# On the minimal extension and structure of weakly group-theoretical braided
fusion categories

Victor Ostrik and Zhiqiang Yu

###### Abstract

We show any slightly degenerate weakly group-theoretical fusion category
admits a minimal extension. Let $d$ be a positive square-free integer, given a
weakly group-theoretical non-degenerate fusion category $\mathcal{C}$, assume
that $\text{FPdim}(\mathcal{C})=nd$ and $(n,d)=1$. If
$(\text{FPdim}(X)^{2},d)=1$ for all simple objects $X$ of $\mathcal{C}$, then
we show that $\mathcal{C}$ contains a non-degenerate fusion subcategory
$\mathcal{C}(\mathbb{Z}_{d},q)$. In particular, we obtain that integral fusion
categories of FP-dimensions $p^{m}d$ such that
$\mathcal{C}^{\prime}\subseteq\text{sVec}$ are nilpotent and group-
theoretical, where $p$ is a prime and $(p,d)=1$.

Keywords: Minimal extension; slightly degenerate fusion category; weakly
group-theoretical fusion category

Mathematics Subject Classification 2020: 18M05 $\cdot$ 18M20

##  1 Introduction

In this paper, we always work over an algebraically closed field $\mathds{k}$
of characteristic zero.

A fusion category $\mathcal{C}$ is a semisimple $\mathds{k}$-linear finite
abelian tensor category, we use $\mathcal{O}(\mathcal{C})$ to denote the set
of isomorphism classes of simple objects of $\mathcal{C}$. Fusion category
$\mathcal{C}$ is a braided fusion category if $\mathcal{C}$ is equipped with a
braiding $c$. Let $\mathcal{D}\subseteq\mathcal{C}$ be a fusion subcategory of
braided fusion category $\mathcal{C}$. Recall that the centralizer
$\mathcal{D}_{\mathcal{C}}^{\prime}$ of $\mathcal{D}$ in $\mathcal{C}$ is the
fusion subcategory generated by simple objects $Y$ of $\mathcal{C}$ such that
$c_{Y,X}c_{X,Y}=\text{id}_{X\otimes Y}$, for all objects
$X\in\mathcal{O}(\mathcal{D})$, see [16]; in particular, we call
$\mathcal{C}^{\prime}:=\mathcal{C}_{\mathcal{C}}^{\prime}$ the Müger center of
$\mathcal{C}$.

Let $\mathcal{E}$ be an arbitrary symmetric fusion category, i.e.,
$\mathcal{E}^{\prime}=\mathcal{E}$. Recall that a braided fusion category over
$\mathcal{E}$ is a braided fusion category $\mathcal{C}$ equipped with a
braided tensor embedding $\mathcal{E}\to\mathcal{C}^{\prime}$, see e.g. [4,
2.9]. A braided fusion category over $\mathcal{E}$ is non-degenerate over
$\mathcal{E}$ if the embedding $\mathcal{E}\to\mathcal{C}^{\prime}$ is an
equivalence. In the special cases when $\mathcal{E}=\text{Vec}$ or
$\mathcal{E}=\text{sVec}$ is the category of finite dimensional vector or
super vector spaces over $\mathds{k}$ this specializes to the notions of non-
degenerate and slightly degenerate braided fusion categories.

A non-degenerate braided fusion category $\mathcal{A}$ is called a minimal
extension of a braided fusion category $\mathcal{C}$ if
$\mathcal{C}\subseteq\mathcal{A}$ and
$\mathcal{C}_{\mathcal{A}}^{\prime}=\mathcal{C}^{\prime}=:\mathcal{E}$. It was
conjectured that these minimal extensions exist for an arbitrary braided
fusion category $\mathcal{C}$ [16, Conjecture 5.2]. However, counterexamples
posed by Drinfeld [5] shows that this conjecture fails for some non-degenerate
fusion categories over certain Tannakian fusion categories $\mathcal{E}$
(recall that $\mathcal{E}$ is Tannakian if it is braided equivalent to
representation category $\text{Rep}(G)$ for some finite group $G$, see e.g.
[8]). The following special case of the minimal extension conjecture is still
open and is of great interest:

######  Conjecture 1.1.

Any slightly non-degenerate fusion category admits a minimal extension.

Note that it is proved in [15] and [2] that the set of all minimal extensions
(up to a suitable equivalence) of a given slightly non-degenerate category is
a torsor over group ${\mathbb{Z}}/16$. Conjecture 1.1 states that this torsor
is non-empty.

Our first main result is a proof of Conjecture 1.1 for a class of slightly
degenerate fusion categories. Recall that a fusion category $\mathcal{C}$ is
weakly group-theoretical if $\mathcal{C}$ is Morita equivalent to a nilpotent
fusion category $\mathcal{D}$ [12] (see section 2 for the definition of
nilpotency of fusion categories), equivalently, there exists a braided
equivalence between Drinfeld centers
$\mathcal{Z}(\mathcal{C})\cong\mathcal{Z}(\mathcal{D})$ [12, Theorem 1.3].

######  Theorem 1.2 (Theorem 3.5).

Every slightly degenerate weakly group-theoretical fusion category admits a
minimal extension.

In fact, let $\mathcal{C}$ be a braided fusion category such that
$\mathcal{C}^{\prime}=\mathcal{E}$. In Theorem 3.2, we show $\mathcal{C}$
admits a minimal extension if and only if $\mathcal{C}$ is Witt equivalent to
$\mathcal{E}\boxtimes\mathcal{D}$ for some non-degenerate fusion category
$\mathcal{D}$, the precise definition of Witt equivalence can be found in
section 2.

Our second main result is to give a sufficient condition when a non-degenerate
fusion category of FP-dimension $nd$ contains a pointed non-degenerate fusion
subcategory of FP-dimension $d$, where $n$ is a positive integer and $d$ a
square-free integer such that $(n,d)=1$. Explicitly,

######  Theorem 1.3 (Theorem 4.5).

Assume that $\mathcal{C}$ is a non-degenerate fusion category with
$\text{FPdim}(\mathcal{C})=nd$, if $\mathcal{C}$ contains a Tannakian
subcategory $\mathcal{E}=\text{Rep}(G)$ satisfying
$\mathcal{C}_{G}^{0}\cong\mathcal{C}(\mathbb{Z}_{d},q)\boxtimes\mathcal{A}$,
moreover for any object $V\in\mathcal{O}(\mathcal{C})$,
$(\text{FPdim}(V)^{2},d)=1$. Then
$\mathcal{C}\cong\mathcal{C}(\mathbb{Z}_{d},q)\boxtimes\mathcal{C}(\mathbb{Z}_{d},q)_{\mathcal{C}}^{\prime}$,
where $\mathcal{C}(\mathbb{Z}_{d},q)$ is the non-degenerate fusion category
determined by metric group $(\mathbb{Z}_{d},q)$.

Here, $\mathcal{C}_{G}^{0}$ is the de-equivariantization of
$\mathcal{E}_{\mathcal{C}}^{\prime}$ by $\mathcal{E}$ [8]. According to [17,
Theorem 3.1], integral non-degenerate weakly group-theoretical fusion
categories $\mathcal{C}$ of FP-dimension $nd$ satisfy conditions of Theorem
1.3, see Theorem 4.2. Besides, combined with Theorem 1.2, we can also extend
the conclusion of Theorem 1.3 to slightly degenerate weakly group-theoretical
fusion categories, see Corollary 4.8.

The paper is organized as follows. In section 2, we recall some basic
properties of fusion categories that we use throughout. In section 3, we give
a partial answer on minimal extension of slightly degenerate fusion categories
in Theorem 3.5. In section 4, we classify non-degenerate weakly group-
theoretical fusion categories of particular FP-dimensions in Theorem 4.5. In
particular, given a prime $p$ such that $(p,d)=1$, if
$\mathcal{C}^{\prime}\subseteq\text{sVec}$, then we obtain group-theoretical
property of integral fusion categories $\mathcal{C}$ of FP-dimensions $p^{m}d$
in Corollary 4.9.

This paper was written during a visit of the second named author at University
of Oregon supported by China Scholarship Council (Grant no.201806140143), he
appreciates the Department of Mathematics for their warm hospitality. The work
of V. O. was partially supported by the HSE University Basic Research Program,
Russian Academic Excellence Project ’5-100’ and by the NSF grant DMS-1702251.

##  2 Preliminaries

In this section, we recall some definitions and properties of fusion
categories and braided fusion categories, we refer the reader to [8, 9, 10,
12]. Let $\mathds{k}^{*}:=\mathds{k}\backslash{\\{0}\\}$,
$\mathbb{Z}_{r}:=\mathbb{Z}/r$, $r\in\mathbb{N}$.

###  2.1 Fusion categories

Let $\mathcal{C}$ be a fusion category, the cardinal of
$\mathcal{O}(\mathcal{C})$ is called rank of $\mathcal{C}$, and denoted by
$\text{rank}(\mathcal{C})$. Let $\text{Gr}(\mathcal{C})$ be the Grothendieck
ring of $\mathcal{C}$. Then there is a unique ring homomorphism FPdim(-) from
the Grothendieck ring $\text{Gr}(\mathcal{C})$ to $\mathds{k}$ such that
$\text{FPdim}(X)\geq 1$ is an algebraic integer for all objects
$X\in\mathcal{O}(\mathcal{C})$ [10, Theorem 8.6], and $\text{FPdim}(X)$ is
called the Frobenius-Perron dimension of object $X$. The Frobenius-Perron
dimension $\text{FPdim}(\mathcal{C})$ of fusion category $\mathcal{C}$ is
defined by

$\displaystyle\text{FPdim}(\mathcal{C}):=\sum_{X\in\mathcal{O}(\mathcal{C})}\text{FPdim}(X)^{2}.$

A simple object $X$ of fusion category $\mathcal{C}$ is invertible if
$\text{FPdim}(X)=1$. Fusion category $\mathcal{C}$ is a pointed fusion
category, if simple objects of $\mathcal{C}$ are all invertible. In this case,
$\mathcal{C}\cong\text{Vec}_{G}^{\omega}$, the category of finite-dimensional
$G$-graded vector spaces of $\mathds{k}$ where $G=\mathcal{O}(\mathcal{C})$ is
a finite group induced by fusion rules of $\mathcal{C}$ and $\omega\in
Z^{3}(G,\mathds{k}^{*})$ is a $3$-cocylce. In the following, we use
$G(\mathcal{C})$ to denote the group of isomorphism classes of invertible
objects of fusion category $\mathcal{C}$.

A fusion category $\mathcal{C}$ is weakly integral if
$\text{FPdim}(\mathcal{C})\in\mathbb{Z}$; $\mathcal{C}$ is integral if
$\text{FPdim}(X)\in\mathbb{Z}$, $\forall X\in\mathcal{O}(\mathcal{C})$. Let
$\mathcal{C}_{\text{ad}}$ be the adjoint fusion subcategory of $\mathcal{C}$,
i.e., $\mathcal{C}_{\text{ad}}$ is generated by simple objects $Y$ such that
$Y\subseteq X\otimes X^{*}$ for some simple object $X\in\mathcal{C}$. If
$\mathcal{C}$ is weakly integral then $\mathcal{C}_{\text{ad}}$ is integral
[10, Proposition 8.27].

Let $G$ be a finite group, for a $G$-graded fusion category
$\mathcal{C}=\oplus_{g\in G}\mathcal{C}_{g}$, we use the following notation to
denote the Frobenius-Perron dimension of component $\mathcal{C}_{g}$

$\displaystyle\text{FPdim}(\mathcal{C}_{g}):=\sum_{X\in\mathcal{O}(\mathcal{C}_{g})}\text{FPdim}(X)^{2}.$

If the $G$-grading of $\mathcal{C}$ is faithful, that is, for any $g\in G$,
the component $\mathcal{C}_{g}\neq 0$, then

$\displaystyle\text{FPdim}(\mathcal{C})=|G|\text{FPdim}(\mathcal{C}_{g})$

by [10, Proposition 8.20]. Any fusion category $\mathcal{C}$ has a faithful
grading with $\mathcal{C}_{\text{ad}}$ being its trivial component, the
grading is called the universal grading of $\mathcal{C}$. Moreover, for any
faithful $G$-grading of $\mathcal{C}=\oplus_{g\in G}\mathcal{C}_{g}$, we have
$\mathcal{C}_{\text{ad}}\subseteq\mathcal{C}_{e}$ [13, Corollary 3.7].

