Fully extended topological field theories and topological phases are intimately related. This motivates one to consider not only "group-like" symmetries, but also symmetries induced by the action of monoidal categories. To generalize this to higher dimensions, one must consider n-categories, and algebraic models quickly become intractable beyond dimension 3 or 4. Instead, we could formalize n- categories as categories (homotopy coherently) enriched in (n-1)-categories. This is to say, categories whose collections of morphisms are themselves (n-1)-categories.

This gives us an inductive definition for the n-categories we were hoping for, but there is no reason to stop there. To build a theory of condensation which can handle enrichment in (n-1)-categories, one really should build a theory which can handle much more general enrichment. As a result, you could also consider condensation of infinity categories, including derived categories, or perhaps other "higher algebraic" enrichments. Since infinity categories "remember" homotopy type, we also end up with something that naturally treats condensation of continuous symmetries. Many fundamental results of enriched infinity category theory are extremely recent, and so it takes some work to put all these pieces together. There are two reasonable ways one can proceed with this.

The first is to build fusion n-categories in this language and develop the formalism of Gaiotto and Johnson-Freyd. In the enriched world, an object satisfying the right analogue of dualizability is called atomic. The key insight here is that (atomic) condensation is an absolute colimit: a colimit that is preserved by all functors. A fusion n- category is then defined to be an (atomic) condensate of the category of fusion (n-1)-categories. One recovers the nice finiteness or dualizability properties from the assumption that everything we are working with is atomic, and so every fusion n-category defines a fully-dualizable object, and so also a fully extended TFT, by using the cobordism hypothesis.

There is another approach that we can take, building on the iterative condensation procedure of Kong, Zhang, Zhao, and Zheng. Here, one notices that a 0-form symmetry defines a defect from one vacuum to itself, and a 1-form symmetry defines a defect from the invisible 0-form defect to itself, and so on. So an i-form symmetry algebra is exactly an Ei-algebra in the i-th looping of your "category of vacua" at the chosen vacuum. One now works their way back down, one step at a time. By condensing an i-form symmetry, one obtains an (i-1)-form symmetry, and so on, until one arrives at a (-1)-form symmetry: another vacuum. Building an analogue of this construction really relies on the fact that the functor taking codimension-i operators to codimension-(i-1) operators is monoidal, so that if one puts in a symmetry algebra, they really do get back another symmetry algebra with codimension reduced by 1. This approach has one additional upside: we didn't have to make any dualizability assumptions, so we can use this to "condense" symmetries that are non-semisimple, for example. It turns out that a special case of this in 3 dimensions is given by the theory of vertex operator algebra (VOA) extensions, where one really hopes to be able to study logarithmic VOAs. These are exactly the ones that don't define fusion categories, and so don't satisfy the dualizability conditions that we needed before, but we can still apply this more general theory to obtain a notion of non-invertible crossed-braided category (which is really just a monoidal 2-category), which gives a sort of 2-categorical refinement of my earlier work on weak braiding.

There is still lots of work to do here though! The theory of Ei-modules, centers and centralizers still has not caught up to the fusion-categorical world, and more importantly, there are a lot of interesting examples that one could try to calculate with the flexibility of the enriched theory, which we hope to explore further.

Citations

• Condensations in higher categories- Gaiotto and Johnson Freyd
• On the classification of topological orders- Johnson Freyd
• Higher condensation theory- Kong, Zhang, Zhao, and Zheng
• Tensor categories for vertex super algebra extensions- Creutzig, Kanade, and McRae
• A 3-categorical perspective on G-crossed braiding- Jones, Penneys, and Reutter