Anomalous Simple Categories (ASCies)

• ASCies are tensor categories defined by the absence of any non-trivial normal anomaly-free subcategories, representing irreducible anomalous symmetry.
• They are characterized through short exact sequences and a fiber functor condition that ensures the absence of ’t Hooft anomalies.
• Their formulation using SymTFT links fusion category quotients to RG interfaces, clarifying anomaly matching and symmetry-breaking patterns.

Searching arXiv for the cited papers and related work on Anomalous Simple Categories. First search: ASCies / "Categorical Anomaly Matching". Second search: anomalous actions of groups on tensor categories. Anomalous Simple Categories, or ASCies, are tensor categories—more generally, fusion or higher fusion categories—that have no non-trivial normal subcategory, so that after quotienting a symmetry category by any maximal normal anomaly-free sector, the residual quotient is an irreducible anomalous symmetry. In the framework of categorical anomaly matching, ASCies are introduced as the fundamental building blocks of categorical anomalies: a given symmetry category may support multiple ASCies, each encoding distinct anomalous features, and these structures arise naturally in the Symmetry Topological Field Theory (SymTFT) description of renormalization-group interfaces (Antinucci et al., 1 Aug 2025).

1. Definition through short exact sequences of tensor categories

The basic input is a short exact sequence of tensor categories

$$N \stackrel{I}{\longrightarrow} C \stackrel{P}{\longrightarrow} S,$$

where $I$ is a tensor functor that is an embedding, $P$ is a surjective normal tensor functor, and exactness means

$\operatorname{im}(I)=\ker(P).$

Here $\ker(P)$ is the full tensor subcategory generated by objects $D$ with $P(D)\cong n\,1$ for some $n$, so they map to purely trivial objects in the target. In addition, the sequence is required to satisfy the fiber functor condition

$P\circ I=f,\qquad f:N\to Vec,$

with $f$ a tensor functor to $I$0, so that $I$1 has no ’t Hooft anomaly (Antinucci et al., 1 Aug 2025).

A normal subcategory $I$2 is one for which such an exact sequence exists. A maximal normal subcategory $I$3 is a normal subcategory not contained in any larger normal subcategory $I$4 that still fits into a sequence

$I$5

A tensor category $I$6 is then called an ASCy if it has no non-trivial normal subcategory, equivalently if the only normal subcategory is $I$7, or equivalently if it does not admit any short exact sequence of the above form with $I$8 (Antinucci et al., 1 Aug 2025).

This definition makes “simple” highly specific. It does not mean semisimple, rank-minimal, or generated by a small set of simples. It means that there is no non-trivial normal anomaly-free tensor subcategory that can be consistently factored out while preserving the remaining symmetry structure. In this sense, an ASCy is the anomalous quotient that remains after all possible normal anomaly-free sectors have been removed.

2. Normal subcategories, anomalous quotients, and irreducible anomaly content

Given a symmetry category $I$9, the central operation is to identify maximal normal anomaly-free subcategories $P$0 and factor them out through surjective normal tensor functors

$P$1

The quotients $P$2 are ASCies, and they represent the anomalous part of the symmetry $P$3. The associated set of anomalous simple quotients is

$P$4

A given $P$5 can support several inequivalent pairs $P$6, so the anomaly content is not, in general, exhausted by a single quotient (Antinucci et al., 1 Aug 2025).

This formalism separates anomaly-free structure from irreducible anomalous structure. The subcategory $P$7 is a maximal anomaly-free sub-symmetry that can be mapped to the trivial category via a fiber functor, whereas $P$8 captures one aspect of the anomaly that cannot be eliminated by quotienting. For fusion $P$9-categories, the short exact sequence also obeys the dimensional relation

$\operatorname{im}(I)=\ker(P).$0

A recurring point is that normality is stronger than anomaly-freeness. The data explicitly note that many categories contain anomaly-free invertible subcategories that are not normal, hence do not appear as kernels of surjective normal functors and cannot be consistently forgotten in a quotient respecting the full symmetry. This is why ASCies are defined using normal subcategories rather than arbitrary anomaly-free subcategories (Antinucci et al., 1 Aug 2025).

