Crossed Squares in Group Theory

• Crossed squares of groups are algebraic structures defined by a commutative square of homomorphisms with compatible group actions and a Peiffer lifting, modeling interactions between crossed modules.
• They are categorically equivalent to cat²-groups and internal double group-groupoids, providing a robust framework for understanding connected homotopy 3-types.
• Practical implementations in software like GAP’s XMod demonstrate their effectiveness in classifying low-order groups and in applications such as lattice gauge theory.

Searching arXiv for recent and foundational papers on crossed squares of groups and related equivalences. A crossed square of groups is a commutative square of group homomorphisms equipped with compatible actions and a Peiffer lifting, designed to encode the interaction of two crossed-module directions simultaneously. In the literature surveyed here, crossed squares appear as algebraic models for connected homotopy $3$-types, as structures equivalent to cat$^2$-groups, as internal categorical objects in the category of crossed modules, and as objects equivalent to crossed modules over group-groupoids and hence to internal double group-groupoids; more recently they also arise as strict $3$-group data governing higher symmetries and anomalies in $2$-dimensional lattice gauge theory (Temel et al., 2018, Arvasi et al., 2019, Martins-Ferreira et al., 2018, Kapustin et al., 22 Jul 2025).

1. Definition and formal data

In the Brown–Loday formulation, a crossed square consists of a commutative diagram

$\xymatrix@R=20pt@C=20pt{ L \ar[r]^{\kappa} \ar[d]_{\lambda} & M \ar[d]^{\mu}\ N \ar[r]_{\nu} & P }$

together with left actions of $P$ on $L$, $M$, and $N$, and a crossed pairing

$\bt : M \times N \longrightarrow L.$

The commutativity means that

$^2$0

The standard axiomatization requires that, with the given actions, the homomorphisms $^2$1 are crossed modules, that $^2$2 and $^2$3 are $^2$4-equivariant, and that the pairing $^2$5 satisfies the identities

$^2$6

$^2$7

$^2$8

$^2$9

These identities express, respectively, multiplicativity in each variable up to the induced actions, boundary control of the pairing, compatibility with the two maps out of $3$0, and equivariance under the ambient $3$1-action (Arvasi et al., 2019).

Equivalent notational systems are common. One source writes the horizontal and vertical maps as $3$2 and the pairing as $3$3; another writes the square as

$3$4

with Peiffer lifting $3$5. In the group-groupoid approach, group laws are written additively rather than multiplicatively, but the structural content is unchanged (Martins-Ferreira et al., 2018, Kapustin et al., 22 Jul 2025, Temel et al., 2018).

2. Peiffer lifting, orientation, and standard constructions

The decisive extra datum in a crossed square is the Peiffer lifting. It lifts the two Peiffer commutators determined by the induced actions of $3$6 on $3$7 and of $3$8 on $3$9. In the notation of the operator-algebraic formulation,

$2$0

so $2$1 is an element of $2$2 whose two boundaries recover the corresponding commutator data in $2$3 and $2$4. This is the sense in which crossed squares internalize Peiffer commutator calculus one dimension higher than ordinary crossed modules (Kapustin et al., 22 Jul 2025).

There is also an orientation issue. An oriented crossed square has a transpose $2$5 obtained by interchanging $2$6 and $2$7; its crossed pairing $2$8 is given by

$2$9

According to the computational treatment, an oriented crossed square represents an equivalence class up to this transposition (Arvasi et al., 2019).

Several canonical examples recur throughout the literature. If $\xymatrix@R=20pt@C=20pt{ L \ar[r]^{\kappa} \ar[d]_{\lambda} & M \ar[d]^{\mu}\ N \ar[r]_{\nu} & P }$0, then the inclusion square

$\xymatrix@R=20pt@C=20pt{ L \ar[r]^{\kappa} \ar[d]_{\lambda} & M \ar[d]^{\mu}\ N \ar[r]_{\nu} & P }$1

with $\xymatrix@R=20pt@C=20pt{ L \ar[r]^{\kappa} \ar[d]_{\lambda} & M \ar[d]^{\mu}\ N \ar[r]_{\nu} & P }$2-conjugation actions and commutator pairing

$\xymatrix@R=20pt@C=20pt{ L \ar[r]^{\kappa} \ar[d]_{\lambda} & M \ar[d]^{\mu}\ N \ar[r]_{\nu} & P }$3

is a crossed square. A related construction replaces $\xymatrix@R=20pt@C=20pt{ L \ar[r]^{\kappa} \ar[d]_{\lambda} & M \ar[d]^{\mu}\ N \ar[r]_{\nu} & P }$4 by $\xymatrix@R=20pt@C=20pt{ L \ar[r]^{\kappa} \ar[d]_{\lambda} & M \ar[d]^{\mu}\ N \ar[r]_{\nu} & P }$5 and uses the pairing $\xymatrix@R=20pt@C=20pt{ L \ar[r]^{\kappa} \ar[d]_{\lambda} & M \ar[d]^{\mu}\ N \ar[r]_{\nu} & P }$6; this form is convenient when verifying the crossed-square identities directly from commutator identities (Arvasi et al., 2019, Martins-Ferreira et al., 2018).

