nLab
core-natural transformation

Core-natural transformations

Core-natural transformations

Idea

A canonical morphism is a morphism which is equivariant under an automorphism group, with actions mediated by two functors. It may be seen as a natural transformation in a particular context. Conversely, canonical morphisms can be combined into a notion of transformation more general than a natural transformation, although this also may be seen as a natural transformation in a particular context.

This usage of the word ‘canonical’ is due to Jim Dolan. In general, this term is often used in mathematics to mean that the result of a construction may be specified using only the data at hand, without making arbitrary choices. The idea behind using the word here is, roughly, that only canonical morphisms may be specified (in the situations in which they appear) without violating the principle of equivalence. (Both avoiding arbitrary choices and avoiding violation of the principle of equivalence are related to avoiding the axiom of choice, but that does not seem to be directly relevant.) However, there are certainly also uses of ‘canonical’ in mathematics that do not fall under this definition.

Arguably, ‘natural’ would be a better term for this intuition, but canonical morphisms are more general than the natural transformations that appear in the same contexts, so that word is taken. Another possible term is ‘core-natural’ or ‘groupoid-natural’, since (as will be seen below) canonical morphisms may be interpreted as natural transformations between functors restricted to the core (underlying groupoid) of a given category. The terms ‘basis/coordinate–free/invariant’ and ‘generally covariant’ also capture the same intuition, although these tend to be restricted to certain disciplines (linear algebra, geometry, physics); see also principle of equivalence, definable set (in model theory), and parametric polymorphism (in type theory).

Definitions

Given categories CC and DD, functors F,G:CDF, G\colon C \to D, and an object xx of CC, a canonical morphism from F(x)F(x) to G(x)G(x) is a morphism h:F(x)G(x)h\colon F(x) \to G(x) in DD such that the diagram

F(x) h G(x) F(u) G(u) F(x) h G(x) \array { F(x) & \overset{h}\to & G(x) \\ F(u) \downarrow & & \downarrow G(u) \\ F(x) & \underset{h}\to & G(x) }

commutes for any automorphism uu of xx in CC.

As FF and GG are only ever applied to isomorphisms, this definition makes sense even when they are defined only on the core of CC. In fact, as they are applied only to xx and its automorphisms, the definition makes sense when they are defined only on Aut C(x)Aut_C(x), the automorphism group of xx in CC. In that case, FF and GG are representations of Aut C(x)Aut_C(x) in DD, and a canonical morphism is precisely an intertwiner between these representations.

In the other direction, we can consider a family of canonical morphisms, one for each object of CC, which are coherent in the sense that

F(x) h x G(x) F(u) G(u) F(y) h y G(y) \array { F(x) & \overset{h_x}\to & G(x) \\ F(u) \downarrow & & \downarrow G(u) \\ F(y) & \underset{h_y}\to & G(y) }

commutes for every isomorphism u:xyu\colon x \to y in CC. Such a family may be called a canonical transformation from FF to GG, although this should not be confused with a canonical coordinate transformation. Note that every natural isomorphism is canonical, but not conversely. The main use of having a term like ‘canonical’ at all is to say that an expression for a morphism in DD, involving a variable for an object in CC, is ‘canonical’ in that variable, as a weaker condition than saying that the expression is ‘natural’ in that variable.

By the axiom of choice, if there exists a canonical morphism from F(x)F(x) to G(x)G(x) for every object xx, then there exists a canonical transformation from FF to GG (the converse is obvious). Actually, this does not require the full axiom of choice; it uses only … (that groupoid-relevant version, I need to find its name).

Alternative characterisations

As remarked above, intertwiners between representations of groups may be seen as canonical morphisms between functors defined on (the delooping of) a group. Conversely, we may define a canonical morphism from F(x)F(x) to G(x)G(x) to be an intertwiner between the restrictions of FF and GG to the automorphism group of xx, thought of as representations of that group.

Simlarly, we may define a canonical transformation from FF to GG to be a natural transformation between the restrictions of FF and GG to the core C˜\tilde{C} of CC. This is the origin of the alternative term ‘core-natural transformation’.