3. Tensor functors, anomaly matching, and the SymTFT criterion

For a $\operatorname{im}(I)=\ker(P).$1-dimensional QFT with symmetry category $\operatorname{im}(I)=\ker(P).$2, the symmetry is modeled as a tensor $\operatorname{im}(I)=\ker(P).$3-category. If a UV theory with symmetry $\operatorname{im}(I)=\ker(P).$4 flows to an IR theory with symmetry $\operatorname{im}(I)=\ker(P).$5, anomaly matching is encoded by a tensor functor

$\operatorname{im}(I)=\ker(P).$6

equipped with coherent monoidal structure

$\operatorname{im}(I)=\ker(P).$7

compatible with associators. In the group case this reduces to a group homomorphism $\operatorname{im}(I)=\ker(P).$8 together with the anomaly pullback condition

$\operatorname{im}(I)=\ker(P).$9

Within this framework, ASCies are the quotient symmetries that encode the irreducible anomalous content seen by such tensor functors (Antinucci et al., 1 Aug 2025).

The SymTFT formulation makes this concrete. A symmetry category $\ker(P)$0 determines a $\ker(P)$1-dimensional TQFT $\ker(P)$2 with a symmetry boundary $\ker(P)$3 and a physical boundary $\ker(P)$4. An RG interface between $\ker(P)$5 and $\ker(P)$6 is represented by a topological interface $\ker(P)$7, and the Matching Equation is

$\ker(P)$8

For fusion categories in $\ker(P)$9 dimensions, after folding one obtains a Lagrangian algebra $D$0 and a map on bulk anyons

$D$1

so that the Matching Equation becomes

$D$2

The paper states that, for fusion $D$3- and $D$4-categories, this condition is equivalent to the existence of a tensor functor $D$5 (Antinucci et al., 1 Aug 2025).

ASCies admit a sharp SymTFT characterization. A symmetry category $D$6 is an ASCy iff its center $D$7 has no non-trivial magnetic Lagrangian algebra, equivalently no condensable algebra $D$8 satisfying

$D$9

except the trivial one. This is the categorical refinement of the statement that there is no symmetric gapped phase preserving a non-trivial piece of the symmetry. The only way to trivialize an ASCy is to break the full symmetry.

4. Canonical examples and explicit quotient constructions

The basic low-rank examples listed for ASCies are $P(D)\cong n\,1$0, $P(D)\cong n\,1$1, the Fibonacci category $P(D)\cong n\,1$2, the Ising categories $P(D)\cong n\,1$3, and the Haagerup category $P(D)\cong n\,1$4. These are simple in the normal-subcategory sense used here, although some contain anomaly-free invertible subcategories that are not normal (Antinucci et al., 1 Aug 2025).

Two families of worked constructions are especially prominent: anomalous cyclic groups and Tambara–Yamagami categories.

Category or family	Exact-sequence output	Resulting ASCies
$P(D)\cong n\,1$5 with $P(D)\cong n\,1$6 and suitable restriction condition	$P(D)\cong n\,1$7	Quotients $P(D)\cong n\,1$8
$P(D)\cong n\,1$9	$n$0	$n$1
$n$2	$n$3 and $n$4	$n$5
$n$6	$n$7	$n$8
Ising $n$9	no non-trivial normal quotient	ASCy already

For cyclic groups, the explicit necessary condition is that the restriction of $P\circ I=f,\qquad f:N\to Vec,$0 to $P\circ I=f,\qquad f:N\to Vec,$1 be trivial. In the notation of the data, with $P\circ I=f,\qquad f:N\to Vec,$2,

$P\circ I=f,\qquad f:N\to Vec,$3

Under that condition there exist surjective functors

$P\circ I=f,\qquad f:N\to Vec,$4

and the functor $P\circ I=f,\qquad f:N\to Vec,$5 acts non-trivially on junctions through

$P\circ I=f,\qquad f:N\to Vec,$6

The example

$P\circ I=f,\qquad f:N\to Vec,$7

shows explicitly that one symmetry category can support more than one ASCy (Antinucci et al., 1 Aug 2025).

