equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
Sensible reasoning in mathematics in general, and particularly in higher structures, such as (higher) category theory, homotopy theory, homotopy type theory, etc., should be suitably invariant under the respective notion of equivalence. This is a common issue whenever some structure is presented by other structures, since the notion of equivalence of the presentation can be finer than that of the notion being presented.
For instance, a category can be presented by a simplicial set (the nerve of a corresponding strict category, see there), but isomorphism of simplicial sets is much finer than equivalences of their corresponding categories. It is generally a mistake to mix up these two levels and, for instance, assign to a category properties that are shared only by some of the simplicial sets representing it, say, by distinguishing between isomorphic objects. This breaks the equivalence invariance. (However, see below.)
The ideas here generalize in many directions. For example, not only properties, but also constructions involving categories and functors, can fail to preserve equivalences.
Michael Makkai proposed the Principle of Isomorphism, “all grammatically correct properties of objects of a fixed category are to be invariant under isomorphism”, in his paper
“Towards a categorical foundation of mathematics”, in Johann A. Makowsky, Elena V. Ravve, eds., Logic Colloquium ‘95: Proceedings of the Annual European Summer Meeting of the Association of Symbolic Logic, held in Haifa, Israel, August 9-18, 1995 (Berlin: Springer-Verlag, 1998), 153-190.
Peter Aczel mentioned a similar concept, the Structure Identity Principle, “isomorphic structures have the same structural properties” in Oberwolfach report 52/2011. This seems a priori a little weaker to me, but if we demand that objects should be seen as only having structural properties (as opposed to the category of ZF-sets), then we look like we get back the Principle of Isomorphism.
A very precise way of stating this idea is encapsulated in Vladimir Voevodsky‘s univalence axiom, which is a fundamental part of homotopy type theory as a foundation for mathematics. By identifying equivalences/isomorphisms with inhabitants of an identity type, it ensures that all properties and structure which can be expressed within the formal type theory are invariant under such (see AhrensNorth18).
Floating around the web (and maybe the $n$Lab) is the idea of half-jokingly referring to a breaking of equivalence invariance as “evil”. This is probably meant as a pedagogical way of amplifying that it is to be avoided.
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Draw an analogy with vector spaces (maybe just finite-dimensional ones?). We can use a basis to study a vector space, but the basis is not part of the vector space. Even if we use a foundation of mathematics in which the basis literally is part of the vector space (which arguably was the case for 18th-century linear algebra, which studied only Cartesian spaces; and is still the case today if we study finite-dimensional vector spaces using a foundational type theory with propositions as types, but probably this is too obscure), anything that we consider to be a property of the vector space should be independent of the basis. We can make this precise by comparing the groupoids $Vect_\sim$ and $Vect_b$.
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In material set theory based on ZFC, any two objects are pure sets and can be compared for equality. Yet this is often mathematical nonsense which depends on precisely how one writes definitions; for example, the ordered pair $(0,0)$ is equal to the natural number $1$ by some standards and equal to $2$ by other standards (while $(1,1) = 2$ by yet other standards), but this makes no difference mathematically. In contrast, structural set theory either implicitly (as in ETCS) or explicitly (as in SEAR or ETCS with elements) considers only equality between elements of a given set (or functions or binary relations between two given sets, etc). This can be built into the logic using dependent type theory (which can serve as a foundation all by itself). The result is that, depending on precisely how one writes definitions, it may or may not be possible to compare two things for equality. For example, the ordered pair $(1, 1)$ is equal to $2$ under triangular encoding, equal to $3$ under binary encoding, and equal to $1$ under Chinese Remainder encoding. But if we take ordered pairs as primitive, it makes no sense to compare $(1, 1)$ to $2$. (Of course, we could do this in material set theory as well; but allowing non-set primitives there would defeat the traditional attraction of the theory.)
Does it make sense, then, to ask whether two arbitrary sets are equal? In material set theory, of course it does; two sets are equal if they have the same elements (by the axiom of extensionality). But structurally, this definition is meaningless. All the same, a structural set theory built on first-order logic with equality automatically gives this question (whether two arbitrary sets are equal) a meaning, as do the most common forms of foundational type theory, but this is again never used in mathematics. ($SEAR$, which is based on first-order logic without equality, does not give this question a meaning and does not suffer for it.) So just as we don't need to ask whether $(0,0) = 2$, we don't need to ask whether $\{(0,0)\} = \{2\}$ as sets. (Note that this is a completely different question, structurally, from whether, say, $\{(0,0)\} = \{2\}$ as subsets of $\mathbb{N} \coprod (\mathbb{N} \times \mathbb{N})$.)
Now let us move from the foundations of set theory to those of category theory. Usually, category theory is founded on set theory, or something very much like it; if the objects of a category don't form a set, then this is only because of size issues, and they still form a proper class. Accordingly, it makes sense to compare them for equality. However, if the category of sets is itself an example of a category, then by the previous paragraph, it does not have to make sense (and is mathematically meaningless) to test the objects of this category for equality. So we get the idea that we cannot compare objects of a given category for equality in general (although this may make sense for particular categories). In other words, a category is not by default a strict category (a category equipped with the extra stuff of an explicit equality predicate on its objects).