A fusion category $\mathcal{C}$ is said to be nilpotent if there exists a
natural number $n$ such that $\mathcal{C}^{(n)}=\text{Vec}$ [13], where Vec is
the category of finite-dimensional vector spaces over $\mathds{k}$, and

$\displaystyle\mathcal{C}^{(0)}:=\mathcal{C},~{}\mathcal{C}^{(1)}:=\mathcal{C}_{\text{ad}},~{}\mathcal{C}^{(j)}:=(\mathcal{C}^{(j-1)})_{\text{ad}},~{}j\geq
1.$

Obviously, pointed fusion categories are nilpotent, and fusion categories of
prime power FP-dimensions are also nilpotent [10, Theorem 8.28], for example.
A fusion category $\mathcal{C}$ is weakly group-theoretical if it is Morita
equivalent to a nilpotent fusion category $\mathcal{D}$; $\mathcal{C}$ is
group-theoretical if it is Morita equivalent to a pointed fusion category, see
[12] for details.

###  2.2 Braided fusion categories

A braided fusion category $\mathcal{C}$ is a fusion category with a braiding

$\displaystyle c_{X,Y}:X\otimes Y\overset{\sim}{\to}Y\otimes X,~{}\forall
X,Y\in\mathcal{C}.$

Two objects $X,Y\in\mathcal{C}$ are said to centralize each other, if

$\displaystyle c_{Y,X}c_{X,Y}=\text{id}_{X\otimes Y}.$

Let $\mathcal{D}\subseteq\mathcal{C}$ be a fusion subcategory. Then the
centralizer of $\mathcal{D}$ in $\mathcal{C}$ is the fusion subcategory
generated by objects of $\mathcal{C}$ that centralize every object of
$\mathcal{D}$. We call
$\mathcal{C}^{\prime}:=\mathcal{C}_{\mathcal{C}}^{\prime}$ the Müger center of
$\mathcal{C}$ [16]. Moreover, braided fusion category $\mathcal{C}$ is a pre-
modular fusion category if $\mathcal{C}$ is a spherical fusion category [9].

A braided fusion category $\mathcal{E}$ is symmetric if
$\mathcal{E}=\mathcal{E}^{\prime}$. For any symmetric fusion category
$\mathcal{E}$, there exists a finite group $G$ and a central element $u\in G$
such that $\mathcal{E}\cong\text{Rep}(G,u)$ [5], where $\text{Rep}(G,u)$ is
the category of finite-dimensional representation of $G$ and $u$ acts as
parity automorphism for any $X\in\mathcal{O}(\mathcal{E})$. A symmetric fusion
category $\mathcal{E}$ is a Tannakian fusion category if
$\mathcal{E}\cong\text{Rep}(G)$, the braiding of $\text{Rep}(G)$ is given by
reflection of vector spaces. A Tannakian subcategory
$\mathcal{E}\subseteq\mathcal{C}$ is maximal, if $\mathcal{E}$ is not
contained in any other Tannakian subcategory of $\mathcal{C}$.

We say that $\mathcal{C}$ is a fusion category over $\mathcal{E}$ if there
exists a braided tensor functor $F:\mathcal{E}\to\mathcal{Z}(\mathcal{C})$
such that $\mathcal{E}$ is mapped faithfully to $\mathcal{C}$ via functor
$\text{Forg}\circ F$, where
$\text{Forg}:\mathcal{Z}(\mathcal{C})\to\mathcal{C}$ is the forgetful functor
[8, Definition 4.16]. A fusion category $\mathcal{C}$ is a braided fusion
category over $\mathcal{E}$, if $\mathcal{C}$ is braided and
$\mathcal{E}\subseteq\mathcal{C}^{\prime}$. In particular, $\mathcal{C}$ be is
a non-degenerate fusion category over $\mathcal{E}$ if
$\mathcal{C}^{\prime}=\mathcal{E}$.

Given a braided fusion category $\mathcal{C}$, let
$\mathcal{E}=\text{Rep}(G)\subseteq\mathcal{C}$ be a non-trivial Tannakian
subcategory, then there exists a $G$-graded fusion category $\mathcal{C}_{G}$,
which is called the de-equivariantization of $\mathcal{C}$ by $\mathcal{E}$,
such that $\mathcal{C}\cong(\mathcal{C}_{G})^{G}$ as braided fusion
categories. In general, $\mathcal{C}_{G}$ is a braided $G$-crossed fusion
category, but the trivial component
$\mathcal{C}_{G}^{0}\cong\mathcal{E}^{\prime}_{G}$ is a braided fusion
category, where $\mathcal{E}^{\prime}$ is the centralizer of $\mathcal{E}$ in
$\mathcal{C}$. In particular, if $\mathcal{E}\subseteq\mathcal{C}^{\prime}$,
then $\mathcal{C}_{G}$ is a braided fusion category, and
$(\mathcal{C}_{G})^{\prime}\cong(\mathcal{C}^{\prime})_{G}$, see [8, section
4].

Let $\mathcal{C}$ be a braided fusion category. Then $\mathcal{C}$ is said to
be non-degenerate if its Müger center $\mathcal{C}^{\prime}=\text{Vec}$. A
pre-modular fusion category $\mathcal{C}$ is a modular fusion category if
$\mathcal{C}$ is non-degenerate. It is known that pointed non-degenerate
fusion categories are in bijective correspondence with metric groups $(G,q)$,
where $G$ is an abelian group, $q:G\to\mathds{k}^{*}$ is a non-degenerate
quadratic form, see [8, 9] for details. We use $\mathcal{C}(G,q)$ to denote
the pointed non-degenerate fusion category determined by the metric group
$(G,q)$ below.

A braided fusion category $\mathcal{C}$ is slightly degenerate if
$\mathcal{C}^{\prime}=\text{sVec}$, where sVec is the symmetric category of
finite-dimensional super vector spaces over $\mathds{k}$. We say a slightly
degenerate fusion category $\mathcal{C}$ splits, if there exists a braided
equivalence $\mathcal{C}\cong\text{sVec}\boxtimes\mathcal{D}$, where
$\mathcal{D}$ is a non-degenerate fusion category. Slightly degenerate pointed
fusion categories split [12, Proposition 2.6], for example. $\mathcal{C}$ is a
super-mdoular fusion category if $\mathcal{C}^{\prime}=\text{sVec}$ and
$\mathcal{C}$ is spherical. In particular, for any slightly degenerate fusion
category $\mathcal{C}$, if $\mathcal{C}$ contains a Tannakian subcategory
$\mathcal{E}=\text{Rep}(G)$, then $\mathcal{C}_{G}^{0}$ is again slightly
degenerate [8, Propositon 4.30].

###  2.3 Witt equivalence and Witt groups

Let $\mathcal{C}$ be a braided fusion category, $(A,m)\in\mathcal{C}$ a
commutative algebra, where $m:A\otimes A\to A$ is the multiplication such that
$m=m\circ c$. $A$ is a connected étale algebra if
$\text{Hom}_{\mathcal{C}}(I,A)\cong\mathds{k}$ and the category of right
$A$-modules $\mathcal{C}_{A}$ in $\mathcal{C}$ is a semisimple category [3,
Proposition 2.7, Definition 3.1]. In addition, we say two étale algebras
$A,B\in\mathcal{C}$ are centralizing each other if $c_{B,A}\circ
c_{A,B}=\text{id}_{A\otimes B}$.

Let $\mathcal{E}$ be a symmetric fusion category. By [3, Proposition 3.22],
the symmetric category $\mathcal{E}\boxtimes\mathcal{E}$ contains a connected
étale algebra $A$ such that
$(\mathcal{E}\boxtimes\mathcal{E})_{A}\cong\mathcal{E}$ as braided fusion
category. In fact, the étale algebra

$\displaystyle
A=\mathcal{I}(I):=\oplus_{X\in\mathcal{O}(\mathcal{E})}X\boxtimes X^{*},$

where $\mathcal{I}$ is the right adjoint functor of the surjective braided
tensor functor

$\displaystyle\otimes:\mathcal{E}\boxtimes\mathcal{E}\to\mathcal{E},~{}X\boxtimes
Y\mapsto X\otimes Y,~{}\forall X,Y\in\mathcal{E}.$

Then for any braided fusion categories $\mathcal{C},\mathcal{D}$ over
$\mathcal{E}$,

$\displaystyle\mathcal{C}\boxtimes_{\mathcal{E}}\mathcal{D}:=(\mathcal{C}\boxtimes\mathcal{D})_{A},$

is also a braided fusion category over $\mathcal{E}$, where
$(\mathcal{C}\boxtimes\mathcal{D})_{A}$ is the category of right $A$-modules
of $A$ in $\mathcal{C}\boxtimes\mathcal{D}$.

Recall that two non-degenerate fusion categories $\mathcal{C},\mathcal{D}$
over $\mathcal{E}$ is Witt equivalent if and only if there exists a fusion
category $\mathcal{A}$ over $\mathcal{E}$ such that there is a braided
equivalence

$\displaystyle\mathcal{C}\boxtimes_{\mathcal{E}}\mathcal{D}^{\text{rev}}\cong\mathcal{Z}(\mathcal{A},\mathcal{E}),$

where $\mathcal{Z}(\mathcal{A},\mathcal{E})$ is the centralizer of
$\mathcal{E}$ in Drinfeld center $\mathcal{Z}(\mathcal{A})$,
$\mathcal{D}^{\text{rev}}:=\mathcal{D}$ as fusion category, the braiding
$c^{\text{rev}}$ of $\mathcal{D}^{\text{rev}}$ is given by

$\displaystyle c^{\text{rev}}_{X,Y}:=c_{Y,X}^{-1},~{}\forall
X,Y\in\mathcal{D}^{\text{rev}}.$

The Witt equivalence of $\mathcal{C}$ will be denoted by $[\mathcal{C}]$, and
the Witt equivalence classes of non-degenerate fusion categories over
$\mathcal{E}$ form the Witt groups group $\mathcal{W}(\mathcal{E})$, we denote
$\mathcal{W}:=\mathcal{W}(\text{Vec})$ and
$s\mathcal{W}:=\mathcal{W}(\text{sVec})$, see [3, 4] for detail.

Given a braided fusion category $\mathcal{C}$ with
$\mathcal{C}^{\prime}\cong\mathcal{E}$, M. Müger conjectured that there is a
non-degenerate fusion category $\mathcal{A}$ such that
$\mathcal{C}\subseteq\mathcal{A}$ as braided fusion subcategory and
$\mathcal{C}_{\mathcal{A}}^{\prime}\cong\mathcal{E}$ [16, Conjecture 5.2], and
$\mathcal{A}$ is called a minimal extension of $\mathcal{C}$. Obviously, if a
slightly degenerate category $\mathcal{C}$ splits, then it has a minimal
extension. This fails for some non-trivial Tannakian fusion category
$\mathcal{E}$ by V. Drinfeld [5], however. Let $\mathcal{C}$ be a super-
modular fusion category, assume that $\mathcal{C}$ admits a minimal modular
extension $\mathcal{A}$, i.e., the minimal extension $\mathcal{A}$ of
$\mathcal{C}$ is a modular fusion category, then it was showed in [2, 15] that
$\mathcal{A}$ has exactly 16 modular equivalence classes, and these minimal
modular extensions are also Witt inequivalent [4].

##  3 On the minimal extension conjecture

In this section, we study the minimal extension for braided fusion categories,
and we show that every slightly degenerate weakly group-theoretical fusion
category admits a minimal extension.

To begin with, we need the following lemma, which is a direct result of [3,
Corollary 3.26].

######  Lemma 3.1.

Let $\mathcal{C}$ be a non-degenerate fusion category over $\mathcal{E}$, and
$\mathcal{D}$ a braided fusion category. Then we have a braided equivalence
$\mathcal{C}\boxtimes\mathcal{D}\cong\mathcal{C}\boxtimes_{\mathcal{E}}(\mathcal{E}\boxtimes\mathcal{D})$.

