2-Quasi crossed modules of commutative algebras
UDC 512.6 We define 2-quasi crossed modules of commutative algebras obtained by relaxing some 2-crossed module conditions. Moreover, we prove that there exists a functorial relationship between these two structures which enables us to construct the coproduct object in the category of 2-crossed modul...
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2022
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507032869666816 |
|---|---|
| author | Kadir, Emir Kadir, Emir Kadir, Emir |
| author_facet | Kadir, Emir Kadir, Emir Kadir, Emir |
| author_sort | Kadir, Emir |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:44:52Z |
| description | UDC 512.6
We define 2-quasi crossed modules of commutative algebras obtained by relaxing some 2-crossed module conditions. Moreover, we prove that there exists a functorial relationship between these two structures which enables us to construct the coproduct object in the category of 2-crossed modules of commutative algebras. |
| doi_str_mv | 10.37863/umzh.v74i3.467 |
| first_indexed | 2026-03-24T02:02:52Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i3.467
UDC 512.6
Kadir Emir (Dep. Math. and Stat., Masaryk Univ., Brno, Czech Republic)
2-QUASI CROSSED MODULES OF COMMUTATIVE ALGEBRAS
2-КВАЗI СХРЕЩЕНI МОДУЛI КОМУТАТИВНИХ АЛГЕБР
We define 2-quasi crossed modules of commutative algebras obtained by relaxing some 2-crossed module conditions.
Moreover, we prove that there exists a functorial relationship between these two structures which enables us to construct
the coproduct object in the category of 2-crossed modules of commutative algebras.
Дано визначення 2-квазi схрещених модулiв комутативних алгебр на базi послаблення деяких умов для 2-схрещених
модулiв. Крiм того, доведено, що iснує функторне спiввiдношення мiж цими двома структурами, яке дозволяє
збудувати об’єкт ко-добутку у категорiї 2-схрещених модулiв комутативних алгебр.
1. Introduction. Crossed modules of groups [14] are given by a group homomorphism \partial : E \rightarrow G,
together with an action \triangleleft of G on E, such that the following Peiffer relations:
XM1: \partial (g \triangleleft e) = g \partial (e) g - 1, XM2: \partial (e) \triangleleft f = e f e - 1
are satisfied, for all e, f \in E, g \in G. Without the second condition, we call it a precrossed module.
2-crossed modules of groups [7] are given by a group complex L
\delta - \rightarrow E
\partial - \rightarrow G, satisfying certain
conditions together with the actions of G on L and E, making it a complex of G-modules, where
G acts on itself by conjugation. The first Peiffer relation for the map \partial : E \rightarrow G automatically
holds, thus \partial : E \rightarrow G is a precrossed module. The second Peiffer relation does not hold in general.
However the Peiffer lifting \{ - , - \} : E \times E \rightarrow L measures how far the second Peiffer relation is
from being satisfied, namely: \delta (\{ e, f\} ) =
\bigl(
efe - 1
\bigr) \bigl(
\partial (e) \triangleleft f - 1
\bigr)
, for all e, f \in E. The category of
2-crossed modules is equivalent to a reflexive subcategory of the category of simplicial groups with
Moore complex of length two [11].
As for the group case, 2-crossed modules of commutative algebras are introduced in [8] to
obtain a method for computing the (co)homology groups of a commutative algebra with coefficients
which coincides with the Andre – Quillen theory for n = 0, 1, 2, 3. Consequently, without simplicial
theory, they get the Jacobi – Zariski sequence. The construction of 2-crossed modules of commutative
algebras depends on, essentially switching actions by automorphisms to actions by multipliers under
certain conditions. A 2-crossed module of commutative algebras L
\partial 2 - \rightarrow E
\partial 1 - \rightarrow R has an underlying
complex of commutative algebras and the following data: we have the algebra actions \triangleleft of R on E,
L; and a Peiffer lifting map \{ - , - \} : E \times E \rightarrow L satisfying the conditions given in Definition 2.4.