We can found mathematics on type theory without identity types, and then it is literally true that (in general) it makes no sense to compare objects of a given type for equality; one has to define a relevant equality predicate when there is one. From the other direction, we can found mathematics on (weak) $\infty$-infinity-groupoid theory, i.e. homotopy theory (as in homotopy type theory), where the automatic notion of equality between object of a category is actually only isomorphism.
However, these are not commonly used foundations. Therefore, category theory is usually written in a language in which it does literally make sense to ask whether two objects of a given category are equal (meaning something stronger than that they are isomorphic), whether two functors between two given categories are equal, etc. If we want these definitions to make sense in the general mathematical situation (where $Set$ is an example of a category, even though comparing two arbitrary sets for equality makes no mathematical sense), then we need to check that our definitions are compatible with the principle of equivalence.
(Strictly speaking, it does not even make sense to ask if two arbitrary sets are isomorphic; going back to our earlier example, $\{(0,0)\}$ is isomorphic to $\{2\}$ as objects of $Set$ because they are both singletons, but they are not isomorphic as objects of the poset of subsets of $\mathbb{N} + \mathbb{N}\times \mathbb{N}$, because neither is a subset of the other.)
In Tractatus Logico-Philosophicus, 2.02331, Wittgenstein writes
Either a thing has properties that nothing else has, in which case we can immediately use a description to distinguish it from the others and refer to it; or, on the other hand, there are several things that have the whole set of their properties in common, in which case it is quite impossible to indicate one of them. For if there is nothing to distinguish a thing, I cannot distinguish it, since otherwise it would be distinguished after all.
Given that a main theme of the Tractatus is on the limits of language, it is tempting to consider this as a description not just for ‘the world’ (as in Tractatus 1.), but for any given language and the objects about which one reasons in that language. This seems to encapsulate Makkai’s motivation alluded to above.
If $X$ is an $\infty$-groupoid, then a property $P$ of objects of $X$ is compatible with equivalence if, whenever $P(a)$ holds for an object $a$ of $X$ and $b$ is equivalent (as an object of $X$) to $a$, then $P(b)$ holds. Alternatively, an operation $f$ from objects of $X$ to objects of (another $\infty$-groupoid) $Y$ is compatible with equivalence if, whenever $a$ and $b$ are equivalent objects of $X$, $f(a)$ and $f(b)$ are equivalent (as objects of $Y$).
(Note that everything is an object of some $\infty$-groupoid. For example, a $2$-morphism in some random $5$-category is an object of a hom-$3$-category, which has an underlying $3$-groupoid, which is a special kind of $\infty$-groupoid. For some purposes, this $2$-morphism may instead need to be seen as an object in a $2$-arrow $5$-category, which has an underlying $\infty$-groupoid as well; exactly which $\infty$-groupoid is relevant depends on the context. Also note that we really do need $\infty$-groupoids here, rather than $\infty$-categories, since dualities do respect equivalences.)
The operation-version of the principle of equivalence gives the property-version if you think of a property as an operation taking values in the $\infty$-groupoid (in fact a $0$-groupoid, or set) of truth values. (The property-version also gives the operation-version, although that is a little more involved.) An operation will obey the principle of equivalence if it is given (as the object-operation) by a functor (’$\infty$-functor’, if you prefer). A property adheres to the principle of equivalence precisely if it's a functor to the $\infty$-groupoid of truth values.
This definition should be the conclusion of a theorem that using certain language (including avoiding equations between objects of a category) makes it impossible to say anything that violates the principle of equivalence. Michael Makkai works on such a language, FOLDS (‘first-order logic with dependent sorts’), which does not include an axiomatic notion of equality at all. This pertains to the mathematical foundations of category theory. To learn more, see his paper First Order Logic with Dependent Sorts, with Applications to Category Theory.
A common misconception in category theory is that strict categories violate the principle of equivalence. However, this is not true; strict categories are the mathematical structures in which the equivalence of objects are precisely equality on objects: important examples of strict categories which do not violate the principle of equivalence include posets and the walking parallel pair. In fact, every strict category is equivalent to a setoid-valued presheaf on the simplex category or the full subcategory of the arrow category of the (2,1)-category of weak categories, consisting of essentially surjective functors whose domain is a setoid.
Instead, what does break the principle of equivalence is to say that certain concrete categories with non-trivial automorphisms such as Set or Grp are strict categories, as the objects of the categories are sets, and equality of objects is equality of sets. In this case, one has to instead say that they are isomorphic and then (usually) impose some coherence relation on the relevant family of isomorphisms. But of course, this is more complicated!
The concept of an identity-on-objects functor appears in category theory as well, particularly when defining Freyd categories and dagger categories. The traditional definition goes as follows:
An identity-on-objects functor $F: A\to B$ between categories $A$ and $B$ is a functor between categories with the same collection of objects, and has as its underlying object function $F_{ob}$ the identity function on the collection of objects.