###### Proof.

Notice that we have an injective braided tensor functor

$\displaystyle\iota:\mathcal{C}\boxtimes\mathcal{D}\to\mathcal{C}\boxtimes(\mathcal{E}\boxtimes\mathcal{D}),~{}X\boxtimes
Y\mapsto X\boxtimes I\boxtimes Y,$

and a surjective braided tensor functor
$F:\mathcal{C}\boxtimes(\mathcal{E}\boxtimes\mathcal{D})\to\mathcal{C}\boxtimes_{\mathcal{E}}(\mathcal{E}\boxtimes\mathcal{D})$.
Then the composition
$F\circ\iota:\mathcal{C}\boxtimes\mathcal{D}\to\mathcal{C}\boxtimes_{\mathcal{E}}(\mathcal{E}\boxtimes\mathcal{D})$
is a braided tensor functor. By definition [4], we have

$\displaystyle\mathcal{C}\boxtimes_{\mathcal{E}}(\mathcal{E}\boxtimes\mathcal{D})=(\mathcal{C}\boxtimes\mathcal{E}\boxtimes\mathcal{D})_{A}$

where $A:=\mathcal{I}(I)=\oplus_{X\in\mathcal{O}(\mathcal{E})}X\boxtimes
X^{*}\boxtimes I$, and $\mathcal{I}$ is the right adjoint functor of tensor
functor $\otimes:\mathcal{E}\boxtimes\mathcal{E}\to\mathcal{E}$. While
$(\mathcal{C}\boxtimes\mathcal{D})\cap A\cong I$, therefore $F\circ\iota$ is
injective [15, Proposition 3.4], so $F\circ\iota$ is a braided equivalence by
[9, Proposition 6.3.3], as we have the following equations

$\displaystyle\text{FPdim}(\mathcal{C}\boxtimes\mathcal{D})=\text{FPdim}(\mathcal{C}\boxtimes_{\mathcal{E}}(\mathcal{E}\boxtimes\mathcal{D}))=\text{FPdim}(\mathcal{C})\text{FPdim}(\mathcal{D}),$

this finishes the proof. ∎

For any connected étale algebra $A\in\mathcal{C}$, a right $A$-module
$(M,\mu)$ is a dyslectic (or, local) $A$-module, if $\mu\circ c_{A,M}\circ
c_{M,A}=\mu$, where $\mu:M\otimes A\to M$ is the $A$-module structure of $M$.
We use $\mathcal{C}_{A}^{0}$ to denote the category of dyslectic modules of
$A$ [3].

######  Theorem 3.2.

Let $\mathcal{B}$ and $\mathcal{C}$ be two Witt equivalent non-degenerate
fusion categories over a symmetric fusion category $\mathcal{E}$, moreover
$\mathcal{B}\cong\mathcal{E}\boxtimes\mathcal{D}$ where $\mathcal{D}$ is non-
degenerate. Then $\mathcal{C}$ admits a minimal extension $\mathcal{A}$. In
addition, $[\mathcal{C}]=[\mathcal{E}\boxtimes\mathcal{A}]$.

###### Proof.

By assumption, there exists a fusion category $\mathcal{A}_{1}$ over
$\mathcal{E}$ such that we have a braided equivalence
$\mathcal{C}\boxtimes_{\mathcal{E}}\mathcal{B}^{\text{rev}}\cong\mathcal{Z}(\mathcal{A}_{1},\mathcal{E})$.
On the one hand, by Lemma 3.1, we have the following braided equivalences

$\displaystyle\mathcal{C}\boxtimes\mathcal{D}^{\text{rev}}\cong\mathcal{C}\boxtimes_{\mathcal{E}}\mathcal{B}^{\text{rev}}\cong\mathcal{Z}(\mathcal{A}_{1},\mathcal{E}).$

On the other hand, [8, Theorem 3.13] says that
$\mathcal{Z}(\mathcal{A}_{1})\cong\mathcal{D}^{\text{rev}}\boxtimes\mathcal{A}$,
where $\mathcal{A}$ is the centralizer of $\mathcal{D}^{\text{rev}}$ in
$\mathcal{Z}(\mathcal{A}_{1})$, and then $\mathcal{A}$ is also non-degenerate.
By [8, Theorem 3.10],

$\displaystyle\text{FPdim}(\mathcal{Z}(\mathcal{A}_{1}))=\text{FPdim}(\mathcal{E})\text{FPdim}(\mathcal{Z}(\mathcal{A}_{1},\mathcal{E})),$

hence we have equations

$\displaystyle\text{FPdim}(\mathcal{A})=\frac{\text{FPdim}(\mathcal{Z}(\mathcal{A}_{1}))}{\text{FPdim}(\mathcal{D})}=\frac{\text{FPdim}(\mathcal{E})\text{FPdim}(\mathcal{Z}(\mathcal{A}_{1},\mathcal{E}))}{\text{FPdim}(\mathcal{D})}=\text{FPdim}(\mathcal{E})\text{FPdim}(\mathcal{C}).$

We claim that $\mathcal{A}$ is a minimal extension of $\mathcal{C}$. Note that
we have an injective braided tensor functor from
$\mathcal{C}\boxtimes\mathcal{D}^{\text{rev}}$ to
$\mathcal{Z}(\mathcal{A}_{1})$, by definition $\mathcal{C}$ and
$\mathcal{D}^{\text{rev}}$ are centralizing each other in braided fusion
category $\mathcal{C}\boxtimes\mathcal{D}^{\text{rev}}$, hence they are still
centralizing each other in $\mathcal{Z}(\mathcal{A}_{1})$. Therefore, we have
an inclusion
$\mathcal{D}^{\text{rev}}\subseteq\mathcal{C}^{\prime}_{\mathcal{Z}(\mathcal{A}_{1})}$,
where $\mathcal{C}^{\prime}_{\mathcal{Z}(\mathcal{A}_{1})}$ is the centralizer
of $\mathcal{C}$ in $\mathcal{Z}(\mathcal{A}_{1})$. While centralizer of
$\mathcal{D}^{\text{rev}}$ in $\mathcal{Z}(\mathcal{A}_{1})$ is $\mathcal{A}$,
by [8, Corollary 3.11] $\mathcal{C}\subseteq\mathcal{A}$.

Let $A$ be the connected étale algebra such that
$(\mathcal{E}\boxtimes\mathcal{E})_{A}\cong\mathcal{E}$. Denote
$\mathcal{M}:=\mathcal{E}\boxtimes\mathcal{A}$, since $A\cap\mathcal{E}=I$,
$\mathcal{M}_{A}^{0}$ is also non-degenerate over $\mathcal{E}$ by [4,
Corollary 4.6], and we have the following braided equivalence

$\displaystyle\mathcal{M}\boxtimes_{\mathcal{E}}(\mathcal{M}_{A}^{0})^{\text{rev}}\cong\mathcal{Z}(\mathcal{M}_{A},\mathcal{E}).$

In particular,
$\text{FPdim}(\mathcal{M}_{A}^{0})=\frac{\text{FPdim}(\mathcal{M})}{\text{FPdim}(\mathcal{E})^{2}}=\text{FPdim}(\mathcal{C})$.
While $\mathcal{C}\cong\text{Vec}\boxtimes\mathcal{C}$ as a fusion subcategory
of $\mathcal{M}$ and $\mathcal{C}\cap A=I$, so there is a injective braided
tensor functor $F:\mathcal{C}\to\mathcal{M}_{A}^{0}$ [15, Proposition 3.4],
which must be an equivalence by [9, Proposition 6.3.3]. That is,

$\displaystyle\mathcal{M}\boxtimes_{\mathcal{E}}\mathcal{C}^{\text{rev}}\cong\mathcal{Z}(\mathcal{M}_{A},\mathcal{E}),$

so $[\mathcal{E}\boxtimes\mathcal{A}]=[\mathcal{C}]$ by definition, this
finishes the proof. ∎

######  Remark 3.3.

Let $\mathcal{C}$ and $\mathcal{D}$ Witt equivalent braided fusion categories.
If $\mathcal{C}$ has a minimal extension $\mathcal{A}$, then Theorem 3.2
implies that $\mathcal{D}$ is also Witt equivalent to
$\mathcal{E}\boxtimes\mathcal{A}$. Hence, $\mathcal{D}$ has a minimal
extension. Consequently, the property whether a braided fusion category admits
a minimal extension is a Witt invariant.

In addition, if $\mathcal{C}$ is a pre-modular fusion category, we don’t know
whether its minimal extensions $\mathcal{A}$ admit a spherical structure,
however.

Therefore, to prove minimal extension conjecture for non-degenerate fusion
categories over a symmetric fusion category $\mathcal{E}$ is equivalent to
show that the group homomorphism
$S_{\mathcal{E}}:\mathcal{W}\to\mathcal{W}(\mathcal{E})$ is surjective, where

$\displaystyle
S_{\mathcal{E}}:[\mathcal{A}]\mapsto[\mathcal{E}\boxtimes\mathcal{A}],~{}\text{for
all}~{}[\mathcal{A}]\in\mathcal{W},$

$[\mathcal{A}]$ is the Witt equivalence class of non-degenerate fusion
category $\mathcal{A}$, see also [4, Question 5.15]. When
$\mathcal{E}=\text{sVec}$, it is well-known
$\text{ker}(S_{\text{sVec}})=\mathcal{W}_{\text{Ising}}\cong\mathbb{Z}_{16}$
[4, Proposition 5.14], where $\mathcal{W}_{\text{Ising}}\subseteq\mathcal{W}$
is the subgroup generated by the Witt equivalence class of Ising fusion
category.

Assume that $\mathcal{E}=\text{Rep}(G)$ is a Tannakian fusion category, then
for any non-degenerate fusion category $\mathcal{C}$ over $\mathcal{E}$,
$\mathcal{C}_{G}$ is a non-degenerate fusion category [8, Proposition 4.56].
Thus, there is a well-defined group homomorphism $\phi_{\mathcal{E}}$ between
Witt groups $\mathcal{W}(\mathcal{E})$ and $\mathcal{W}$, where

$\displaystyle\phi_{\mathcal{E}}:\mathcal{W}(\mathcal{E})\to\mathcal{W},~{}[\mathcal{C}]\mapsto[\mathcal{C}_{G}],~{}\text{for
all}~{}[\mathcal{C}]\in\mathcal{W}(\mathcal{E}),$

see [4, Remark 5.8]. For any non-degenerate fusion category $\mathcal{A}$,
note that $(\mathcal{E}\boxtimes\mathcal{A})_{G}\cong\mathcal{A}$ as braided
fusion category, so
$\phi_{\mathcal{E}}(S_{\mathcal{E}}([\mathcal{A}]))=[\mathcal{A}]$. Therefore,

######  Corollary 3.4.

Let $\mathcal{E}=\text{Rep}(G)$ be a Tannakian fusion category. Then
$\phi_{\mathcal{E}}\circ S_{\mathcal{E}}=\text{id}$, so there exists a split
exact sequence

$\displaystyle
1\to\text{ker}(\phi_{\mathcal{E}})\overset{i}{\to}\mathcal{W}(\mathcal{E})\underset{S_{\mathcal{E}}}{\overset{\phi_{\mathcal{E}}}{\rightleftarrows}}\mathcal{W}\to
1.$

Hence $S_{\mathcal{E}}$ is injective for all Tannakian fusion categories
$\mathcal{E}$. However, Drinfeld’s counterexample [5] shows that
$S_{\mathcal{E}}$ is not surjective for some Tannakian categories
$\mathcal{E}$, in general. In [18], we will continue to consider the
structures of subgroup $\text{ker}(\phi_{\mathcal{E}})$ and Witt group
$s\mathcal{W}$.

Recall that a braided fusion category $\mathcal{C}$ is said to be anisotropic
if $\mathcal{C}$ does not contain any non-trivial Tannakian subcategory; while
$\mathcal{C}$ is weakly anisotropic, if $\mathcal{C}$ does not contain a non-
trivial Tannakian subcategory (if exists) that is invariant under all braided
auto-equivalences of $\mathcal{C}$ [8].

######  Theorem 3.5.

Let $\mathcal{C}$ be a slightly degenerate weakly group-theoretical fusion
category. Then $\mathcal{C}$ admits a minimal extension.

###### Proof.

By Theorem 3.2, it suffices to show that $\mathcal{C}$ is Witt equivalent to a
split slightly degenerate fusion category. When $\mathcal{C}$ is a pointed
fusion category, then [12, Proposition 2.6] means
$\mathcal{C}\cong\text{sVec}\boxtimes\mathcal{D}$, where $\mathcal{D}$ is a
pointed non-degenerate fusion category. Assume $\mathcal{C}$ is not a pointed
fusion category below.

