pairs of theories. If the goal is to offer a weaker notion of equivalence then much more needs to be said about what features the standard establishes equivalence with regards to. This is what empirical equivalence offered: equivalence with regard to the predictions made by two theories, without implying “full” equivalence. One possible line, here, would be to say that categorical equivalence and its various elaborations are attempting to capture a kind of “structural equivalence”, though I think more needs to be said about just what that means.
- 45.
In particular, one might worry that the third approach brushes too many foundational questions aside, and that although it is pragmatically attractive, we should still be interested in the answers to those questions—which the first two approaches may yet yield.
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Acknowledgements
I am grateful to Thomas Barrett, Lukas Barth, Neil Dewar, Ben Eva, Ben Feintzeig, Hans Halvorson, Laurenz Hudetz, David Malament, Toby Meadows, and Sarita Rosenstock for many helpful conversations in connection with this material, and to Hajnal Andréka, Thomas Barrett, István Németi, and an anonymous referee for detailed comments on a previous draft. A version of the paper was presented at a workshop at the Munich Center for Mathematical Philosophy; I am grateful to the organizers and the audience for their valuable feedback.
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Weatherall, J.O. (2021). Why Not Categorical Equivalence?. In: Madarász, J., Székely, G. (eds) Hajnal Andréka and István Németi on Unity of Science. Outstanding Contributions to Logic, vol 19. Springer, Cham. https://doi.org/10.1007/978-3-030-64187-0_18
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