I guess what I am saying is that in the context in which this arises in Chris’s application, it might be more worthwhile to call $f \circ i$ and $g \circ i$ themselves by their proper name. Why is it that we may want to restrict an $n$-dimensional extended QFT this way? Where does this come from abstractly? If this were identified, we’djust speak of a transformation between these restrictions.

Here is maybe a hint: for 3d TQFT $Z : 3Cob \to 3Vect$ we know what it means to consider the restriction along $3Cob_{3Core} \hookrightarrow 3 Cob$ to invertible 3-morphisms. That’s essentially what’s called the “modular functor” on surfaces. That plays a big role in existing theory. Maybe it would be useful to understand the generalization of this notion for any $n$-d QFT. Then the “supernatural transformations” that we are talking about might be just called “transformations between the modular $n$-functors underlying given $n$-functors”. That would seem to me to be a more insightful terminology.

]]>strictly speaking there is no real need for a new term, since a subnatural transformation $f \to g$ is just a natural transformation $f \circ i \to g \circ i$

Sure, we’ve already noticed that with core-natural transformations. But you still want some term; a ??? $f \to g$.

]]>if it is the case that the filtration is always by a genuine sub-$(n,r)$-category $i : C_0 \hookrightarrow C_1$ then strictly speaking there is no real need for a new term, since a subnatural transformation $f \to g$ is just a natural transformation $f \circ i \to g \circ i$.

I felt and still feel that this is the best way to go about it – *if* it is true in your examples that the filtration is always by a genuine sub-category.

Notably I feel that thinking about natural transformations $f \circ i \to g \circ i$ suggests more vividly to us that there is something to be understood here, conceptually: instead of just giving this construction a name and moving on, we should ask ourselves what it is here that should make us want to define transformation between TFTs after restricting them.

Not sure if you have seen it, but not long ago I created that entry titled (grandiosely so and intentionally in this case ;-) holographic principle of higher category theory which is about the nature of transformations between $(n,r)$-functors between $(n,r)$-categories of cobordisms.

For the kind of story that this entry tries to tell, it would be of great interest to have a general abstract *reason* for why one would want to consider transformations only after restriction.

I mean, I understand well that one good reason is that this way the result is interesting. But there should also be a good *a priori*-reason for it.

I prefer ‘subnatural transformation’ to ‘subtransformation’, although I think that ‘core-natural transformation’ is even clearer. We shouldn’t have problem with the red herring principle here, since ‘core’ is clearly modifying ‘natural’ (or at least it’s clear to me!).

What is the more general concept? A ‘filtered-natural transformation’?

]]>Subnatural sounds good to me.

]]>I know this thread is mostly dead, but I just discovered it and want to throw a few more coins into the wishing well. Somewhere along the line Urs mentioned how I had discovered this and related concepts in the wild and suggested the name “unnatural transformation”. This was at an Olberwolfach conference. Mainly, this was a wild suggestion to a friendly audience, mostly meant to stir up discussion and generate ideas about better terminology. I think I also suggested “supernatural transformation”. Neither of these are very satisfying, in my opinion. I think Jim’s canonical transformation, which I learned about later, is much better.

I also like the term “subtransformation”. Especially so because “transformation” seems to be identified with “natural transformation” in many people’s minds. So if you just use an adjective, like “canonical” or “unnatural” or “core”, to modify transformation people might be confused that this is some sort of special kind of natural transformation, when in reality the opposite is true.

In the examples I was considering there were two important aspects. One was the filtration (which for the sake this discussion is the inclusion of the core). But the other is the importance of lax transformations. For a long time I’ve been stuck with this project because I didn’t know how to deal with lax transformation in a model invariant, (infinity, n)-manner. I think I’ve finally come to grips with later point and am ready to return to the first aspect. Whence I discovered this thread.

It seems like you can define these sort of “less than natural” transformations (and higher generalizations) whenever you have an n+1 step filtration of your n-category (I remember talking to Urs about this way back in Olberwolfach. He may have suggested it to me). So a traditional natural transformation would correspond to the trivial constant filtration, while Jim’s canonical transformation would correspond to the core filtration. In general you have something more exotic. The inclusionary aspect of the filtration seems to jive with the “sub” of “subtransformation”, which I find pleasant. Another term which has been suggested to me is “subnatural transformation”, which shares many of these benefits.