However, in material set theories or material class theories, “having the same collection of objects” is sometimes defined as $Ob = ob(A) = ob(B)$. This definition involves equality of sets, which violates the principle of equality.
Instead, what category theorists usually mean by two categories having the same objects is actually having a single $(Set \times Set)$-enriched category, where morphisms are in families of sets of pairs $(f,g):A(x,y) \times B(x,y))$ for objects $x$ and $y$, and composition is composition of pairs of morphisms $(h,k)\circ(f,g) = (h \circ f,k \circ g)$ for $f:A(x,y)$, $g:B(x,y)$, $h:A(y,z)$, and $k:B(y,z)$.
Another problem occurs in set theories with certain concrete categories like Rel, Hilb, and the permutation category $\mathbb{P}$: the objects of the categories are sets, and so saying that the underlying function $F_ob$ on the objects is the identity function violates the principle of equality, because by definition $F_ob(x) = x$ for all objects $x$, and because objects are sets, one is talking about equality of sets.
Instead, when category theorists are talking about two categories with the same collection of objects and and identity-on-objects functor, what they really are talking about is an $Set^I$-enriched category, where $Set^I$ is the arrow category of $Set$.
A †-category is a category $C$ with a function $(-)^\dagger: Hom_C(A,B) \to hom_C(B,A)$ for every object $A,B \in Ob(C)$, such that
However, there is another definition of a †-category, as a category $C$ with a functor
which is the identity on objects and has $F^2 = 1$.
This second definition break equivalence-invariance: it imposes equations between objects, so we cannot transport a dagger-category structure along an equivalence of categories.
It was once believed that there was no known way to express the idea without equations between objects. This stemmed from the fact that category theorists were using functors to define the dagger. It was only later that it was recognized that one could include the dagger in the definition of dagger categories in the same way that one includes composition of morphisms in the definition of category, which resulted in a definition of dagger category that doesn’t violate the principle of equivalence.
A discussion about this is archived on the nForum.
It violates the principle of equivalence to state that two morphisms in a $2$-category are equal, because these morphisms are objects in a hom-category, but does not violate the principle of equivalence to state that that two $2$-morphisms are equal, given a common source and target. And so on. In an $\infty$-category, every claim of equality break equivalence-invariance.
Defining higher categorial structures using such equalities tends to lead to strict concepts; avoiding them and imposing coherence relations leads to weak concepts. Sometimes there is a coherence theorem showing that every weak concept can be strictified, which justifies using equality as a figure of speech. See bicategory, Gray-category, and model category for examples of this in action.
Since Hilb and nCob are dagger-categories, the discussion above is relevant, particularly for TQFT.
See at
The principle of equivalence in general relativity (see equivalence principle (physics)) is a special case of this principle of equivalence; the objects of the $\infty$-groupoid are physical “fields” (not to be confused with algebraic fields!), and the isomorphisms are coordinate transformations. So physical properties and operations should not depend on the choice of coordinate system, and this is enforced by allowing only functorial operations, which are specifically taken to be generated by traces, tensor products, and linear operations acting on the pseudo-Riemannian metric, and the covariant derivative.
See at
Just as we can make use of bases in linear algebra, so we may make use of strict categories to discuss category theory. Philosophically, the concept of strict category is not in itself a breaking of equivalence-invariance; what does break equivalence-invariance is to say that $Set$ (and other well-known categories such as Grp, etc) are strict categories. Mathematically, strict categories form a $1$-groupoid $Str Cat_\sim$ that is different from the $2$-groupoid $Cat_\sim$, but there is still a canonical pseudo functor from $Str Cat_\sim$ to $Cat_\sim$ that we may find useful.
Much as the axiom of choice tells us that every vector space has a basis, so the global axiom of choice tells us that every category may be given the structure of a strict category. Then we can use this extra structure (in either case), as long as we prove that the result is independent of the structure chosen. Even if we don't wish to accept the axiom of choice, we can still prove a theorem about those vector spaces that have bases or those categories that can be given a strict structure.
Perhaps the extreme case of this is using model categories to study homotopy theory, which is (from the nPOV) really about $(\infty,1)$-categories. Even if model categories are not taken to be strict categories, they still form a $2$-groupoid and thus are still far more strict than $(\infty,1)$-categories, which only form an $\infty$-groupoid. Nevertheless, they are quite useful (at least assuming the axiom of choice?).
A discussion of the principle of equivalence in the very foundations of mathematics by replacing ZFC by homotopy type theory is in
Vladimir Voevodsky, Foundations/formalization of mathematics and homotopy theory, talk at IAS (video)
Thierry Coquand, Nils Anders Danielsson, Isomorphism is equality (pdf)
Steve Awodey, Structuralism, Invariance, and Univalence (2014) (pdf)
Benedikt Ahrens, Paige Randall North, Univalent foundations and the equivalence principle, (pdf)
Last revised on March 25, 2023 at 04:41:14. See the history of this page for a list of all contributions to it.