Geometric constructions are equally standard. For a triad of pointed spaces $\xymatrix@R=20pt@C=20pt{ L \ar[r]^{\kappa} \ar[d]_{\lambda} & M \ar[d]^{\mu}\ N \ar[r]_{\nu} & P }$7, $\xymatrix@R=20pt@C=20pt{ L \ar[r]^{\kappa} \ar[d]_{\lambda} & M \ar[d]^{\mu}\ N \ar[r]_{\nu} & P }$8, one obtains a crossed square from the long exact sequence in homotopy. In the specific case $\xymatrix@R=20pt@C=20pt{ L \ar[r]^{\kappa} \ar[d]_{\lambda} & M \ar[d]^{\mu}\ N \ar[r]_{\nu} & P }$9 with $P$0 and $P$1 the upper and lower hemispheres, the crossed pairing on the free group $P$2 is

$P$3

This example is one of the basic illustrations of the role of crossed squares in homotopy $P$4-type theory (Arvasi et al., 2019).

A further source of examples comes from topological crossed modules $P$5. Passing to fundamental groupoids yields a crossed module of group-groupoids $P$6, and the kernels of the endpoint maps then produce a crossed square; the paper explicitly identifies such constructions with fundamental crossed squares (Temel et al., 2018).

3. Equivalences with cat$P$7-groups, crossed modules, and internal groupoids

A central structural fact is that crossed squares sit inside a web of categorical equivalences. One such equivalence identifies the category $P$8 of crossed squares with the category $P$9 of cat$L$0-groups. If $L$1 is a cat$L$2-group, then the associated crossed square is extracted from the kernel/image diagram

$L$3

with the pairing induced by the commutator in $L$4. Conversely, from a crossed square one forms the semidirect-product object

$L$5

and the crossed pairing enters the action by

$L$6

These constructions are inverse up to natural isomorphism (Arvasi et al., 2019).

Another important equivalence identifies crossed squares with crossed modules over group-groupoids, or $L$7-groups. In that setting, if $L$8 is a crossed module in the category of group-groupoids, then one extracts a crossed square by taking

$L$9

with

$M$0

and Peiffer function

$M$1

The resulting functor yields an equivalence

$M$2

and, combined with the equivalence between crossed modules over group-groupoids and internal double group-groupoids, gives

$M$3

Thus crossed squares correspond to internal double group-groupoids (Temel et al., 2018).

A third formulation, developed categorically through Whitehead sequences, identifies crossed squares with internal groupoids in the category of crossed modules. In that treatment, crossed squares are precisely Whitehead sequences in the category of actions of crossed modules; under the action-system machinery, and using the Smith–Huq condition in $M$4, they are equivalent to internal categories, hence internal groupoids, in $M$5. Since Loday had already identified internal categories in crossed modules with cat$M$6-groups, this recovers the classical equivalence from a different categorical angle (Martins-Ferreira et al., 2018).

A recurrent point of confusion is the diagonal structure in a cat$M$7-group. The computational paper stresses that the diagonal $M$8 need not be a cat$M$9-group; it may only be a pre-cat$N$0-group, and an explicit example occurs for $N$1 (Arvasi et al., 2019).

4. Split extensions, $N$2-groups, and internal double group-groupoids

The passage from crossed modules over group-groupoids to crossed squares is mediated by split extensions and semidirect products. Given a split extension of group-groupoids

$N$3

with splitting $N$4, one obtains the derived action

$N$5

This action is compatible with the groupoid structure, and the semidirect product law

$N$6

turns $N$7 into a semidirect product group-groupoid $N$8. The paper proves that such an action is derived from a split extension if and only if this semidirect-product structure exists (Temel et al., 2018).

A crossed module over group-groupoids is then a morphism $N$9 equipped with such an action, subject to the internalized crossed-module identities

$\bt : M \times N \longrightarrow L.$0

together with the analogous identities on objects. Proposition 3.11 reduces these requirements to ordinary crossed-module axioms at the level of arrows and objects (Temel et al., 2018).

From such data one constructs an internal double group-groupoid. The square object is $\bt : M \times N \longrightarrow L.$1, the horizontal structure has

$\bt : M \times N \longrightarrow L.$2

and the vertical structure is defined similarly on $\bt : M \times N \longrightarrow L.$3. The internal double group-groupoid satisfies the interchange and linearity laws

$\bt : M \times N \longrightarrow L.$4

$\bt : M \times N \longrightarrow L.$5

$\bt : M \times N \longrightarrow L.$6

and its compositions can be written purely in terms of the ambient group structure: $\bt : M \times N \longrightarrow L.$7 These formulas are the internal mechanism behind the equivalence $\bt : M \times N \longrightarrow L.$8 (Temel et al., 2018).