On the one hand, if $\mathcal{C}$ is weakly anisotropic, then $\mathcal{C}$ is
braided equivalent to $\mathcal{B}\boxtimes\mathcal{D}$ by [17, Theorem 1.1],
where $\mathcal{B}\cong\text{Vec}$ or $\mathcal{B}$ is equivalent to an Ising
category $\mathcal{I}$, $\mathcal{D}$ is a pointed fusion category. So
$\mathcal{B}$ is non-degenerate and $\mathcal{D}$ is slightly degenerate.
While [12, Proposition 2.6] implies
$\mathcal{D}\cong\text{sVec}\boxtimes\mathcal{D}_{0}$, where $\mathcal{D}_{0}$
is a pointed non-degenerate fusion category, therefore
$\mathcal{C}\cong\text{sVec}\boxtimes(\mathcal{B}\boxtimes\mathcal{D}_{0})$.
That is, $\mathcal{C}$ splits and admits a minimal extension.

On the other hand, if $\mathcal{C}$ is not a weakly anisotropic fusion
category, then $\mathcal{C}$ contains a maximal Tannakian subcategory
$\text{Rep}(G)$ such that slightly degenerate fusion category
$\mathcal{C}_{G}^{0}$ is a weakly anisotropic fusion category by [8, Corollary
5.19]. By [4, Corollary 4.6], we have a braided fusion categories tensor
equivalence
$\mathcal{C}\boxtimes_{\text{sVec}}(\mathcal{C}_{G}^{0})^{\text{rev}}\cong\mathcal{Z}(\mathcal{C}_{G},\text{sVec})$.
Thus, $\mathcal{C}$ and $\mathcal{C}_{G}^{0}$ are Witt equivalent by
definition, consequently $\mathcal{C}$ admits a minimal extension. ∎

Hence, let $\mathcal{D}$ be a slightly degenerate fusion category, if
$\mathcal{D}$ is Witt equivalent to a weakly group-theoretical fusion
category, then Theorem 3.2 and Theorem 3.5 together show $\mathcal{D}$ admits
a minimal extension.

######  Remark 3.6.

In fact, the proof of Theorem 3.2 implies that the minimal extension
$\mathcal{A}$ in Theorem 3.5 is a braided fusion subcategory of
$\mathcal{Z}(\mathcal{C}_{G})$. Moreover, $\mathcal{A}$ is weak also a weakly
group-theoretical fusion category [12, Proposition 4.1]. In particular,
$\mathcal{A}$ is integral if $\mathcal{C}$ is integral.

##  4 Classifications of certain non-degenerate weakly group-theoretical
fusion categories

In this section, we study the structure of non-degenerate and slightly
degenerate weakly group-theoretical fusion categories of particular Frobenius-
Perron dimensions.

Let $G$ be a finite group. Given a $G$-graded fusion category
$\mathcal{C}=\oplus_{g\in G}\mathcal{C}_{g}$, assume
$\mathcal{C}_{e}\cong\text{Vec}_{N}^{\omega}$, where $N$ is a finite group and
$\omega\in Z^{3}(N,\mathds{k}^{*})$ is a $3$-cocycle. By definition, the
$G$-grading structure of $\mathcal{C}$ induces an action of group $N$ on sets
$\mathcal{O}(\mathcal{C}_{g})$ for all $g\in G$.

The following lemma is trivial, we include it for the reader’s interest.

######  Lemma 4.1.

Let $\mathcal{C}=\oplus_{g\in G}\mathcal{C}_{g}$ be a faithful $G$-graded
fusion category, and $\mathcal{C}_{e}\cong\text{Vec}_{N}^{\omega}$ is pointed,
where $G,N$ are finite groups and $\omega\in Z^{3}(N,\mathds{k}^{*})$ is a
$3$-cocycle. Then for any element $g\in G$, N acts transitively on
$\mathcal{O}(\mathcal{C}_{g})$.

###### Proof.

By definition, for any $g\in G$, let $X\in\mathcal{C}_{g}$ be an arbitrary
simple object, there exists a subgroup $H$ of $N$ such that $X\otimes
X^{*}=\oplus_{h\in H}h$, then $\text{FPdim}(X)^{2}=|H|$. Note that the orbit
of $X$ has a size of $[N:H]$, while by [10, Proposition 8.20]

$\displaystyle\text{FPdim}(\mathcal{C}_{g})=\text{FPdim}(\mathcal{D})=|N|=[N:H]\text{FPdim}(X)^{2},$

which then shows the orbit of $X$ is exactly $\mathcal{O}(\mathcal{C}_{g})$. ∎

Now we are ready to give our first classification theorem of non-degenerate
fusion categories.

######  Theorem 4.2.

Let d be a square-free positive integer. Assume that $\mathcal{C}$ is an
integral non-degenerate weakly group-theoretical fusion category such that
$\text{FPdim}(\mathcal{C})=nd$ and $(n,d)=1$. Then
$\mathcal{C}\cong\mathcal{C}(\mathbb{Z}_{d},q)\boxtimes\mathcal{C}(\mathbb{Z}_{d},q)_{\mathcal{C}}^{\prime}$
as braided fusion category.

###### Proof.

Assume that $\mathcal{C}$ is not pointed, otherwise the result is trivial.
Since $\mathcal{C}$ is a weakly group-theoretical integral fusion category, by
[17, Theorem 3.1] $\mathcal{C}$ contains a non-trivial Tannakian subcategory.
Let $\mathcal{E}=\text{Rep}(G)$ be a maximal Tannakian subcategory of
$\mathcal{C}$, then $\mathcal{C}_{G}$ is a $G$-crossed braided fusion category
whose trivial component is braided equivalent to $\mathcal{C}_{G}^{0}$.
Moreover, the $G$-grading of $\mathcal{C}_{G}$ is faithful and
$\mathcal{C}_{G}^{0}$ is a non-degenerate fusion category by [8, Proposition
4.56]. Meanwhile, [8, Corollary 5.19] says $\mathcal{C}_{G}^{0}$ is weakly
anisotropic, therefore, [17, Theorem 1.1] means $\mathcal{C}_{G}^{0}$ is a
pointed fusion category. So $\mathcal{C}_{G}^{0}$ contains a pointed non-
degenerate fusion category $\mathcal{C}(\mathbb{Z}_{d},q)$, since $|G|^{2}$
divides $\text{FPdim}(\mathcal{C})$.

Let $\mathcal{B}:=\mathcal{C}_{G}$ and $\mathcal{B}=\oplus_{g\in
G}\mathcal{B}_{g}$ be the $G$-grading of $\mathcal{B}$, then
$\mathcal{B}_{\text{ad}}\subseteq\mathcal{B}_{e}$ by universal property of
$\mathcal{B}_{\text{ad}}$, so $\mathcal{B}$ is nilpotent. Hence, for any $g\in
G$ and any object $X\in\mathcal{O}(\mathcal{B}_{g})$, we have that
$\text{FPdim}(X)^{2}$ divides $\text{FPdim}(\mathcal{B}_{e})$ [13, Corollary
5.3], so $(\text{FPdim}(X),d)=1$. By Lemma 4.1
$\text{rank}(\mathcal{B}_{g})=\frac{\text{FPdim}(\mathcal{B}_{e})}{\text{FPdim}(X)^{2}}$,
in particular $d|\text{rank}(\mathcal{B}_{g})$. Since
$\mathcal{B}_{e}=\mathcal{C}_{G}^{0}$ is a non-degenerate fusion category, by
[14, Lemma 10.7] $\text{rank}(\mathcal{B}_{g})$ is equal to the cardinal of
$\mathcal{O}(\mathcal{C}^{0}_{G})^{g}$, the set of isomorphism classes of
simple objects of $\mathcal{C}^{0}_{G}$ that are fixed by $g$. Note that $g$
acts as a braided tensor equivalence on $\mathcal{C}_{G}^{0}$, thus
$\mathcal{O}(\mathcal{C}_{G}^{0})^{g}$ is an abelian group and contains
$\mathbb{Z}_{d}$ as a subgroup, which then implies that $G$ acts trivially on
$\mathcal{C}(\mathbb{Z}_{d},q)$. For $(d,|G|)=1$ and
$H^{i}(\mathbb{Z}_{d},G)=0$ for all non-negative integer $i$, fusion category
$\mathcal{E}^{\prime}=(\mathcal{C}_{G}^{0})^{G}\subseteq\mathcal{C}$ contains
a fusion subcategory
$\mathcal{C}(\mathbb{Z}_{d},q)^{G}\cong\text{Rep}(G)\boxtimes\mathcal{C}(\mathbb{Z}_{d},q)$.

Therefore, $\mathcal{C}$ contains a non-degenerate fusion subcategory
$\mathcal{C}(\mathbb{Z}_{d},q)$. By [8, Theorem 3.13] we have a braided tensor
equivalence
$\mathcal{C}\cong\mathcal{C}(\mathbb{Z}_{d},q)\boxtimes\mathcal{D}$, where
$\mathcal{D}$ is the centralizer of $\mathcal{C}(\mathbb{Z}_{d},q)$, and
$\mathcal{D}$ is also a non-degenerate fusion category. ∎

Next we will generalize Theorem 4.2 to strictly weakly integral fusion
categories. While in this case, it was shown in [17, Theorem 1.1] that if a
braided weakly group-theoretical fusion category $\mathcal{C}$ is weakly
anisotropic, $\mathcal{C}$ needs not be a pointed fusion category. Hence, the
proof of Theorem 4.2 fails in general.

Let $\mathcal{E}=\text{Rep}(G)$ be a Tannakian fusion category. Assume that

$\displaystyle
B:=\mathcal{I}(I)=\oplus_{X\in\mathcal{O}(\mathcal{E})}X\boxtimes X^{*}$

is the connected étale algebra such that
$(\mathcal{E}\boxtimes\mathcal{E})_{B}\cong\mathcal{E}$ as braided fusion
categories [4, Remark 2.11]. Let $\mathcal{B}$ and $\mathcal{C}$ be braided
fusion categories, which contain $\mathcal{E}$ as a fusion subcategory. Since
$\mathcal{E}\boxtimes\mathcal{E}\subseteq\mathcal{C}\boxtimes\mathcal{B}$, we
have a braided fusion category $(\mathcal{C}\boxtimes\mathcal{B})_{B}^{0}$ by
[8, Proposition 4.30], we denote it by
$\mathcal{C}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B}$, for simplification
of notation.

Given two $G$-crossed braided fusion categories $\mathcal{C}_{G}$ and
$\mathcal{D}_{G}$, notice that the fusion category
$\mathcal{C}_{G}\boxtimes\mathcal{B}_{G}$ is graded by group $G\times G$ with
trivial component $\mathcal{C}_{G}^{0}\boxtimes\mathcal{B}_{G}^{0}$. Let
$\mathcal{C}_{G}\boxtimes_{G}\mathcal{B}_{G}$ denote the fusion subcategory of
$\mathcal{C}_{G}\boxtimes\mathcal{B}_{G}$ graded by group $H$ of diagonal
elements of $G\times G$, so $H\cong G$. Then there exists a well-defined
action of $G$ on the braided fusion subcategory
$\mathcal{C}_{G}^{0}\boxtimes\mathcal{B}_{G}^{0}$, and naturally
$\mathcal{C}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B}\cong(\mathcal{C}_{G}\boxtimes_{G}\mathcal{B}_{G})^{G}$
as braided fusion categories by [11, Theorem 7.12].

In addition, $\mathcal{C}\boxtimes_{\mathcal{E}}\mathcal{B}$ contains a
Tannakian subcategory
$(\mathcal{E}\boxtimes\mathcal{E})_{B}\cong\text{Rep}(G)$, then it follows
from [8, Lemma 4.32] that there is a braided fusion category equivalence

$\displaystyle(\mathcal{C}\boxtimes_{\mathcal{E}}\mathcal{B})_{G}^{0}\cong(\mathcal{C}\boxtimes\mathcal{B})_{G\times
G}^{0},$

which then is equivalent to $\mathcal{C}^{0}_{G}\boxtimes\mathcal{B}^{0}_{G}$.

