December 15, 2006

Frobenius Algebroids with Invertible Products

Posted by Urs Schreiber

Sometimes, banalities fool us into ignoring important structure. That happens when the banality is the degenerate case of something that is in general more interesting.

One banality is this.

An associative monoidoid (I mean any enriched category, please see the comments below)

with invertible product (this invertibility is the degeneracy)

is naturally a Frobenius monoidoid

only that you would tend to ignore this fact. It’s like ordering Pizza Tonno without Tuna.

But notice this:

when we compute gerbe holonomy in local data, we are inclined to choose a dual triangulation of our surface and decorate it with the transition data – which amounts to decorating it in a monoidoid with invertible product.

On the other hand, when we compute amplitudes in 2-dimensional field theory using state sum models, we are inclined to choose a dual triangulation of our surface and decorate it in a Frobenius algebroid.

I want to invite everybody to think of these two procedures as two versions of the same underlying principle. I used to call this principle locally trivialized 2-transport. But apparently I should start referring to it in terms of 2-anafunctors.

So when I give two talks in Toronto # – even though I have only one point to make – it is because there are two aspects to this one point: 2-anafunctors come in two flavors: with pseudo transitions or with lax transitions.

I try to tell parts of this story here. The plot follows this table:

Posted at December 15, 2006 11:04 AM UTC

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Re: Frobenius Algebroids with invertible Products

What the heck is a “monoidoid”? It’s not good to use such quirky terminology without defining it.

An “algebroid” is a category enriched over Vect; this terminology is justified by the fact that a one-object algebroid is an algebra.

A “ringoid” is a category enriched over AbGp; this terminology is justified by the fact that a one-object ringoid is a ring.

So, I’d have to guess that:

A “monoidoid” is a category enriched over Set; this terminology is justified by the fact that a one-object monoidoid is a monoid.

But, a category enriched over Set is just a category!

So, I’m forced to guess that “monoidoid” is your quirky new terminology for “category”.

But, I can’t believe you’d be that perverse.

Posted by: John Baez on December 16, 2006 12:28 AM | Permalink | Reply to this

Re: Frobenius Algebroids with invertible Products

What I wrote above makes sense for any category enriched over some 2-category.

Part of my point was that where some people are familiar with decorating dual triangulations in algebroids, we might want to enlarge the point of view to more general -oids, those that do not necessarily come from enriching over $\mathrm{Vect}$.

For instance, for 2-bundle holonomy, we want to decorate in a category enriched over a 2-group.

And the only reason why in the first case we explicitly have a Frobenius-property in the game, while in the second we have not, is that in the second case the Frobenius property is automatically satisfied. So we may not realize that it’s there.

Okay, so that’s what I wanted to say. Shouldn’t have used the work “monoidoid”, probably. (Somehow I enjoyed writing it, was chuckling to myself when I did.)

Posted by: Urs on December 17, 2006 2:18 PM | Permalink | Reply to this
Read the post Frobenius Algebroids with invertible Products
Weblog: The n-Category Café
Excerpt: Thinking of algebroids with invertible products as Frobenius algebroids.
Tracked: December 17, 2006 2:26 PM

Re: Frobenius Algebroids with Invertible Products

Is there a Universal -oid?

The nice table clarified my meager understanding of Quantum Field Theory, functorially.

Posted by: Jonathan Vos Post on December 20, 2006 5:21 PM | Permalink | Reply to this

Re: Frobenius Algebroids with Invertible Products

Is there a Universal -oid?

John was quite right, I shouldn’t have use the word monoidoid the way I did.

So, let me try to clarify the situation:

whenever we have any structure $C$ which comes with a kind of product map

(1)$C×C\to C$

we say $C$ is “monoidal”.

The reason is that we can imagine every element $c$ of $C$ (if $C$ is in fact something that can be thought of as consisting of elements) to be like an arrow starting and ending at an abstract point $•$.

(2)$•\stackrel{c}{\to }•\phantom{\rule{thinmathspace}{0ex}}.$

The product operation which takes two such elements and sends it to a third can then be thought of as describing the composition of these arrows.

That’s the very starting point of category theory, in a way (not historically, but still): once we talk about composing arrows from $•$ to $•$, there is nothing more natural than considering arrows that go between more kinds of objects, like

(3)$•\stackrel{{c}_{1}}{\to }\nabla \stackrel{{c}_{2}}{\to }\square \phantom{\rule{thinmathspace}{0ex}}.$

Whenever we have a structure like that, with many different objects and such that the arrows

(4)$•\stackrel{{c}_{•}}{\to }•$
(5)$\nabla \stackrel{{c}_{\nabla }}{\to }\nabla$
(6)$\square \stackrel{{c}_{\square }}{\to }\square$

that start and end at one and the same object behave under composition like elements of $C$, then we say that the entire structure is a $C$-oid.

For instance: and algebra is a vector space with a product. The corresponding many-object entity is a category all whose spaces of morphisms between a given pair of objects is a vector space, such that composition is bilinear. This is hence called an algebroid. It is the same thing as a category enriched over vector spaces.

Now, when we have a collection of things and don’t specify any additional structure on it, like a vector space structure, but still have a way to compose two things to get a third, we just say we have a monoid.

Playing with these terms, it turns out that a monoid-oid is just something with many objects and a rule to compose arrows going between them. So it’s a category. Or some enriched category, if you like.

As John rightly pointed out, saying “monoid-oid” is perverse, and I have to apologize for doing so on a family blog like ours.

Posted by: urs on December 20, 2006 5:46 PM | Permalink | Reply to this
Read the post QFT of Charged n-Particle: Towards 2-Functorial CFT
Weblog: The n-Category Café
Excerpt: Towards a 2-functorial description of 2-dimensional conformal field theory. A project description.
Tracked: August 3, 2007 10:24 PM

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