I’m very curious to hear people’s opinions on this. This is a chance to create a new piece of terminology and I think it is important to try to get it right at the beginning.

]]>Re #52: Now also the examples have been redone.

]]>You missed out G. In any case, it's not the canonical morphism that is a represenation of the group Aut(x) but rather F and G which are; then the canonical morphism is an intertwiner between these representations.

However, Jim typically uses the term when F and G are entire functors, defined on all of C rather than just on x and Aut(x). Then there is no standard term for it (other than ‘intertwiner between the induced representations of Aut(x)’).

]]>That's funny, since the idea should be precisely what is *not* evil. So if you put your finger on it, then please let me know!

For the record, the context of the definition of canonical morphism consists of the variables *C*, *D*, *F*, *G*, and *x* (typed as indicated there).

I've reworked the article to focus on canonical morphisms as the primary concept, making the link both to natural transformations on the one hand and to intertwiners on the other. (I still have to go through the section of Examples.)

]]>I appear to be completely failing to communicate my point.

Yes, it seems like you and I are completely failing to communicate, Mike. Most of your last message makes almost no sense to me in the context of the preceding discussion, and it is as clear to me that you don't understand what I'm trying to say and the reverse seems to be clear to you.

maybe with Jordan's name in front to disambiguate

For what it's worth, I agree that I can't possibly be understanding you, since the need to disambiguate here seems to undermine your position. (In fact, I presumably just have no idea what your position is.)

Anyway, I will work on the article a bit more this week, and you can tell me what you think of it then.

]]>I use canonical myself when referring to the existence of something (like a functor) that is the 'obvious and best' choice, something that falls into my lap, so to speak, even though it may be defined up to isomorphism/equivalence. In other words, it is not really a choice because a) it should be obvious (to a practitioner) and b) any other such object requires additional (arbitrary) choices, possibly using AC, in its construction/definition. It seems to me to be similar to having a completely presented set - technical use of AC is not an issue because there is an obvious choice, indicated either conceptually or syntactically (in a non-technical sense, more like 'write down the obvious formula'). ]]>

Sigh.

I appear to be completely failing to communicate my point. (No, I didn't mean that the word "canonical" *literally* always follows "the".) If there's anyone else still listening to this discussion, do you have any idea what I'm trying to say?

Wherever I have seen "canonical" used in mathematics (or elsewhere, except for the Ubuntu software company), it refers to something which is determined uniquely by the construction of something, or by some natural (in the non-technical sense) or desirable requirement. (I have never encountered Wikipedia's supposed specific meaning of "coordinate-free.") Anything you can describe can of course be determined uniquely by *some* requirement, namely that it be equal to the thing described by that description, but that doesn't make it canonical. There is also definitely a role of societal convention in deciding what is canonical and what isn't.

If I'm interested in studying topologies on categories for which the representables are sheaves, then naturally the "canonical" topology will play an important role. There are other topologies that exist on every category and can be defined "globally," such as the trivial topology or the atomic topology or the largest topology for which the representables are separated, but we don't call those "canonical," because, well, we don't. We either have other names for them (like "trivial") or we don't talk about them enough to merit having their own names. When I construct the quotient of a group by a normal subgroup, the "canonical" homomorphism (as referenced by Wikipedia) is determined by the definition of the quotient group as a set of equivalence classes in the group itself, or if you're a category theorist, by the universal property it satisfies. The Jordan canonical form is a particularly nice form for any matrix; obviously it's not the only "form" into which every matrix can be put (just like there are more topologies on a category than the "canonical" one), but matrix theorists have decided that it's important enough to call "canonical," maybe with Jordan's name in front to disambiguate.

The set-theoretic usage mentioned at the end of the Wikipedia section also fits this usage, but seems to be diametrically opposed to the notion of 'equivariant transformation'. Once you've chosen a representative of each equivalence class, you can declare that representative to be canonical (societal convention again, this time in a "for the noonce" form rather than globally). But choosing an element out of a collection of sets is exactly what you're saying *can't* be done 'canonically.'