######  Proposition 4.3.

Let $A,B$ be two connected étale algebras in a non-degenerate fusion category
$\mathcal{C}$. If $A\cap B=I$, moreover $A,B$ are centralizing each other,
then we have the following braided fusion categories equivalences

$\displaystyle(\mathcal{C}_{B}^{0})_{A\otimes B}^{0}\cong\mathcal{C}_{A\otimes
B}^{0}\cong(\mathcal{C}_{A}^{0})_{A\otimes B}^{0}.$

###### Proof.

As étale algebras $A,B$ are centralizing each other, so $A\otimes B$ is also a
commutative algebra; since étale algebras are self-dual [3, Remark 3.4],
$\text{Hom}_{\mathcal{C}}(I,A\otimes B)\cong\text{Hom}_{\mathcal{C}}(A,B)$ is
one-dimensional, hence $A\otimes B$ is connected. It is easy to see that
$A\otimes B$ is a commutative algebra over $A$, hence we have an abelian
category equivalence $(\mathcal{C}_{A})_{A\otimes B}\cong\mathcal{C}_{A\otimes
B}$, so $A\otimes B$ is a connected étale algebra by [3, Proposition 2.7,
Proposition 3.16]. By [3, Corollary 3.30, Corollary 3.32]
$(\mathcal{C}_{A}^{0})_{A\otimes B}^{0}$ and $(\mathcal{C}_{B}^{0})_{A\otimes
B}^{0}$ are non-degenerate fusion categories with same Frobenius-Perron
dimension. It follows from [9, Proposition 6.3.3] and [3, Corollary 3.26] that
$\mathcal{C}_{A\otimes B}^{0}\cong(\mathcal{C}_{A}^{0})_{A\otimes B}^{0}$ as
braided fusion categories, then $(\mathcal{C}_{B}^{0})_{A\otimes
B}^{0}\cong\mathcal{C}_{A\otimes B}^{0}\cong(\mathcal{C}_{A}^{0})_{A\otimes
B}^{0}$. ∎

######  Lemma 4.4.

Let $\mathcal{C},\mathcal{D}$ be braided fusion categories. Assume that there
exists a Tannakian category $\text{Rep}(G)$ such that
$\mathcal{C}_{G}^{0}\cong\mathcal{D}_{G}^{0}$, then
$\mathcal{C}\cong\mathcal{D}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{Z}(\text{Vec}_{G}^{\omega})$
for some $\omega\in Z^{3}(G,\mathds{k}^{*})$.

###### Proof.

Since we have a braided equivalence
$\mathcal{C}_{G}^{0}\cong\mathcal{D}_{G}^{0}$, the $G$-crossed braided fusion
categories $\mathcal{C}_{G}$ and $\mathcal{D}_{G}$ are differed from a
3-cocycle $\omega\in Z^{3}(G,\mathds{k}^{*})$ by [11, Theorem 8.9]. While, by
definition, the associator of
$\mathcal{D}_{G}\boxtimes_{G}\text{Vec}_{G}^{\omega}$ differs from that of
$\mathcal{D}_{G}$ by the $3$-cocylce $\omega$. That is, we have an equivalence
of $G$-crossed braided fusion categories
$\mathcal{D}_{G}\boxtimes_{G}\text{Vec}_{G}^{\omega}\cong\mathcal{C}_{G}$.
Consequently, we have the following braided tensor equivalences

$\displaystyle\mathcal{D}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{Z}(\text{Vec}_{G}^{\omega})\cong(\mathcal{D}_{G}\boxtimes_{G}\text{Vec}_{G}^{\omega})^{G}\cong(\mathcal{C}_{G})^{G}\cong\mathcal{C},$

as desired. This finishes the proof of the lemma. ∎

For any braided fusion category $\mathcal{C}$, the equivalence classes of
invertible $\mathcal{C}$-module categories form a group, which is called the
Picard group $\text{Pic}(\mathcal{C})$ of $\mathcal{C}$ [11, section 4]. In
addition, if $\mathcal{C}$ is a non-degenerate fusion category, then [11,
Theorem 5.2] says that the group
$\text{Aut}^{\text{br}}_{\otimes}(\mathcal{C})$ of braided tensor equivalences
of $\mathcal{C}$ is isomorphic to $\text{Pic}(\mathcal{C})$.

######  Theorem 4.5.

Let $\mathcal{C}$ be a non-degenerate fusion category with
$\text{FPdim}(\mathcal{C})=nd$, where $n$ is a positive integer, and $d$ is a
square-free integer such that $(n,d)=1$. Assume that $\mathcal{C}$ contains a
Tannakian subcategory $\mathcal{E}=\text{Rep}(G)$ satisfying
$\mathcal{C}_{G}^{0}\cong\mathcal{C}(\mathbb{Z}_{d},q)\boxtimes\mathcal{A}$,
moreover for any $V\in\mathcal{O}(\mathcal{C})$, $(\text{FPdim}(V)^{2},d)=1$.
Then
$\mathcal{C}\cong\mathcal{C}(\mathbb{Z}_{d},q)\boxtimes\mathcal{C}(\mathbb{Z}_{d},q)_{\mathcal{C}}^{\prime}$
as braided fusion category.

###### Proof.

Note that $G$ acts as braided equivalence on non-degenerate fusion category
$\mathcal{C}_{G}^{0}$, then it preserves group $\mathbb{Z}_{d}$, which then
induces a group homomorphism

$\displaystyle\rho:G\to\text{Pic}(\mathcal{C}(\mathbb{Z}_{d},q))\cong\text{Aut}^{\text{br}}_{\otimes}(\mathcal{C}(\mathbb{Z}_{d},q)).$

Since $|G|^{2}$ divides $\text{FPdim}(\mathcal{C})$ and $d$ is square-free, we
have $(|G|,d)=1$, and then $H^{m}(G,\mathbb{Z}_{d})=0$ for all nonnegative
integers $m$. Particularly, the $3$-obstruction $o_{3}(\rho)$ in
$H^{3}(G,\mathbb{Z}_{d})$ vanishes, so we can lift $\rho$ to a
$2$-homomorphism
$\underline{\rho}:\underline{G}\to\underline{\text{Pic}(\mathcal{C}(\mathbb{Z}_{d},q))}$
by [11, Corollary 7.9]. Meanwhile fusion subcategory $\mathcal{A}$ is exactly
the centralizer of $\mathcal{C}(\mathbb{Z}_{d},q)$ in $\mathcal{C}_{G}^{0}$,
hence $G$ also preserves $\mathcal{A}$, hence we have another well-defined
$2$-homomorphism $\underline{G}\to\underline{\text{Pic}(\mathcal{A})}$. Then
there exists an action of $G\times G$ on
$\mathcal{C}(\mathbb{Z}_{d},q)\boxtimes\mathcal{A}$, with $(g,h)\mapsto
g\boxtimes h$, $g,h\in G$.

To obtain a $G$-crossed extension of $\mathcal{C}(\mathbb{Z}_{d},q)$, by [11,
Theorem 7.12] we have to lift $\rho$ to a $3$-homomorphism
$\underline{\underline{\rho}}:\underline{\underline{G}}\to\underline{\underline{\text{Pic}(\mathcal{C}(\mathbb{Z}_{d},q))}}$,
which results in a $4$-obstruction $o_{4}(\rho)$ in the group
$H^{4}(G,\mathds{k}^{*})$, but the obstruction vanishes by [11, Theorem 8.16].
Let $\widetilde{\mathcal{C}(\mathbb{Z}_{d},q)}$ be the faithful braided
$G$-crossed extension of $\mathcal{C}(\mathbb{Z}_{d},q)$ determined by
$\underline{\underline{\rho}}$. Then it follows from [8, Proposition 4.56]
that $\widetilde{\mathcal{C}(\mathbb{Z}_{d},q)}^{G}$ is a non-degenerate
fusion category. Let
$\mathcal{B}:=\widetilde{\mathcal{C}(\mathbb{Z}_{d},q^{-1})}^{G}$, where
$q^{-1}$ is the inverse quadratic form of $q$ on $\mathbb{Z}_{d}$, that is,
$q^{-1}(v):=q(v)^{-1}$, for all $v\in\mathbb{Z}_{d}$. Obviously,
$\mathcal{C}(\mathbb{Z}_{d},q^{-1})\cong\mathcal{C}(\mathbb{Z}_{d},q)^{\text{rev}}$
and
$\mathcal{B}^{\text{rev}}\cong\widetilde{\mathcal{C}(\mathbb{Z}_{d},q)}^{G}$,
where $\mathcal{C}(\mathbb{Z}_{d},q)^{\text{rev}}$ and
$\mathcal{B}^{\text{rev}}$ are non-degenerate fusion categories with reverse
braiding, respectively.

Next we show that there exists a non-degenerate fusion category $\mathcal{D}$
of FP-dimension $nd$ which also contains $\mathcal{C}(\mathbb{Z}_{d},q)$ such
that $\mathcal{D}_{G}^{0}\cong\mathcal{C}_{G}^{0}$. To do this, we first show
that there exists a non-degenerate fusion category $\widetilde{\mathcal{A}}$
such that $\widetilde{\mathcal{A}}_{G}^{0}\cong\mathcal{A}$. Since
$\mathcal{E}\boxtimes\mathcal{E}\subseteq\mathcal{C}\boxtimes\mathcal{B}$, we
have a non-degenerate fusion category
$\mathcal{C}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B}:=(\mathcal{C}\boxtimes\mathcal{B})_{B}^{0}$
by [3, Corollary 3.30]. Note that
$\mathcal{C}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B}$ contains a Tannakian
subcategory $\mathcal{E}\cong\text{Rep}(G)$, moreover we show that
$\mathcal{C}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B}$ contains a
$G$-equivariant connected étale algebra $A$ of FP-dimension $d$. In fact, by
definition
$\mathcal{C}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B}\cong(\mathcal{C}_{G}\boxtimes_{G}\mathcal{B}_{G})^{G}$,
and

$(\mathcal{C}^{0}_{G}\boxtimes\mathcal{B}^{0}_{G})\cong\mathcal{A}\boxtimes\mathcal{C}(\mathbb{Z}_{d},q)\boxtimes\mathcal{C}(\mathbb{Z}_{d},q^{-1})\cong\mathcal{A}\boxtimes\mathcal{Z}(\mathcal{C}(\mathbb{Z}_{d},q)),$

invertible diagonal simple objects of
$\mathcal{C}(\mathbb{Z}_{d},q)\boxtimes\mathcal{C}(\mathbb{Z}_{d},q^{-1})$
generate a Tannakian subcategory $\mathcal{H}\cong\text{Rep}(\mathbb{Z}_{d})$,
which obviously admits a $G$-equivariant structure by [11, Theorem 9.5]. Hence
we have an exact sequence of finite groups
$1\to\mathbb{Z}_{d}\to\widetilde{G}\to G\to 1$, which splits as $(d,|G|)=1$,
so $\widetilde{G}\cong\mathbb{Z}_{d}\rtimes G$. Consequently, there is a
braided equivalence $\mathcal{H}^{G}\cong\text{Rep}(\widetilde{G})$, étale
algebra $A$ is exactly the regular algebra $\text{Fun}(\widetilde{G}/G)$,
which is a $G$-equivariant connected étale algebra. Then there is another non-
degenerate fusion category
$(\mathcal{C}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B})_{A}^{0}$ by [3,
Corollary 3.30].