For Tambara–Yamagami categories, the Ising category $P\circ I=f,\qquad f:N\to Vec,$8 is itself an ASCy. By contrast, $P\circ I=f,\qquad f:N\to Vec,$9 admits two inequivalent quotient constructions, one detecting an anomalous $f$0 $f$1-form symmetry and the other an anomalous $f$2. The category $f$3 exhibits the opposite pattern: its invertible $f$4 subcategory is anomaly-free, but the second obstruction remains non-trivial, and the ASCy quotient is $f$5.

5. Relation to group-cohomological and higher-categorical anomalies

The ASCy formalism sits alongside a second categorical language for anomalies based on anomalous group actions on tensor categories. In that setting, for a group $f$6 and a $f$7-cocycle $f$8, a $f$9-anomalous action of $I$00 on a $I$01-linear monoidal category $I$02 is a $I$03-linear monoidal $I$04-functor

$I$05

and the cohomology class $I$06 is called the anomaly. The source $I$07 has trivial associator on $I$08-morphisms but a nontrivial pentagonator determined by $I$09, so the anomaly appears as a higher associativity defect rather than as an ordinary strict action (Lanier, 3 May 2025).

The main construction theorem in that setting states that if one has a surjection $I$10 with kernel $I$11, a normalized representative $I$12, a cochain $I$13 satisfying

$I$14

and an ordinary action of $I$15 on a tensor category $I$16, then there exists a $I$17-anomalous action of $I$18 on the twisted crossed product tensor category

$I$19

The explicit coherence condition is

$I$20

A plausible implication is that ASCies and $I$21-anomalous actions organize complementary aspects of anomalous symmetry. ASCies isolate the irreducible anomalous quotient symmetry after factoring out maximal normal anomaly-free sectors, whereas $I$22-anomalous actions describe the higher-categorical obstruction preventing an honest strict $I$23-action. The reinterpretation supplied with the anomalous-actions paper states that natural candidates for ASCies are fusion or tensor categories equipped with nontrivial monoidal $I$24-functors

$I$25

so that the symmetry necessarily lives at the $I$26-group level rather than at the level of an ordinary group action (Lanier, 3 May 2025).

6. Conceptual scope, common misconceptions, and open directions

Several misunderstandings are explicitly ruled out by the formalism. First, an ASCy need not be “anomalous” in the sense of lacking every anomaly-free substructure: $I$27 itself is listed as an ASCy because the criterion is the absence of non-trivial normal subcategories, not the absence of anomaly-free objects or subcategories. Second, the presence of an anomaly-free invertible subcategory does not imply the existence of a quotient. The Ising category contains an anomaly-free $I$28 invertible subcategory, but it is not normal, so there is no surjective tensor functor with kernel $I$29, and the full symmetry cannot be reduced by quotienting that sector (Antinucci et al., 1 Aug 2025).

The physical role of ASCies is correspondingly precise. For each quotient $I$30, there is a pullback on module categories

$I$31

This identifies symmetry-breaking patterns enforced by anomalies: a symmetric gapped phase of $I$32 pulls back to a specific pattern of $I$33-symmetry breaking, and for anomaly-free $I$34 a fiber functor $I$35 recovers the unique fully symmetric gapped phase. The data further emphasize that ASCies unify anomalous $I$36-form, higher-form, and non-invertible symmetries under one tensor-categorical and SymTFT framework (Antinucci et al., 1 Aug 2025).

The open directions are similarly structural. The mathematical theory of ASCies is identified as a systematic study of simple fusion and higher fusion categories with no non-trivial normal subcategories, their moduli, and their interrelations. Further directions include extension to continuous symmetries, rigorous tensor $I$37-functors for higher fusion categories, non-invertible Lieb–Schultz–Mattis anomalies and mixed anomalies, and the role of ASCies in RG flows with dynamical defects. In the complementary anomalous-action picture, open directions include classifying indecomposable fusion categories admitting a given $I$38-anomalous $I$39-action, extending to braided or modular tensor categories, and relating such anomalous categorical symmetries to $I$40-dimensional symmetry-protected boundaries (Antinucci et al., 1 Aug 2025, Lanier, 3 May 2025).