There are two ways in which higher categories appear in topological field theory: in extended TFTs and in TFTs with defects. How are these appearances related?
According to the Atiyah-Segal axioms, a d-dimensional TFT is a symmetric monoidal functor
\begin{equation*}\text{Bord}_d\rightarrow\mathcal{C}\end{equation*}
where the target category $\mathcal{C}$ is a symmetric monoidal category, typically $\text{Vect}$. The objects of $\text{Bord}_d$ are closed $(d-1)$-manifolds and map to the vector spaces of $\text{Vect}$ under the TFT. The morphisms are d-dimensional bordisms of the closed $(d-1)$-manifolds and map to the linear maps of $\text{Vect}$.
This description can be "extended down" to define a d-dimensional n-extended TFT as a symmetric monoidal $n$-functor
\begin{equation*}\text{Bord}_d^n\rightarrow\mathcal{C}\end{equation*}
where $\mathcal{C}$ is now an symmetric monoidal $n$-category. The objects of $\text{Bord}_d^n$ are $(d-n)$-manifolds (for a "fully extended" TFT, i.e. $n=d$, the bordism objects are points) and map to the objects of $\mathcal{C}$, which can be thought of as $(n-1)$-categories. One should really specify $\mathcal{C}$ as some extension of $\text{Vect}$, but I am ignoring this technicality by thinking of the n-category of all (small) $(n-1)$-categories. More generally, $k$-morphisms are $(d-n+k)$-dimensional bordisms between $(d-n+k-1)$-manifolds and map to $(n+k-1)$-categories. There are also technicalities surrounding how one manages "manifolds with corners" in $\text{Bord}_d^n$, but allow me to gloss over them here.
Meanwhile, one uses higher categories to decorate manifolds with defects. In this picture, $k$-dimensional manifolds are decorated with $k$-dimensional (extended?) TFTs called "$k$-defects." Lower dimensional submanifolds can be decorated with "defects within defects." The $k$-defect assigned to a $k$-dimensional "boundary" between two $(k+1)$-dimensional regions, each with attached $(k+1)$-defects, amounts to a morphism of $(k+1)$-defects. Fusion of defects and sub-defects endows the set of $k$-defects with the structure of a $k$-category. (For details, see http://arxiv.org/abs/1002.0385.) If we again imagine the $k$-category of (small) $(k-1)$-categories, we can understand decoration as an assignment of $(k-1)$-categories to $k$-manifolds.
This construction of TFT with defects feels "upside down" compared to extended TFT: $k$-manifolds are decorated by $(k-1)$-categories and map to $(d-k-1)$-categories under the TFT functor. I realize that decoration is not a functor from a bordism category, but is it a functor in some other sense (from some category where higher degree morphisms are lower dimensional submanifolds)? Defects may have an interpretation as inserted operators (Wilson loops, surface operators, etc); can these two formalisms be combined to compute path integrals with operator insertions in an extended theory? In general, I am curious about the relation between how higher categories are used to define an extended TFT and how they are used to characterize TFTs with defects.
Here is a particular problem from http://arxiv.org/abs/1309.1489. Consider a fully extended 4D TFT. The TFT functor assigns $1$-categories to Riemann surfaces $\Sigma$. Meanwhile, in the defect description, 2D TFTs form a $2$-category. Restricting ourselves to a single object leaves us with a symmetric monoidal category of $1$-dimensional domain walls (essentially boundary conditions) of the 2D theory. Is this category of boundary conditions identified with the category assigned by the TFT functor (as is claimed)? It seems that this identification is only possible since $\dim\Sigma=2=\text{codim }\Sigma$ in dimension four and does not reflect a general relation between extended TFT and TFT with defects.
This post imported from StackExchange MathOverflow at 2014-11-15 11:29 (UTC), posted by SE-user Alex Turzillo
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