We claim that non-degenerate fusion category
$\widetilde{\mathcal{A}}=(\mathcal{C}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B})_{A}^{0}$
is exactly the fusion category that we wanted. In fact, [8, Lemma 4.32] says
that there are braided equivalences of non-degenerate fusion categories

$\displaystyle(\mathcal{C}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B})_{G}^{0}\cong(\mathcal{C}\boxtimes\mathcal{B})_{G\times
G}^{0}\cong\mathcal{C}^{0}_{G}\boxtimes\mathcal{B}^{0}_{G}\cong\mathcal{A}\boxtimes\mathcal{C}(\mathbb{Z}_{d},q)\boxtimes\mathcal{C}(\mathbb{Z}_{d},q^{-1}).$

Since $A\in(\mathcal{C}^{0}_{G}\boxtimes\mathcal{B}^{0}_{G})^{G}$ is a
$G$-equivariant étale algebra, and centralizer of $\mathcal{E}=\text{Rep}(G)$
inside $\mathcal{C}\boxtimes_{\mathcal{E}}\mathcal{B}$ is
$(\mathcal{C}^{0}_{G}\boxtimes\mathcal{B}^{0}_{G})^{G}$, so $\mathcal{E}$ and
$A$ are centralizing each other. Hence, as a braided fusion category,
$((\mathcal{C}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B})_{A}^{0})_{G}^{0}$
is equivalent to
$((\mathcal{C}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B})_{G}^{0})_{A}^{0}$
by Proposition 4.3. Then we have braided equivalences of non-degenerate fusion
categories

$\displaystyle((\mathcal{C}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B})_{A}^{0})_{G}^{0}\cong((\mathcal{C}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B})_{G}^{0})_{A}^{0}\cong(\mathcal{A}\boxtimes\mathcal{C}(\mathbb{Z}_{d},q)\boxtimes\mathcal{C}(\mathbb{Z}_{d},q^{-1}))_{A}^{0}\cong\mathcal{A}.$

Taking
$\mathcal{D}:=(\mathcal{C}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B})_{A}^{0}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B}^{\text{rev}}$,
consequently, [3, Corollary 3.30] shows that $\mathcal{D}$ is also a non-
degenerate fusion category. Moreover, we have the following braided
equivalences of non-degenerate fusion categories

$\displaystyle((\mathcal{C}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B})_{A}^{0}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B}^{\text{rev}})_{G}^{0}\cong((\mathcal{C}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B})_{A}^{0}\boxtimes\mathcal{B}^{\text{rev}})_{G\times
G}^{0}\cong((\mathcal{C}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{B})_{A}^{0})_{G}^{0}\boxtimes(\mathcal{B}^{\text{rev}})_{G}^{0}\cong\mathcal{A}\boxtimes\mathcal{C}(\mathbb{Z}_{d},q).$

By Lemma 4.4,
$\mathcal{C}\cong\mathcal{D}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{Z}(\text{Vec}_{G}^{\omega})$
for some $\omega\in Z^{3}(G,\mathds{k}^{*})$. In the last, we show that
$\mathcal{D}$ then $\mathcal{C}$ contains $\mathcal{C}(\mathbb{Z}_{d},q)$ as a
fusion subcategory.

Now we show that $G$ acts on $\mathcal{C}(\mathbb{Z}_{d},q)$ trivially. If
not, then by [14, Lemma 10.7], there exists a non-trivial component of the
$G$-crossed grading of $\widetilde{\mathcal{C}(\mathbb{Z}_{d},q)}$ which has
rank less than $d$. Since $\mathcal{C}(\mathbb{Z}_{d},q)$ is the trivial
component of $\widetilde{\mathcal{C}(\mathbb{Z}_{d},q)}$, and the $G$-grading
of $\widetilde{\mathcal{C}(\mathbb{Z}_{d},q)}$ is faithful, there exists non-
invertible simple objects of $\widetilde{\mathcal{C}(\mathbb{Z}_{d},q)}$ with
FP-dimensions $\sqrt{t}$ by [13, Corollary 5.3], where $t>1$ is an integer and
$t|d$. While previous equivalence
$\mathcal{C}\cong\mathcal{D}\widehat{\boxtimes}_{\mathcal{E}}\mathcal{Z}(\text{Vec}_{G}^{\omega})$
means that $\mathcal{C}$ contains a simple object $V$ of FP-dimension
$s\sqrt{t}$ for some algebraic integer $s$ with $s^{2}\in\mathbb{Z}$, this
contradicts to the assumption $(\text{FPdim}(V)^{2},d)=1$.

Then $G$ acts on $\mathcal{C}(\mathbb{Z}_{d},q)$ trivially, and each component
of the $G$-crossed grading of $\widetilde{\mathcal{C}(\mathbb{Z}_{d},q)}$ has
rank $d$ by [14, Lemma 10.7]. Consequently,
$\widetilde{\mathcal{C}(\mathbb{Z}_{d},q)}\cong\
\mathcal{C}(\mathbb{Z}_{d},q)\boxtimes\text{Vec}_{G}$ as a $G$-crossed fusion
category by [11, Remark 7.11], then
$\mathcal{B}^{\text{rev}}\cong\mathcal{C}(\mathbb{Z}_{d},q)\boxtimes\mathcal{Z}(\mathcal{E})$
as braided fusion category. And by definition of $\mathcal{D}$,
$\mathcal{C}(\mathbb{Z}_{d},q)\subseteq\mathcal{D}$ as a fusion subcategory.
Therefore, we see that $\mathcal{C}$ also contains a non-degenerate fusion
subcategory $\mathcal{C}(\mathbb{Z}_{d},q)$, and then [8, Theorem 3.13]
implies that we have a braided equivalence of fusion categories
$\mathcal{C}\cong\mathcal{C}(\mathbb{Z}_{d},q)\boxtimes\mathcal{C}(\mathbb{Z}_{d},q)^{\prime}$
as required. ∎

######  Remark 4.6.

In Theorem 4.5, the condition that $(\text{FPdim}(V)^{2},d)=1$ for all simple
objects $V\in\mathcal{C}$ can not be dropped. For example, let $d>1$ be a
square-free odd integer, there exists a non-degenerate fusion category
$\mathcal{C}$ of FP-dimension $4d$, which is braided equivalent to a
$\mathbb{Z}_{2}$-equivariantization of a Tambara-Yamagami fusion category
$\mathcal{TY}(\mathbb{Z}_{d},\tau,\mu)$. It was proved that $\mathcal{C}$
contain a simple object of FP-dimension $\sqrt{d}$ and
$\mathcal{C}_{\text{pt}}=\text{Rep}(\mathbb{Z}_{2})$, see [1, Theorem 3.1] for
details. Obviously,
$\mathcal{C}\ncong\mathcal{C}(\mathbb{Z}_{d},q)\boxtimes\mathcal{C}(\mathbb{Z}_{d},q)_{\mathcal{C}}^{\prime}$
as braided fusion category.

######  Corollary 4.7.

Let $\mathcal{C}$ be a weakly group-theoretical non-degenerate fusion
category, and $d$ be a square-free integer. Assume that
$\text{FPdim}(\mathcal{C})=nd$ and $(n,d)=1$. If $(\text{FPdim}(X)^{2},d)=1$
for all $X\in\mathcal{O}(\mathcal{C})$, then
$\mathcal{C}\cong\mathcal{C}(\mathbb{Z}_{d},q)\boxtimes\mathcal{C}(\mathbb{Z}_{d},q)_{\mathcal{C}}^{\prime}$
as braided fusion category.

###### Proof.

Assume $\mathcal{C}$ is not pointed, otherwise the result is trivial. If
$\mathcal{C}$ is a weakly anisotropic fusion category, it follows from [17,
Theorem 1.1] that $\mathcal{C}$ is braided equivalent to a Deligne tensor
product of an Ising category and a pointed fusion category. If not, let
$\mathcal{E}=\text{Rep}(G)$ be a maximal Tannakian subcategory of
$\mathcal{C}$. By [8, Corollary 5.19] $\mathcal{C}_{G}^{0}$ is a weakly
anisotropic fusion category, as $|G|^{2}$ divides $\text{FPdim}(\mathcal{C})$,
thus [17, Theorem 1.1] says that $\mathcal{C}_{G}^{0}$ satisfies the
conditions of Theorem 4.5, hence $\mathcal{C}$ contains a non-degenerate
fusion category $\mathcal{C}(\mathbb{Z}_{d},q)$. ∎

Moreover, by using Theorem 3.5 and Theorem 4.5, we have the following direct
result.

######  Corollary 4.8.

Let $n$ be an integer, $d$ an odd square-free integer such that $(n,d)=1$. Let
$\mathcal{C}$ be a slightly degenerate weakly group-theoretical fusion
category with $\text{FPdim}(\mathcal{C})=nd$. If there existsa minimal
extension $\mathcal{A}$ is $\mathcal{C}$ such that $(\text{FPdim}(V)^{2},d)=1$
for any $V\in\mathcal{O}(\mathcal{A})$, then
$\mathcal{C}\cong\mathcal{C}(\mathbb{Z}_{d},q)\boxtimes\mathcal{C}(\mathbb{Z}_{d},q)_{\mathcal{C}}^{\prime}$
as braided fusion category.

Recall that a braided fusion category $\mathcal{C}$ of FP-dimension
$p^{a}r^{b}d$ is weakly group-theoretical [19, Proposition 3.14], where $p,r$
are primes, $a,b$ are non-negative integers, $d$ is a square-free integer such
that $(pr,d)=1$; for non-degenerate fusion categories, this conclusion was
first proved in [17, Corollary 5.4]. Therefore, if
$\mathcal{C}^{\prime}\subseteq\text{sVec}$ and $\mathcal{C}$ is integral, then
for all simple objects $X$ of $\mathcal{C}$, $\text{FPdim}(X)^{2}$ divides
$\text{FPdim}(\mathcal{C})$ by [12, Theorem 2.11] and [19, Corollary 3.4],
hence $(\text{FPdim}(X),d)=1$. Thus, Theorem 4.2 shows that
$\mathcal{C}\cong\mathcal{C}(\mathbb{Z}_{m},q)\boxtimes\mathcal{D}$, where
$\mathcal{D}$ is a braided fusion category with Müger center
$\mathcal{D}^{\prime}\subseteq\text{sVec}$, and $m=d$ or $\frac{d}{2}$.

In particular, we have the following corollary, which generalizes the
conclusions of [6, Theorem 4.7] and [19, Proposition 3.8, Corollary 3.12].

######  Corollary 4.9.

Let $\mathcal{C}$ be an integral braided fusion category of FP-dimension
$p^{n}d$ and $\mathcal{C}^{\prime}\subseteq\text{sVec}$, where $p$ is a prime,
$n$ is a nonnegative integer and $d$ is a square-free integer such that
$(p,d)=1$. Then $\mathcal{C}$ is nilpotent and group-theoretical.

###### Proof.

By Theorem 4.5, if $\mathcal{C}^{\prime}=\text{Vec}$, then
$\mathcal{C}\cong\mathcal{D}\boxtimes\mathcal{C}(\mathbb{Z}_{d},q)$ as braided
fusion category, and $\text{FPdim}(\mathcal{D})=p^{n}$. If
$\mathcal{C}^{\prime}=\text{sVec}$, then $\mathcal{C}$ is nilpotent by [19,
Corollary 3.12] when $p$ is odd; if $p=2$, then previous argument shows that
$\mathcal{C}$ contains $\mathcal{C}(\mathbb{Z}_{d},q)$ as a fusion
subcategory, consequently
$\mathcal{C}\cong\mathcal{D}\boxtimes\mathcal{C}(\mathbb{Z}_{d},q)$, where
Müger center $\mathcal{D}^{\prime}\subseteq\text{sVec}$ and
$\text{FPdim}(\mathcal{D})=2^{n}$. Since $\mathcal{C}$ is braided equivalent
to a Deligne tensor product fusion subcategories of prime power FP-dimensions,
it is nilpotent by [10, Theorem 8.28], and $\mathcal{C}$ is group-theoretical
[7, Corollary 6.8]. ∎

## References

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  * [14] A. Kirillov, Jr, On $G$-equivariant modular categories, arXiv:0401119.
  * [15] T. Lan, L. Kong and X. Wen, Modular extensions of unitary braided fusion categories and 2 + 1D topological/SPT orders with symmetries, Comm. Math. Phys. $\mathbf{351}$ (2017), no. 2, 709-739.
  * [16] M. Müger, On the structure of modular categories. Proc. London Math. Soc, $\mathbf{87}$ (2003), no. 3, 291-308.
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Victor Ostrik
Email<EMAIL_ADDRESS>
Department of Mathematical, University of Oregon, Eugene 97402, USA
Laboratory of Algebraic Geometry, National Research University Higher School
of Economics, Moscow, Russia
Zhiqiang Yu
Email<EMAIL_ADDRESS>
School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China


# Temporal Dependencies in Feature Importance for Time Series Predictions

Clayton Rooke  Jonathan Smith  Kin Kwan Leung  Maksims Volkovs  Saba Zuberi

###### Abstract

Explanation methods applied to sequential models for multivariate time series
prediction are receiving more attention in machine learning literature. While
current methods perform well at providing instance-wise explanations, they
struggle to efficiently and accurately make attributions over long periods of
time and with complex feature interactions. We propose WinIT, a framework for
evaluating feature importance in time series prediction settings by
quantifying the shift in predictive distribution over multiple instances in a
windowed setting. Comprehensive empirical evidence shows our method improves
on the previous state-of-the-art, FIT, by capturing temporal dependencies in
feature importance. We also demonstrate how the solution improves the
appropriate attribution of features within time steps, which existing
interpretability methods often fail to do. We compare with baselines on
simulated and real-world clinical data. WinIT achieves $2.47\times$ better
performance than FIT and other feature importance methods on real-world
clinical MIMIC-mortality task. The code for this work is available at
https://github.com/layer6ai-labs/WinIT.