As in the group case, simplicial commutative algebras and 2-crossed modules of commutative
algebras are closely related. A simplicial commutative algebra [2, 10] A = (An, d
i
n, s
i
n), i.e., a simpli-
cial object in the category of commutative algebras, is given by a collection of algebra morphisms din :
An \rightarrow An - 1, i = 0, . . . , n, and sin : An \rightarrow An+1, i = 0, . . . , n, called boundaries and degeneracies
respectively, such that satisfying the well known simplicial identities. The Moore complex of the
simplicial commutative algebra A is the complex
N(A) =
\Bigl(
. . .
d(n+1) - - - - \rightarrow N(A)n
dn - \rightarrow . . .
d3 - \rightarrow N(A)2
d2 - \rightarrow N(A)1
d1 - \rightarrow A0
\Bigr)
,
c\bigcirc KADIR EMIR, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3 323
324 KADIR EMIR
where N(A)n =
\bigcap n - 1
i=0 ker(din) \subset An at level n and the boundary dn : N(A)n \rightarrow N(A)n - 1 is the
restriction of dnn : An \rightarrow A(n - 1). We say that the Moore complex of a simplicial commutative algebra
A has length n if N(A)i is trivial for all i > n.
If A has Moore complex of length one, then N(A)1
d1 - \rightarrow A0 defines a crossed module [12]. One
level further, a simplicial commutative algebra A with Moore complex of length two, corresponds
to a 2-crossed module N(A)2
d2 - \rightarrow N(A)1
d1 - \rightarrow A0 ; see [3] for details. Conversely, one can get the
corresponding simplicial commutative algebra by using a 2-crossed module. This gives an equivalence
between the categories of simplicial commutative algebras with Moore complex of length two, and
that of 2-crossed modules of commutative algebras [9].
The crossed modules of groups with a fixed codomain G are called crossed G-modules. For
any two crossed G-modules of groups \partial : E \rightarrow G and \partial \prime : E\prime \rightarrow G, the coproduct is defined via
the quotient of the free group E \ast E\prime by Brown in [4]. However, we should replace the free group
structure by the semi-direct product when we work in the category of commutative algebras [13].
However, the construction of the coproduct of 2-crossed module is definitely more complicated
than crossed modules. Because 2-crossed modules have much more data than crossed modules. To
overcome this difficulty, it was necessary to define something weaker than a 2-crossed module, yet
with some functorial relations (adjunction) again with 2-crossed modules. For this aim, in this paper,
we first define 2-quasi crossed modules in the category of commutative algebras inspired by [6].
Afterwards, we give an adjunction between the category of 2-crossed modules and the category of
2-quasi crossed modules. This adjunction allow us to define the coproduct object with the category
theoretical point of view.
2. Preliminaries. We fix a commutative ring \kappa , not necessarily with 1. All algebras considered
will be associative and commutative over \kappa , but not necessarily with a multiplicative identity.
If E and R are two algebras, a bilinear map (r, e) \in R \times E \mapsto - \rightarrow r \triangleleft e \in E is called an algebra
action of R on E if, for all e, e\prime \in e and r, r\prime \in R, we have
A1 : r \triangleleft (ee\prime ) = (r \triangleleft e) e\prime = e (r \triangleleft e\prime ), A2 : (rr\prime ) \triangleleft e = r \triangleleft (r\prime \triangleleft e).
Then we get the semidirect product E \rtimes R with
(e, r) (e\prime , r\prime ) = (r \triangleleft e\prime + r\prime \triangleleft e+ ee\prime , rr\prime ),
for all e, e\prime \in E and r, r\prime \in R.
Convention: Let L \rightarrow E \rightarrow R be a chain complex of R-algebras. The actions of R on E and L
will be both denoted by “ \triangleleft "in the rest of the paper. We say that the subalgebra E\prime of E is R-invariant
if r \triangleleft e\prime \in E\prime for all e\prime \in E\prime and r \in R. A function f : L \rightarrow E is said to be R-equivariant if
f(r \triangleleft l) = r \triangleleft f(l), for all l \in L and r \in R. Remark that R has a natural R-algebra structure where
the action is defined via its multiplication.
