Clarified the definition of Frobenius algebras in the category of Vectᴋ by making the non-degenerate pairing more explicit. The previous expression was the map u ↦ ϵ∘μ(1 ⊗ u); however, interpreting the symbols on the right-hand side completely literally in their syntactic form, then by the axioms of a unital associative we have that ϵ(μ(1 ⊗ u)) = ϵ(u) ∈ K. We are trying to have the right-hand side land in V*, which is done by clarifying that the right-hand side should actually be v ↦ ϵ∘μ(v ⊗ u), which is indeed an element of V* and the desired isomorphism of V with V*.

Lillian Ryan Uhl

]]>Added a note that the usual two Frobenius laws follow from the single axiom $(1 \otimes \mu) \circ (\delta \otimes 1) = (\mu \otimes 1) \circ (1 \otimes \delta)$.

]]>Thanks! At least they give it a notation, $\beta_1(A)$.

I notice that their definition of “special Frobenius algebra” is different than the one at Frobenius algebra: we require $\mu\circ\delta = 1$, whereas they require that $\mu\circ\delta$ and $\epsilon\circ\eta$ are a nonzero multiple (in their $\mathbb{C}$-linear context) of the identity. I guess the relationship is that in the latter case, one can multiply $\epsilon$ by the invertible scalar $\mu\circ\delta$ and divide $\delta$ by the same scalar to get another Frobenius structure on the same underlying algebra in which $\mu\circ\delta = 1$ as in our definition, but that still leaves the condition that $\epsilon\circ\eta$ is invertible – is that somehow implied by our definition?

]]>Some such composition occurs, but unnamed, e.g. Fig. 11 of this.

]]>Any Frobenius algebra $A$ has an invariant induced by composing the unit of the multiplication with the counit of the comultiplication, $I \xrightarrow{\eta} A \xrightarrow{\epsilon} I$. What is this called? Is it ever interesting?

]]>Isn’t it? I’m not sure quite what to make of it.

]]>Oh, hey, that’s pretty neat!

]]>I added a bit more, including a claim that the free polycategory containing a Frobenius algebra is the terminal polycategory.

]]>Yes, I went ahead and did it, but then got called away before I could announce it. I also incorporated a couple of “TODO”s from the bottom of the page into Frobenius algebra, and started adding a bit about Frobenius monoids in polycategories.

]]>It looks like this has already been done, but sounds good to me.

]]>I think the first version is probably best.

]]>We have a stubby page Frobenius monoid that seems to be intended to be about an abstract categorical version of a Frobenius algebra. However, much of the page Frobenius algebra is already written in the generality of an arbitrary monoidal category. So I think we should either get rid of Frobenius monoid and redirect it to Frobenius algebra (and add a bit of discussion), or else try to separate the abstract from the concrete versions. My inclination is the former; any other opinions?

]]>Thanks! I have included that at *Cardy condition*.

Ok, I followed the bizarre process for uploading an image (in the Sandbox).

cardy_condition.jpg, screen captured from Lauda & Pfeiffer 2006

I did this on my machine with Adobe Acrobat at 150% magnification which might give different sized bitmaps on machines with different screens. If you think this image should be a different size, or want more of the related images, Capturing a screen dump is trivial, but you also need something simple like MS-Paint to crop and save the image.

]]>Please do! Upload the image to the nLab itself, as described here. Thanks.

]]>So, Urs, see if Aaron has the xypic code still around.

The xypic code is in the TeX source in the ArXiv.

However John probably just “screen captured” (print screen) from a display of the PDF and used something like MS-Paint to crop down the bitmap and save it as a JPEG. I’d do it for you but I don’t have convenient place to host the image for upload into the nLab.

]]>The Cardy condition seems to be in here

Look at 1.14.

So, Urs, see if Aaron has the xypic code still around.

]]>Back then some kind soul provided these cobordism pictures at Frobenius algebra. Is that somebody still around and might easily provide also the picture for the Cardy condition?

I assume you mean such things pictures as commutative_law.jpg (IS THERE ANY WAY TO EMBED IMAGES IN THE NFORUM?)

Those pics come from

How is what you want different?

]]>Back then some kind soul provided these cobordism pictures at *Frobenius algebra*. Is that somebody still around and might easily provide also the picture for the Cardy condition?

Somebody writes by email the following about *Frobenius algebra*. I have no time to look into it right now, maybe somebody who was involved in writing the corresponding part finds a minute:

]]>when discussing the PRO(P)s for various kinds of Frobenius algebras, you have possibly exchanged the roles of Span(FinSet) and Cospan(FinSet) (both symmetric monoidal categories with respect to the coproduct). Indeed, if I understand correctly the reference [Rosenburgh etal. 2005], they prove that Cospan(FinSet) is the PROP for for special commutative Frobenius algebras. (Also, it is not hard to prove direclty that, as a symmetric monoidal category, Cospan(FinSet) is equivalent to the category obtained by quotienting 2Cob by the relation m \circ \delta = id ; but it does not work with Span(FinSet)…)

On the other hand, I can’t quite understand the example of the PROP for bimonoids. (Well, I’m pretty sure now it can’t be Cospan(FinSet)!) Indeed, I would need some more time to understand the reference [Lack 2008], or the reference therein to [Pirashvili, “A PROP to bialgebras”], which seems relevant. But can you confirm to me that (Span(FinSet), \coprod) is the PROP for bialgebras?

Has anyone any ideas how Turaev’s Frobenius crossed $G$-algebras and Ralph Kaufmann’s variant of this fits into the overall picture?

]]>Good question. But I have no idea what to do about it, since it’s the established terminology. Just suck it up, I guess. :-)

]]>I have added a Properties section to Frobenius algebra, in part to create links to the notions of quasi-Frobenius algebra and pseudo-Frobenius algebra. There is plenty of scope for expansion, since one can prove lots of nice properties hold for Frobenius algebras.

]]>