Machine Learning, ICML

##  1 Introduction

Figure 1: Saliency maps for FIT and WinIT explainability methods on Delayed
Spike data (d=2) are compared with the data, per-time step labels and ground
truth feature importance. Here it can be seen that WinIT captures the
important observation despite a delay between the observation and the label
change, unlike FIT, which also overweights all features in the time step with
the label change.

Explaining model predictions is important for transparency, accountability,
and for motivating users to act on data. Good methods for generating
explanations are particularly useful in domains like healthcare and finance,
where explanations are an ethical and legal requirement (Amann et al., 2020).
However, the field of time series explainability for deep neural networks has
only recently seen attention, with the discovery that traditional
explainability methods underperform on deep learning models applied in the
time series domain (Ismail et al., 2020). Recent methods such as Feature
Importance in Time (FIT) (Tonekaboni et al., 2020) and Temporal Saliency
Rescaling (TSR) (Ismail et al., 2020) have improved performance and defined
initial benchmarks but face challenges in the breadth of their application in
real-world scenarios.

In this work we explore time series explainability in the domain where there
may be a delay between important feature shifts and a change in the predictive
distribution. This type of temporal dependency can be important in real-world
settings, where changes in input features may not instantaneously change model
predictions. We demonstrate experimentally that existing state-of-the-art
method FIT fails to extend to the delayed label setting via experiments on a
new synthetic dataset. We propose a new approach, WinIT, to address this
challenge by quantifying the impact of features on the predictive distribution
over multiple instances in a windowed setting. WinIT utilizes a modification
of the instance-wise importance score introduced in FIT, which we refer to as
Inverse FIT, that performs better in the windowed setting. We evaluate WinIT
on real-world clinical data and find that it outperforms FIT by a significant
margin. In summary, our main contributions are:

  * •

Extending FIT to work with lookback-windows that improve performance on
datasets where there is some time delay between the observation of important
features and a corresponding shift in label. We show how to evaluate
performance on the label delay problem with a new synthetic dataset.

  * •

Reformulating the counterfactual explanation method of FIT in a more efficient
manner, suitable for use in a windowed setting.

  * •

Our results show that combining these methods leads to a $2.47\times$
improvement in explanation performance on the real-world clinical MIMIC-
mortality task.

##  2 Background

Traditional perturbation-based and model-based methods have shown limited
success in the time series domain. Gradients, Integrated Gradients,
GradientSHAP, Deep-LIFT (Shrikumar et al., 2017), and DeepSHAP (Lundberg &
Lee, 2017) all leverage model gradients to generate feature importance, but do
not directly consider the temporal nature of the problem. Perturbation-based
methods like feature occlusion (Zeiler & Fergus, 2014) and feature ablation
(Suresh et al., 2017) are model-agnostic methods which measure how changes to
the input features relate to changes in model prediction. RETAIN learns
attention scores over the input features (Choi et al., 2016). LIME learns
explainable models locally around a prediction, applied at every time step in
the time series domain (Ribeiro et al., 2016).

Recent benchmarks (Ismail et al., 2020; Tonekaboni et al., 2020) evaluate
these traditional explainability methods on time series problems in both
simulated and real-world experiments. By separating the importance calculation
in both the time and feature input dimensions (Ismail et al., 2020) finds that
the performance of the existing methods can be improved. In contrast
(Tonekaboni et al., 2020) proposes a new method, FIT, that measures each
observation’s contribution to the predictive distribution shift of the model
over time in order to provide better explanations in certain settings.
However, FIT is limited to measuring the importance of instantaneous shifts in
the predictive distribution. We define an instantaneous shift as one where the
important observations from the input change the model prediction immediately.
As in, the important data and the prediction change occur on the same time
step. The assumption that feature shift and prediction shift occur
simultaneously does not always hold in practice. In real world applications
there can be a delay between an important feature shift and a change in
outcome. It is important for explanation methods for time series predictions
to be able to perform well given such temporal dependencies. For this reason
we present a new method, WinIT, which reformulates the FIT algorithm to make
it more efficient, while also attributing correct feature importance for non-
instantaneous changes in the predictive distribution.

##  3 Notation

Let $\textbf{X}\in\mathbb{R}^{D\times T}$ be a sample of a multi-variate time
series with $D$ features and $T$ time steps. We denote $[N]$ to be the set
$\\{1,\ldots,N\\}$. We also let
$\textbf{x}_{t}:=\textbf{X}_{\cdot,t}\in\mathbb{R}^{D}$ be the set of all
observations at a particular time $t\in[T]$ and
$\mathbf{X}_{1:t}:=[\mathbf{x}_{1};\mathbf{x}_{2};\ldots;\mathbf{x}_{t}]\in\mathbb{R}^{D\times
t}$. Let $y_{t}\in[K]$ be the label at each time step for a classification
task with $K$ classes. Let $S\subseteq[D]$ be a subset of features of interest
and $\textbf{x}_{S,t}$ be the observations of that subset at time $t$. We also
define $S^{c}$ as the set complement of the features of interest. For a model,
$f_{\theta}$, that estimates the conditional distribution
$p(y_{t}|\textbf{X}_{1:t})$ at each time step, we aim to provide a feature
importance score for each set of observations $\textbf{x}_{S,t}$ using the
observations up to that time step, $\textbf{X}_{1:t}\in\mathbb{R}^{D\times
t}$.

For feature importance methods that calculate scores over a set of time steps,
we let $n\in[N]$ be the lookback window up to a maximum window size of $N$.
Then $\textbf{X}_{S,t-n:t}$ represents the set of observations of the subset
of features of interest over a set of time steps of length $n$ and
$\textbf{X}_{1:t-n-1}$ represents the historical observations for all features
before that window. We also refer to the absmax function which in our
implementation finds the maximum absolute value, but then returns the actual
value, not the absolute value.

##  4 Methods

In this section we introduce our approach WinIT. We first review the FIT
importance score in Section 4.1. We then present Inverse FIT, a modified
version of the importance score in Section 4.2. In Section 4.3 we present
WinIT, which extends Inverse FIT using a windowed approach to computing
feature importance for non-instantaneous changes in the predictive
distribution.

###  4.1 FIT

Proposed by (Tonekaboni et al., 2020), FIT defines an importance score for a
subset of features $S$ at time $t$, given by a set of observations
$\textbf{x}_{S,t}$. It measures how well the partial conditional distribution,
where only a subset of features are observed at time $t$,
$p(y|\textbf{X}_{1:t-1},\textbf{x}_{S,t})$, approximates the full predicted
distribution $p(y|\textbf{X}_{1:t})$. This is characterized by the KL
divergence between these two distributions and is referred to as the
“unexplained” predictive distribution shift. It is measured with respect to
the total “temporal” shift from time $t-1$ to time $t$, given by the KL
divergence between the model prediction at time $t$ and the model prediction
at time $t-1$, $KL(p(y|\textbf{X}_{1:t})||p(y|\textbf{X}_{1:t-1}))$. The FIT
importance score for a set of observations $S$ at time $t$ is then given by
the difference between the “temporal” distribution shift and the “unexplained”
distribution shift:

$I_{FIT}(\textbf{x}_{S},t)=KL(p(y|\textbf{X}_{1:t})||p(y|\textbf{X}_{1:t-1}))-\\\
KL(p(y|\textbf{X}_{1:t})||p(y|\textbf{X}_{1:t-1},\textbf{x}_{S,t}))$ (1)

To compute the partial predictive distribution
$p(y|\textbf{X}_{1:t-1},\textbf{x}_{S,t})$, FIT marginalizes over the
complement feature set at time $t$, $\textbf{x}_{S^{c},t}$, by sampling from
the counterfactual distribution
$p(\textbf{x}_{S^{c},t}|\textbf{X}_{1:t-1},\textbf{x}_{S,t})$ approximated by
a generative model $G$.

###  4.2 Inverse FIT

The FIT algorithm quantifies the predictive distribution shift explained by an
observation $\textbf{x}_{S,t}$ by calculating the difference between the
unexplained distribution shift and the total temporal distribution shift. An
alternative approach is to directly compute the explained distribution shift.
We can measure the importance of features $S$ at time $t$ by quantifying how
well the partial conditional distribution,
$p(y|\textbf{X}_{1:t-1},\textbf{x}_{S^{c},t})$, where only the complement set
of features, $S^{c}$, are observed at time $t$, approximates the true
predictive distribution $p(y|\textbf{X}_{1:t})$.

We call this modification Inverse FIT (IFIT). The new formulation of an
instance-wise feature importance score is:

$I_{IFIT}(\textbf{x}_{S},t)=KL(p(y|\textbf{X}_{1:t})||p(y|\textbf{X}_{1:t-1},\textbf{x}_{S^{c},t}))$
(2)

Similar to FIT, we compute the partial predictive distribution
$p(y|\textbf{X}_{1:t-1},\textbf{x}_{S^{c},t})$ by using Monte-Carlo
integration to marginalize over $\textbf{x}_{S,t}$ by sampling from a
generator $G$ that approximates the distribution
$p(\textbf{x}_{S,t}|\textbf{X}_{1:t-1},\textbf{x}_{S^{c},t})$. This approach
is outlined in Algorithm 1.

It is important to note that the Inverse Fit importance score, Equation 2 is
not equivalent to the FIT score. In particular Inverse FIT does not consider
the overall shift in the predictive distribution from $t-1$ to $t$. This
approach performs well when extended to calculating feature importance over a
window of time steps, as shown in Section 5. Furthermore, we find that Inverse
FIT achieves similar performance to FIT, but is faster as seen in Table 1.
This is due to the different generator that can be used (per-feature, rather
than joint) and is relevant in the case where $|S|=1$ as evaluated in our
experiments. For larger set sizes the runtime may vary.

FIT is limited to measuring instantaneous changes in the predictive
distribution, because only the most recent time step of input is considered
when computing importance for a given prediction. The importance score
$I_{FIT}(\textbf{x}_{S},t)$ in Equation 1 is equal to the predictive
distribution shift from time step $t-1$ to $t$ explained by the observation
$\textbf{x}_{S,t}$. However, for sequential models, the observation
$\textbf{x}_{S,t}$ could also influence any of the predictive distributions
from time $t+1$ onwards. This importance is not captured in the FIT algorithm.
In the next section we extend the IFIT method to address this limitation.

