The n-Category Café

I think some insight can come from writing this a little differently. $$\mu : \mathrm{End}(k) \otimes A \to A \otimes \mathrm{End}(V)$$

Yes, that’s good. In fact, that’s equivalent to writing $$\array{ \bullet &\stackrel{\mathrm{Id}}{\to}& \bullet \\ A\downarrow\;\; &\;\Downarrow \mu& \; \downarrow A \\ \bullet &\stackrel{\mathrm{End}(V)}{\to}& \bullet }$$ in $\Sigma(\mathrm{Bim}(\mathrm{Vect}_k))$:

the identity 1-morphism in $\Sigma(\mathrm{Bim}(\mathrm{Vect}_k))$ is the $k$-algebra such that tensoring any other algebra with it does nothing (up to …). This implies precisely $$\mathrm{Id}_\bullet \simeq k \simeq \mathrm{End}(k) \,.$$

So the above diagram really reads $$\array{ \bullet &\stackrel{\mathrm{End}(k)}{\to}& \bullet \\ A\downarrow\;\; &\;\Downarrow \mu& \; \downarrow A \\ \bullet &\stackrel{\mathrm{End}(V)}{\to}& \bullet } \,.$$

That’s a lesson that I have learned a while ago: whenever you see something involving a twist of roughly the sort we are seeing here in the defintion of amplimorphisms, chances are good that you are really dealing with a pseudonatural transformation of some sort.

That’s what these square diagrams suggest: that they are really components of some sort of transformation of some kind of (2 or 3?)-functor taking values in some world of algebras.

This is vaguely interesting, because I already do know # that such transformations do control correlators of CFTs – and these amplimorphisms are supposed to know about symmetries of such CFTs.

But even if there is a relation, I don’t see it yet in detail.

Arrows between arrows do in general not correspond to surfaces!

That’s right!

Just as maps between sets do not in general correspond to curves – and are still usually displayed in a “1-dimensional” notation: $$f : S \to T \,.$$

Since you post this as a comment to the above entry, you might notice that, incidentally, above we do discuss an example of 2-morphisms that do not correspond to surfaces – but to bimodules.

But what this also shows is: the notation for these 2-morphisms is very naturally one that makes full use of the 2-dimensionality of paper and screen.

You might want to read an account of the history of the term “higher dimensional algebra”. Maybe

Higher Dimensional Algebra and its Applications

by Tim Porter.

But, I suspect that some nice algebraic idea is being developed in a somewhat ‘squashed’ way, and mathematicians will need to unfold it a bit to see its inner beauty.

Yes, exactly. My diagrams above were supposed to be a first step in that direction. Compare with the formulation of the definition in the literature to see the difference…

Am I supposed to think of an amplimorphism $$A \to A \otimes \mathrm{End}(V)$$ as a funny sort of morphism from $A$ to itself

Yes. That seems to be what it is used for in applications.

I have added an update to the end of the above entry two days ago which adresses this vaguely:

The point is apparently that for many examples of infinite-dimensional algebras tensoring with a matrix algebra does not change anything $$A \otimes \mathrm{End}(V) \simeq A \,.$$ Then in this case the concept of an amplimorphism reduces to that of an ordinary endomorphism $A \to A$.

In some parts of the literature one finds hence comments along the lines that amplimorphisms are a way to do with finite dimensional algebras what is otherwise only possible with infinite dimensional algebras.

So people working with these seem to have an intuition for why it must be precisely this way. But it remains a little obscure to the outsider.

The only solid fact that I could extract is this:

it is possible to put a braiding on the category of amplimorphisms by using some braiding on the underlying algebras. This way amplimorphisms for an algebra $A$ form a braided monoidal category.

Fact: if we choose $A$ to be a certain Hopf algebra, then this braided monoidal category (which we should think of as a category of reps of that Hopf algebra) is equivalent to the braided monoidal category of local observables in 2-dimensional AQFT.

That was the point of introducing amplimorphisms: people were looking for an analog of the Doplicher-Robberts statement for the case that the category in question has nontrivial braiding.

Still, when I look at the original literature, the definition of amplimorphisms still seems to come out of the blue sky, justifying itself only by this end result.

I am hoping that there is also a nicer explanation of the raison d’etre of amplimorphisms.

Sounds like you need ps2pdf.com!

That ps-file does not contain much beyond the above figure and a little introduction for how to read it.

What is worse, for me, is that this insight that amplimorphisms have an interpretation in terms of components of lax-natural transformations hasn’t really, so far, helped me to see what they really are. But I didn’t look into this question for a while and am only now coming back to it.

Concerning the geometric representation theory picture (aka groupoidification) of the path integral obtained from “taking sections”: that’s something I want to get back to as soon things on my side here have become less hectic. I am thinking this might happen in about two weeks or so.

What generally might help more than converting old ps files to new pdf files is that you post concrete questions, if any, on what I wrote. Then I can try to reply and we can try to walk through what I said in small steps. Or else, walk through your ideas in small steps. Or through anyone else’s ideas we find worth looking at, for that matter. :-)