Higher Categorical Symmetries and Defect Algebras: Examples and Counterexamples

Let $F:{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{\mathbb{C}}^2\to {\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{\mathbb{C}}^2$ have matrix $\left(\begin{array}{cc}\mathbb{C}& 0\\ \mathbb{C}& \mathbb{C}\end{array}\right)$ and let $G$ have matrix $\left(\begin{array}{cc}\mathbb{C}& \mathbb{C}\\ 0& \mathbb{C}\end{array}\right)$. With rows indexed by outputs and columns by inputs, $(GF{)}_{ki}={⨁}_j{G}_{kj}\otimes F_{ji}$, so $GF=\left(\begin{array}{cc}\mathbb{C}\oplus \mathbb{C}& \mathbb{C}\\ \mathbb{C}& \mathbb{C}\end{array}\right)\mathrm{.}$ Each summand is the contribution of one intermediate simple label.

Let $\eta :F\Rightarrow F^{\mathrm{\prime }}$ scale $F_{21}$ by $2$, let ${\eta }^{\mathrm{\prime }}:F^{\mathrm{\prime }}\Rightarrow F^{\mathrm{\prime }\mathrm{\prime }}$ scale the same entry by $7$, let $\theta :G\Rightarrow G^{\mathrm{\prime }}$ scale $G_{12}$ by $3$, and let ${\theta }^{\mathrm{\prime }}:G^{\mathrm{\prime }}\Rightarrow G^{\mathrm{\prime }\mathrm{\prime }}$ scale the same entry by $5$; all other entries and components are identities. On the second intermediate-label summand of the $(1,1)$ entry of $GF$, horizontal composition first gives scalar $3\cdot 2=6$, and the next horizontal 2-cell gives $5\cdot 7=35$, so vertical composition gives $35\cdot 6=210$. Vertically composing first gives scalar $5\cdot 3=15$ in $G_{12}$ and $7\cdot 2=14$ in $F_{21}$, whose horizontal tensor is $15\cdot 14=210$. The first intermediate-label summand is unchanged in both paths. Thus the two interchange composites agree entrywise, and hence agree as natural transformations.

If $f:A\to B$ and $g:B\to C$, their categorical composite has type $g\circ f:A\to C$. A point junction $\alpha :f\Rightarrow f^{\mathrm{\prime }}$ lies in the single hom category $\mathcal{B}(A,B)$, and vertical composition is defined only when the target 1-morphism of one 2-cell is the source 1-morphism of the next.

For $\alpha :f\Rightarrow f^{\mathrm{\prime }}$ in $\mathcal{B}(A,B)$ and $\beta :g\Rightarrow g^{\mathrm{\prime }}$ in $\mathcal{B}(B,C)$, functorial horizontal composition gives $\beta \ast \alpha :g\circ f\Rightarrow g^{\mathrm{\prime }}\circ f^{\mathrm{\prime }}$ in $\mathcal{B}(A,C)$. The object tensor $A\otimes C$ has a different type from the composite $g\circ f$; no physical realization or identification of these operations follows from this typing calculation.

Finite-dimensional complex vector spaces form a finite semisimple category, their tensor unit $\mathbb{C}$ is simple, and vector-space duals give rigidity; thus they satisfy the fusion conditions in [F2]. The flip $v\otimes w\mapsto w\otimes v$ is natural and invertible, and on triples both hexagon composites perform the same adjacent transpositions to reverse the order past one tensor factor; hence it is a braiding as in [F3]. The element $E=e_1\otimes e_1+e_2\otimes e_2$ satisfies $m_R(E)=e_1+e_2=1_R$ and $rE=Er$ for every $r\in R$, so it is a separability idempotent. On $M_{ij}\otimes M_{kl}\cong \mathbb{C}$, the projector ${\sum }_{s=1}^2(r_{e_s}\otimes l_{e_s})$ acts by the scalar ${\sum }_s{\delta }_{j,s}{\delta }_{k,s}$; this is $1$ when $j=k$ and $0$ when $j\mathrm{\ne }k$. By the image description, $M_{ij}{\otimes }_R{M}_{kl}\cong \{\begin{array}{ll}{M}_{il},& j=k,\\ 0,& j\mathrm{\ne }k\mathrm{.}\end{array}$

Applying the rule twice gives $(M_{ij}{\otimes }_R{M}_{kl}){\otimes }_R{M}_{mn}\cong {\delta }_{j,k}{\delta }_{l,m}{M}_{in}\cong M_{ij}{\otimes }_R(M_{kl}{\otimes }_R{M}_{mn}),$ where ${\delta }_{u,v}$ is $1$ if $u=v$ and $0$ otherwise. The regular bimodule decomposes as $R=M_{11}\oplus M_{22}$ and acts as a unit on every $M_{ij}$. These explicit products verify the associativity and unit rules in this finite split-semisimple example.

The cochain $\beta$ is normalized because it equals $1$ whenever any input is $0$. At $(g,h,k,l)=(1,2,3,1)$ in $C_4$, the five factors in [F1] are respectively $\beta (2,3,1)=1$, $\beta (3,3,1{)}^{-1}=1$, $\beta (1,1,1)=-1$, $\beta (1,2,0{)}^{-1}=1$, and $\beta (1,2,3)=1$. Thus $\omega (1,2,3,1)=-1$, so the resulting 4-cochain is not the constant cochain.

If $g=0$, the first two factors in $\delta \beta$ are equal with opposite exponents and the remaining three have an input $0$, so all remaining normalized values are $1$. If $h=0$, the second and third factors cancel and the other factors are normalized; if $k=0$, the third and fourth factors cancel and the other factors are normalized; if $l=0$, the fourth and fifth factors cancel and the other factors are normalized. Thus $\omega =\delta \beta$ is normalized. Applying the degree-four coboundary to this expression gives a product of the 30 composites $d_j{d}_i\beta$ from the six degree-four faces and five degree-three faces. For each $0\le i<j\le 5$, the simplicial face identity $d_j{d}_i=d_i{d}_{j-1}$ pairs two occurrences with opposite exponents; the interior face identities use associativity in $C_4$, and the endpoint identities delete the same first or last entry. All 15 pairs cancel, so $\delta \omega =1$ and $\omega$ is a 4-cocycle.

Theorem [F1] applies to $\omega$ and ${\omega }^{\mathrm{\prime }}=1$ with cochain ${\beta }^{-1}$ (whose coboundary is ${\omega }^{-1}$), and gives a monoidal 2-equivalence $2{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{C_4}^{\omega }\simeq 2{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{C_4}$. The example checks a nonconstant representative and its coherent untwisting; it does not assert a nonzero class in $H^4(C_4;{\mathbb{C}}^{\times })$.

Since $G$ has one element, every tuple in $G^4$ is $(e,e,e,e)$. Normalization forces $\omega (e,e,e,e)=1$, so the convolution grading has one summand and the pentagonator is the identity. Thus the underlying monoidal 2-category is the usual 2-category of finite semisimple $\mathbb{C}$-linear categories, linear functors, and natural transformations with Deligne tensor product.

The simple-object list in [F2] becomes the singleton $\{{\delta }_e\}$, which is also the monoidal unit. A finite semisimple category with $n$ simple labels is the direct sum of $n$ copies of ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{\mathbb{C}}$, so this is exactly the ordinary 2-vector-space example and its unique simple-object class.