Finally, we note that just as every natural transformation between functors F,G:CDF,G\colon C \to D defines a functor from CC to the arrow category ArrDArr D (and conversely), so a canonical transformation between such functors defines a functor from C˜\tilde{C} to ArrDArr D (and conversely if we allow the functors to be defined only on isomorphisms).

Examples

Let GG be a group, let CC be the delooping BG\mathbf{B}G of GG (that is GG thought of as a 11-object category), let xx be the object of CC. Then the functors F,G:CDF,G\colon C \to D are simply representations of GG in DD, and a canonical morphism from F(x)F(x) to G(x)G(x) is an intertwiner between these representations.

More generally, let CC be a groupoid. Then a canonical (or core-natural) transformation from FF to GG is simply a natural transformation from FF to GG, which should make sense since CC is its own core. If CC is any category whatsoever, then a natural transformation from FF to GG is still a canonical transformation, although in general there are other canonical transformations.

We now consider an example that may serve to further motivate the term ‘canonical’. Given a 22-element set xx, there are 44 functions from xx to itself, the identity function, the non-identity involution, and the two constant functions. However, there is no way to specify either constant function without using some property of the specific elements of the set in question; there is no ‘canonical’ way to define either of those. Correspondingly, if we take CC and DD to be FinSet, FF and GG to be the identity functor on FinSetFin Set, and xx to be this 22-element, then there are only 22 canonical morphisms from F(x)=2F(x) = 2 to G(x)=2G(x) = 2: the identity function and the non-identity involution.

If we wish to extend this example to an entire canonical transformation between these F,Gid FinSetF,G \coloneqq id_{Fin Set}, then this is actually the only choice that we can make. This is because, if xx has 33 or more elements, then we have no way to choose among the 22 or more elements that are not equal to any given element, so only the identity function on xx is canonical. Thus we have 22 core-natural transformations from id FinSetid_{Fin Set} to itself, only one of which (the identity natural transformation) is natural.

On the other hand, if we let CC be the category FinOrdFin Ord of finite well-ordered sets, then every element of every such set is uniquely identifiable. Accordingly, given any well-ordered set xx, every function from (the underlying set of) xx to itself is a canonical morphism from F(x)F(x) to G(x)G(x), where now F,GF,G are both the forgetful functor from FinOrdFin Ord to FinSetFin Set. Now there are infinitely many canonical transformations from this functor to itself, but still only the identity transformation is natural.

Now consider the operation of ordinal addition on FinOrdFin Ord. We add two well-ordered sets xx and yy by taking their disjoint union as sets and placing the elements of xx before the elements of yy. This operation is associative up to a unique isomorphism, which is natural and so makes FinOrdFin Ord into a monoidal category. The operation is also commutative up to a unique isomorphism, but the commutativity transformation is not natural. Nevertheless, it is canonical (as it must be, being unique).

The examples above are all of canonical isomorphisms. However, we can adjust the last example for a canonical transformation that is not an isomorphism. Consider two functors from FinOrd×OrdFin Ord \times Ord to OrdOrd, given by ordinal addition in the two possible orders. The canonical isomorphism in the last paragraph exends to a canonical transformation from (x,yx+y)(x, y \mapsto x + y) to (x,yy+x)(x, y \mapsto y + x). As the canonical isomorphism above encapsulates the equation x+y=y+xx + y = y + x for finite ordinal numbers, so this canonical transformation encapsulates the equation x+yy+xx + y \leq y + x for ordinal numbers when xx is finite.

References

An analogous notion for higher functors makes a prominent appearance in Chris Schommer-Pries‘s work on FQFT with defects/bi-branes. See slide 81 (the penultimate page) of:

  • Chris Schommer-Pries, Topological defects and classifying local topological field theories in low dimension (pdf)

See holographic principle of higher category theory for more on that.

On the terminology ‘canonical’, see:

  • A 1993 Usenet thread may be the first public introduction of the contrast between ‘natural’ and ‘canonical’ by Jim Dolan (posting as ‘Robert Scott’); see particularly posts 9&10.

  • A 2010 MathOverflow question about the meaning of ‘canonical’, with many different answers, including this one.

Last revised on April 22, 2017 at 05:47:53. See the history of this page for a list of all contributions to it.