This framework recovers standard examples. The inclusion of a normal subgroup-groupoid gives a crossed module by conjugation; a crossed module of groups $\bt : M \times N \longrightarrow L.$9 yields a crossed module of group-groupoids $^2$00; and a topological crossed module yields, via fundamental groupoids, a crossed module in $^2$01, hence both a double group-groupoid and a crossed square. The paper also notes that Brown–Spencer special double groupoids arise as a special case in which horizontal and vertical groupoids coincide and the object set is a singleton (Temel et al., 2018).

5. Homotopy-theoretic role and computation

Crossed squares are $^2$02-dimensional algebraic structures refining crossed modules. In the programme of higher-dimensional group theory, crossed modules and cat$^2$03-groups model connected homotopy $^2$04-types, while crossed squares and cat$^2$05-groups model connected homotopy $^2$06-types. The computational paper places this explicitly in the context of algebraic models arising from triads of spaces and from the Brown–Loday theory of diagrams of spaces (Arvasi et al., 2019).

The operator-algebraic treatment gives a closely related description in terms of a nonabelian complex. From a crossed square $^2$07, let $^2$08 act on $^2$09 by $^2$10 and form

$^2$11

Then define

$^2$12

$^2$13

The associated homotopy groups are identified as

$^2$14

The same source shows that the Peiffer lifting determines a quadratic refinement

$^2$15

defined using

$^2$16

and proves that $^2$17 is quadratic in the sense that the polarization $^2$18 is bilinear and $^2$19. The paper interprets this as part of the Postnikov data of the $^2$20-type (Kapustin et al., 22 Jul 2025).

Computation with crossed squares has been implemented in the GAP package $^2$21. The package provides conversions between crossed squares and cat$^2$22-groups via CrossedSquareOfCat2Group and Cat2GroupOfCrossedSquare, constructors such as CrossedSquareByNormalSubgroups, and a suite of cat$^2$23-group functions including AllCat2Groups, AreIsomorphicCat2Groups, AllCat2GroupsUpToIsomorphism, and AllCat2GroupFamilies. The paper reports a complete enumeration of cat$^2$24-group structures on groups of order at most $^2$25, yielding $^2$26 isomorphism classes in total. Among the highlighted counts are $^2$27 cat$^2$28-groups and $^2$29 isomorphism classes for $^2$30, $^2$31 cat$^2$32-groups and $^2$33 classes for $^2$34, $^2$35 cat$^2$36-groups and $^2$37 classes for $^2$38, and $^2$39 cat$^2$40-groups with $^2$41 classes for $^2$42. The same study records that only $^2$43 cat$^2$44-groups in the tables have diagonals that are not cat$^2$45-groups (Arvasi et al., 2019).

6. Higher symmetries and lattice gauge theory

A recent application interprets crossed squares as the natural algebraic packaging of symmetry restrictions in $^2$46-dimensional quantum lattice gauge theory. In that framework one studies automorphisms of the observable $^2$47-algebra generated by finite-depth gauge-invariant circuits and restricts them to nested spatial regions. For a fixed cover of the plane by a lower half-plane and two complementary half-axes, the groups are

$^2$48

$^2$49

$^2$50

$^2$51

The maps $^2$52 and $^2$53 are inclusions, $^2$54 send a local unitary to the corresponding inner automorphism, and the Peiffer lifting is the truncation-defined commutator

$^2$55

The paper proves that this data satisfies the crossed-square axioms and interprets the resulting object as a strict $^2$56-group encoding $^2$57-form symmetry, $^2$58-form symmetry, and anomaly data (Kapustin et al., 22 Jul 2025).

The same formalism makes the anomaly theory explicit through the quadratic map $^2$59. For the vanilla $^2$60 lattice gauge theory, the distinguished $^2$61 generated by a magnetic string pair has trivial quadratic function,

$^2$62

so the $^2$63-form symmetry is anomaly-free. For the twisted $^2$64 theory, one finds

$^2$65

which signals a fermionic $^2$66-form symmetry. More generally, for twisted $^2$67 gauge theory,

$^2$68

and for $^2$69 gauge theory with exchange symmetry,

$^2$70

In these examples, the quadratic refinement is identified with the $^2$71 anomaly datum of the $^2$72-form symmetry (Kapustin et al., 22 Jul 2025).

The same paper proposes a higher-dimensional generalization: from a good cover of $^2$73 by closed cones with contractible bases, one forms groups of approximately localized automorphisms on finite intersections and organizes the restriction maps and commutator liftings into a crossed $^2$74-cube. Specializing to $^2$75 recovers precisely the crossed-square situation. This suggests that crossed squares occupy the $^2$76 instance of a broader nonabelian Čech-type framework for higher symmetries in lattice models and quantum field theory (Kapustin et al., 22 Jul 2025).