The n-Category Café

If you examine Kevin Costello’s paper you’ll see that a 1-object Calabi–Yau $A_\infty$ category is an infinitely categorified version of a Frobenius algebra — see Section 2.

And as we’ve discussed here, a modular tensor category is an example of a ‘once categorified’ Frobenius algebra.

However, the former example doesn’t include the latter — unless there are tricks that I don’t understand, which is entirely possible. The reason is that an $A_\infty$ algebra is a kind of weak monoid in the $(\infty,1)$-category of chain complexes, while a tensor category is a kind of weak monoid in the 2-category of Kapranov–Voevodsky 2-vector spaces.

If you’re interested in categorified Frobenius algebras, you should also check out week268, including the discussion here on the $n$-Café. You’ll see that categorified Frobenius algebras are important in propositional logic and the study of ‘thick planar tangles’, which look like this:

All these applications of categorified Frobenius algebras are deeply related, but the applications to logic have not been integrated with the rest.

Charlie - There are two different roles you mention for Frobenius algebras in 2d TFT: commutative Frobenius algebras arise as the values on the circle, while (not.nec.commutative) Frobenius algebras arise as generators for the category of “branes”, i.e. what an extended TFT assigns to a point or an open-closed TFT has as labels. By “generator” I mean you describe the category in terms of modules for this algebra.

So there are different ways to categorify each. For the case of the circle (the commutative case), we can pass to modular tensor categories as John explains - they arise from extended 3d TFTs (or categorified 2d TFTs) on the circle. We can also “homotopify”, which can be considered a kind of categorification, and replace commutative Frobenius algebras in vector spaces by an analog in chain complexes - this is what arises on the circle in Costello’s TCFTs, e.g. in string topology and the A- and B-models of topological strings.

At the level of a point, a 1-object Calabi-Yau category is (essentially by definition) a Frobenius algebra, while the A_oo version that appears in Costello (as John explains) is a homotopical analog thereof. Any reasonable CY category can be thought of as modules for such a homotopical Frobenius algebra.

But we can also categorify this picture, by looking at what a categorified 2d TFT - or a 3d TFT - assigns to the point.. this is now some kind of 2- (or $(\infty,2)$-) category, and we can describe it in terms of a different categorified notion of Frobenius algebra (or eg if it has one object it will BE such a notion).

An example of this is studied in detail in my paper with David Nadler on Character Theory - we show that the “finite Hecke category”, a categorified analog of the group algebra of the Weyl group (or Artin braid group) of a semisimple Lie group, is such a beast, a categorified (or really $(\infty,2)$-categorified) Frobenius algebra. Its center (defined appropriately) is what you’d assign to the circle in this TFT, and thus a homotopical analog of a modular tensor category (or categorified commutative Frobenius algebra). It’s a categorified analog of the (commutative Frob. algebra of) class functions on a group, which are known as character sheaves.