Algorithm 1 IFIT
Input: $f_{\theta}$: Trained Black-box predictor model, $G$: Trained
generative model, $\textbf{X}_{1:T}\in\mathbb{R}^{D\times T}$: Time series
where $T$ is the max time and $D$ is the number of features, $S$: a subset of
features of interest, $L$: number of Monte Carlo samples
Output: Importance score matrix $I\in\mathbb{R}^{T\times D}$

Train $G$ using $\textbf{X}_{1:T}$

for all $t\in[T]$ do

$p(y|\textbf{X}_{1:t})=f_{\theta}(\textbf{X}_{1:t})$

$p(\textbf{x}_{t}|\textbf{X}_{1:t-1})\approx G(\textbf{X}_{1:t-1})$

for all $l\in[L]$ do

$\text{Sample }\hat{\textbf{x}}_{S,t}^{(l)}\sim
p(\textbf{x}_{S,t}|\textbf{X}_{1:t-1},\textbf{x}_{S^{c},t})$

$p(\hat{y}^{(l)})=f_{\theta}(\textbf{X}_{1:t-1},\textbf{x}_{S^{c},t},\hat{\textbf{x}}_{S,t}^{(l)})$

end for

$p(y|\textbf{X}_{1:t-1},\textbf{x}_{S^{c},t})\approx\frac{1}{L}\sum_{l=1}^{L}p(\hat{y}^{(l)})$

$I(\textbf{x}_{S},t)=KL(p(y|\textbf{X}_{1:t})||p(y|\textbf{X}_{1:t-1},\textbf{x}_{S^{c},t}))$

end for

Algorithm 2 WinIT
Input: $f_{\theta}$: Trained Black-box predictor model, $G$: Trained
generative model, $\textbf{X}_{1:T}\in\mathbb{R}^{D\times T}$: Time series
where $T$ is the max time and $D$ is the number of features, $N$: feature
importance lookback window size, $S$: a subset of features of interest, $L$:
number of Monte Carlo samples
Output: Importance score matrix $I\in\mathbb{R}^{T\times D\times N}$

Train $G$ using $\textbf{X}_{1:T}$

for all $t\in[T]$ do

$p(y|\textbf{X}_{1:t})=f_{\theta}(\textbf{X}_{1:t})$

$KL_{0}=0$

for all $n\in[N]$ do

$p(\textbf{x}_{t-n:t}|\textbf{X}_{1:t-n-1})\approx G(\textbf{X}_{1:t-n-1})$

for all $l\in[L]$ do

$\text{Sample }\hat{\textbf{X}}_{S,t-n:t}^{(l)}\sim$

$\hskip
56.9055ptp(\textbf{X}_{S,t-n:t}|\textbf{X}_{1:t-n-1},\textbf{X}_{S^{c},t-n:t})$

$p(\hat{y}^{(l)})=f_{\theta}(\textbf{X}_{1:t-n-1},\textbf{X}_{S^{c},t-n:t},\hat{\textbf{X}}_{S,t-n:t}^{(l)})$

end for

$p(y|\textbf{X}_{1:t-n-1},\textbf{X}_{S^{c},t-n:t})\approx\frac{1}{L}\sum_{l=1}^{L}p(\hat{y}^{(l)})$

$KL_{n}=KL(p(y|\textbf{X}_{1:t})||p(y|\textbf{X}_{1:t-n-1},\textbf{X}_{S^{c},t-n:t}))$

$W(\textbf{x}_{S},t-n,n)=KL_{n}-KL_{n-1}$

$KL_{n-1}=KL_{n}$

end for

end for

$I_{WinIT}(\textbf{x}_{S},t)=\text{absmax}_{n\in[N]}W(\textbf{x}_{S},t,n)$

Return $I$

###  4.3 WinIT

We formulate an extension of IFIT with a window of past observations when
attributing importance for a given prediction and call this WinIT. For a
prediction at time $t$, with a window size of $N$, we compute importance
scores for the observations $\textbf{X}_{S,t-N:t}$. For a set of observations
$\textbf{x}_{S,t-n}$ the sum of the $I_{IFIT}$ importance scores for all
remaining time steps $t-n+1$ to $t$ is subtracted from the total importance
score for time steps $t-n$ to $t$, to get the observation score at time $t-n$.
When $n=1$ the importance score for remaining time steps is zero. Because the
KL divergences in a sequence of $n$ windows cancel out in subsequent scores
this can be rewritten as the difference between the current time step and the
following time step as seen in in Equation 3. Here $W$ represents the
importance of the feature subset $S$ at time step $t-n$ that affects the
prediction at time step $t$.

Since WinIT generates $N$ scores for each time step, but the explainability
methods were evaluated based on single importance score for each observation
on individual features (no subsets), the final scores must be aggregated. To
generate a single observation score, the absolute maximum value across all of
the $n\in[N]$ windows is computed as shown in Equation 4. This is used in
order to capture the most important contribution of each observation in the
final importance score. In the benchmark experiments, important contributions
tend to be sparse; a different aggregation metric like an average would not
properly capture infrequent but important contributions.

$W(\textbf{x}_{S},t,n)=\\\
KL(p(y|\textbf{X}_{1:t})||p(y|\textbf{X}_{1:t-n-1},\textbf{X}_{S^{c},t-n:t}))-\\\
KL(p(y|\textbf{X}_{1:t})||p(y|\textbf{X}_{1:t-n},\textbf{X}_{S^{c},t-n+1:t}))$
(3) $\displaystyle
I_{WinIT}(\textbf{x}_{S},t)=\text{absmax}_{n\in[N]}W(\textbf{x}_{S},t,n)$ (4)

This leads to a new formulation of a instance-wise feature importance score
now taken over multiple overlapping windows described in Algorithm 2.

##  5 Experiments

For the following experiments, an RNN-based predictor is trained on the
training dataset, and the explainability methods are evaluated on the test
dataset. A recurrent latent variable generator (Chung et al., 2015) is trained
on the training dataset for the FIT and WinIT models.

To evaluate the explainability methods on experiments with simulated data
ground truth importance scores are defined. An observation is given a ground
truth importance score of 1 if it causes the label to change. All other
observations have a ground truth importance score of $0$. Explainability
methods are evaluated against the ground truth using AUROC and AUPRC, where
the ranking score is calculated per-sample by ranking all the feature
instances by importance and then averaged over the entire dataset. For the
real-world clinical data, no ground truth feature importance is available.
Instead, the explainability methods are evaluated based on AUROC drop after
the Top K=50, or Top 5%, of observations with the highest importance score is
removed from the test dataset by carrying forward the previous values.

###  5.1 Simulated Data

Spike is a benchmark experiment presented in (Tonekaboni et al., 2020) which
uses a multivariate dataset composed of 3 random NARMA time series with random
‘spikes’, immediate large increases, added to the samples. The label is 0
until a spike occurs in the first feature, at which point it changes to 1 for
the rest of the sample. As shown in Table 1, WinIT shows similar performance
to FIT on the Spike benchmark, with FIT having the highest AUROC, and the
AUPRC of the two methods being the same within one standard deviation.

Table 1: Explanation performance on Spike, and Delayed Spike (d=2) datasets.
For WinIT we use a window size of 8. All evaluations are conducted over 5
random seeds and averaged.
Spike
---
Method | AUROC | AUPRC | Time (s)
FIT |  0.994 $\pm$ 0.002 |  0.852 $\pm$ 0.098 | 394.78
IFIT |  0.954 $\pm$ 0.006 |  0.844 $\pm$ 0.081 | 70.91
WinIT |  0.965 $\pm$ 0.002 |  0.905 $\pm$ 0.048 | 449.41
Delayed Spike (d=2)
Method | AUROC | AUPRC | Time (s)
FIT |  0.516 $\pm$ 0.035 |  0.002 $\pm$ 0.001 | 340.00
WinIT |  0.970 $\pm$ 0.006 |  0.909 $\pm$ 0.029 | 455.03

To demonstrate the temporal dependency effect, we present a simple experiment
using simulated data as a modification of the Spike data. Three independent
NARMA sequences are generated and two of the features add linear trends.
Spikes are then added following the same procedure as the Spike data. However,
the time step at which the label changes is different. In the Spike data the
label changes immediately to 1 after encountering the first spike. In the
Delayed Spike data the label changes after $d=2$ time steps. To measure the
accuracy of the explainability methods, we define the ground truth importance
score as 1 for the first spike and 0 for all other observations. As shown in
Table 1, since FIT only considers the observations from time steps $t-1$ to
$t$ as they relate to the prediction at $t$, it is unable to assign importance
to the correct observation, which occurs at time step $t-d$. However, since
the spike falls in the window $\textbf{X}_{t-N:t}$, WinIT is able to assign
importance to the correct observation.

We also show performance of the IFIT and WinIT models as an ablation study in
Table 1. This reveals some of the tradeoffs between runtime and performance
for the different methods, with IFIT-based methods taking less time than FIT.
Methods that do not use a lookback window achieve poor results on the Delayed
Spike data, reflected in the low AUPRC of both IFIT and FIT.

###  5.2 Clinical Data

MIMIC III is a multivariate time series clinical dataset with a number of
vital and lab measurements taken over time for around 40000 patients at the
Beth Israel Deaconess Medical Center in Boston, MA (Johnson et al., 2016).
MIMIC III is used in the FIT paper to construct the MIMIC-mortality
experiment, which uses 8 vital and 20 lab measurements hourly over a 48 hour
period to predict patient mortality. As shown in Table 2, the WinIT method is
a significant improvement over FIT and other explainability methods on the
MIMIC-mortality experiment. In fact, we see a $2.47\times$ improvement over
FIT when calculating AUC Drop in the top 5% of features and a $1.36\times$
improvement when calculating AUC Drop in the top 50 features.

Adjusting the window size can lead to different performance in all settings.
WinIT with different lookback windows of size 1, 5, 10, and 15 shows improving
performance in AUC Drop (Top 5%) in the real-world setting of the MIMIC-
mortality task as seen in Figure 2. AUC Drop (Top 50), while outperforming all
other methods, does exhibit more variance and does not improve with window
size. This may be because only a few features benefit from the additional
information related to delays between feature changes and label changes. It
may also be that globally important features from the Top 5% display more
temporal dependence than other features.

Table 2: Explanation performance on MIMIC-mortality task. WinIT uses a window size of 10. At the bottom we show performance improvement against the second-best method (FIT). $\dagger$ indicates results are from (Tonekaboni et al., 2020). Method |  AUC Drop (95-pc) |  AUC Drop (k=50)
---|---|---
AFO† |  0.023 $\pm$ 0.003 |  0.068 $\pm$ 0.003
FO† |  0.028 $\pm$ 0.006 |  0.095 $\pm$ 0.042
Deep-Lift† |  0.045 $\pm$ 0.004 |  0.067 $\pm$ 0.038
IG† |  0.036 $\pm$ 0.003 |  0.056 $\pm$ 0.014
RETAIN† |  0.020 $\pm$ 0.014 |  0.032 $\pm$ 0.019
LIME† |  0.028 $\pm$ 0.000 |  0.032 $\pm$ 0.019
GradSHAP† |  0.036 $\pm$ 0.000 |  0.065 $\pm$ 0.062
FIT |  0.038 $\pm$ 0.005 |  0.138 $\pm$ 0.037
WinIT | 0.094 $\pm$ 0.003 | 0.188 $\pm$ 0.015
(vs. FIT) | $2.47\times$ | $1.36\times$

Figure 2: WinIT performance on the MIMIC-mortality task improves as the window
size increases from 1 to 15 for AUC Drop (Top 5%) (a) and AUC Drop (K=50) (b).

Figure 3: Saliency maps for FIT and WinIT methods on the Spike dataset. Here
it can be seen that both methods capture the important observation, but FIT
also overweights the importance of all features in the important time step and
other time steps for all features, while WinIT only overweights a few other
time steps in the important feature.

Figure 4: Saliency maps for FIT and WinIT methods on an instance from the
MIMIC-mortality task. Here the features identified as most important by WinIT
have stronger correlation with the underlying features while FIT shows an over
weighting of a single time step.

###  5.3 Saliency Maps

As a sanity check on the explanations provided by WinIT we show saliency maps
from instances in the Spike and Delayed Spike datasets in Figure 1 and Figure
3. In the Delayed Spike example it is clear how FIT fails to identify the
important observations, instead providing a higher average score to all
observations. FIT also suffers from a common problem in time series
explanations, where it overweights features that occur in the same time step
as an important observation. This can be seen in both figures. WinIT, on the
other hand, sees failure cases for the Spike dataset when multiple spikes
appear close together.

We also show comparisons of FIT and WinIT saliency maps for an instance from
the MIMIC-mortality task in Figure 4. In this case the overweighting of time
steps is even more apparent with the FIT explanation due to the larger number
of features.

##  6 Conclusion

In this work we propose WinIT, a method for time series explainability that
allows for attributing correct importance to observations for non-
instantaneous changes in a time series model’s predictive distribution. WinIT
uses a windowed approach to computing the feature importance and is based on a
modification of the FIT importance score that performs well in the windowed
setting. WinIT is comparable to FIT on the Spike benchmark, and significantly
outperforms FIT on the proposed Delayed Spike data,where changes in the
model’s predictive distribution are not instantaneous, as well as on the real-
world MIMIC-mortality task.

In the future, we hope to evaluate WinIT on the other benchmark experiments,
as well as new simulated and real-world experiments to help better understand
where temporal dependencies make the greatest impact. The methods we present
can be further optimized through selection and tuning of the generative
methods used and their application to different kinds of real-world data.

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