It also seems to me that in natural language, canonical always

doescome after 'the'.

Well, that statement seems easy to refute (not all hits are relevant, but some seem to be).

Actually the Wikipedia hit seems very much to agree with me (although I had no hand in writing it, which is sometimes the reason). It reminds us in particular of the Jordan canonical form of a matrix, which is surely *a* canonical form but not *the* canonical form. (As such, this is an example of a functor rather than a transformation, of course.)

The best sense that I can make of what you're saying here is that ‘canonical’ should only come after ‘the’, never after ‘a’.

Yes, that's about what I meant.

If so, my reply is that the general notion of ‘canonical’ is used in a grammatically different way (following ‘a’, not ‘the’) which should not cause confusion

I disagree that it should not cause confusion; I already find it confusing. It also seems to me that in natural language, canonical always *does* come after 'the'.

On second thought, if you simply have two actions/representations of a group on objects of a category , then this is two functors from to , and a canonical morphism between these is precisely a -equivariant morphism between the original representations of . The general concept is a straightforward generalistion of this. So there probably is no room for confusion.

]]>The canonical topology is not the

uniquetopology, but it is the unique topology which contains all other topologies for which the representables are sheaves. In that sense it is determined uniquely. By contrast there can be many 'canonical' transformations between two given functors with nothing to single out any of them.

Yes, the canonical topology is not the unique topology, but it is the unique topology satisfying a certain property.

Similarly, the non-identity canonical morphism is not the unique morphism , but it is the unique morphism satisfying a certain property. (There are many ways to state this property: it is a derangement, it is not the identity, it is equal to , etc.) Also, the canonical transformation is not the unique canonical transformation , but it is the unique transformation satisfying a certain property.

In contrast, the two noncanonical morphisms cannot be characterised uniquely using non-evil language (although that is hard to make precise).

The best sense that I can make of what you're saying here is that ‘canonical’ should only come after ‘the’, never after ‘a’. In any given situation, we may (if we wish) state some property that is satisfied by at most one thing, and declare that thing to be ‘canonical’. We can do this completely arbitrarily, as long as the thing that we pick is uniquely determined by the data at hand, but we can only do it once. So it doesn't make sense to define ‘canonical’ in general, since then there are many situations in which we have chosen canonical things. Instead, we must use a family of ad hoc definitions, to pick (at most) one thing out in each context. Is this what you mean?

If so, my reply is that the general notion of ‘canonical’ is used in a grammatically different way (following ‘a’, not ‘the’) which should not cause confusion; but it is related, since only *a* canonical thing can be uniquely specified by the data at hand so that you can define it to be *the* canonical thing.

"Equivariant" is used pretty widely, but I'm only aware of one basic meaning, namely respecting the actions of some group. Are there others?

Not that I know of, but the concept of canonical morphism applies in a more specific situation, when one automorphism group acts (on both sides) on another via a couple of functors. I'm worried that ‘equivariant morphism’ (a term which certainly exists already) might be applied to some other group actions in this situation. But perhaps there are no other group actions likely to be thought of.

]]>http://mathoverflow.net/questions/11069/induced-grothendieck-topology-on-a-presheaf-or-sheaf-category-of-a-site/11222#11222

It's given by a universal property on the category of sheaves on a site, so I would say that it's the canonical topology for a Grothendieck topos. My translation wasn't as extensive as I'd previously thought, but between my answer and Clark's you should get the gist of it. ]]>

I'm reminded of the quote from Through the Looking Glass

]]>"When I use a word," Humpty Dumpty said, in rather a scornful tone, "it means just what I choose it to mean -- neither more nor less."

"The question is," said Alice, "whether you can make words mean so many different things."

The canonical topology is not the *unique* topology, but it is the unique topology which contains all other topologies for which the representables are sheaves. In that sense it is determined uniquely. By contrast there can be many 'canonical' transformations between two given functors with nothing to single out any of them.

"Equivariant" is used pretty widely, but I'm only aware of one basic meaning, namely respecting the actions of some group. Are there others?

]]>