2.1. Crossed modules of algebras.
Definition 2.1. A precrossed module of algebras (E,R, \partial ), is given by an algebra homomorphi-
sm \partial : E \rightarrow R, together with an action \triangleleft of R on E, such that the following relation, called the
“first Peiffer relation”, holds
(XM1) \partial (r \triangleleft e) = r \partial (e), for all e \in E and r \in R.
A crossed module of algebras (E,R, \partial ) is a precrossed module satisfying, furthermore, the “second
Peiffer relation”
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
2-QUASI CROSSED MODULES OF COMMUTATIVE ALGEBRAS 325
(XM2) \partial (e) \triangleleft e\prime = e e\prime , for all e, e\prime \in E.
Example 2.1. Let R be an algebra and E \trianglelefteq R be any ideal of R. Then (E,R, i), where i :
E \rightarrow R is the inclusion map, is a crossed module. We use the multiplication in R to define the action
of R on E.
Definition 2.2. A crossed module morphism (f1, f0) : (E,R) \rightarrow (E\prime , R\prime ) consists of algebra
homomorphisms f0 : R \rightarrow R\prime and f1 : E \rightarrow E\prime such that the following diagram commutes:
E \partial //
f1
��
R
f0
��
E\prime
\partial \prime
//R\prime
and preserve the action, namely f1(r \triangleleft e) = f0(e) \triangleleft f1(e), for all r \in R and e \in E.
Thus we get the category of crossed modules of algebras denoted by XMod.
Definition 2.3. The category of crossed modules with fixed codomain R is the full subcategory
of XMod that is denoted by XMod/R. These crossed modules will be called crossed R-modules.
2.2. 2-crossed modules of algebras.
Definition 2.4. A 2-crossed module of algebras L
\partial 2 - \rightarrow E
\partial 1 - \rightarrow R is given by a chain complex
of R-algebra homomorphisms (\partial 1 \circ \partial 2 = 0), equipped with an R-equivariant bilinear map namely
r \triangleleft \{ e, e\prime \} = \{ r \triangleleft e, e\prime \} = \{ e, r \triangleleft e\prime \} , called Peiffer lifting
\{ - , - \} : E \otimes R E - \rightarrow L,
such that satisfying:
(2XM1) \partial 2\{ e, e\prime \} = ee\prime - \partial 1(e
\prime ) \triangleleft e,
(2XM2) \{ \partial 2(l), \partial 2(l\prime )\} = ll\prime ,
(2XM3) \{ e, e\prime e\prime \prime \} = \{ ee\prime , e\prime \prime \} + \partial 1(e
\prime \prime ) \triangleleft \{ e, e\prime \} ,
(2XM4) \{ e, \partial 2(l)\} - \{ \partial 2(l), e\} = \partial 1(e) \triangleleft l,
for all l, l\prime \in L, e, e\prime , e\prime \prime \in E, and r \in R.
Remark 2.1. Note that \partial 2 : L \rightarrow E is a crossed module, where E acts on L with
e \triangleleft \prime l = \{ e, \partial 2(l)\} .
However, \partial 1 : E \rightarrow R is a precrossed module in general. The Peiffer lifting in E measures exactly
the failure of \partial 1 : E \rightarrow R to be a crossed module.
Example 2.2. Let (E,R, \partial ) be a precrossed module. ker(\partial )
i - \rightarrow E
\partial - \rightarrow R, where i : ker(\partial ) \rightarrow
\rightarrow E is the inclusion map, is a 2-crossed module, where
\{ - , - \} : (e, e\prime ) \in E \otimes R E \mapsto - \rightarrow \{ e, e\prime \} = ee\prime - \partial (e) \triangleleft e\prime \in ker(\partial ).
Notation. Any 2-crossed module of algebras will be denoted by (L,E,R, \partial 1, \partial 2).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
326 KADIR EMIR
Definition 2.5. Given 2-crossed modules (L,E,R, \partial 1, \partial 2) and (L\prime , E\prime , R\prime , \partial \prime
1, \partial
\prime
2), a 2-crossed
module morphism consists of algebra homomorphisms f0 : R \rightarrow R\prime , f1 : E \rightarrow E\prime and f2 : L \rightarrow L\prime ,
making the diagram
L
\partial 2 //
f2��
E
\partial 1 //
f1��
R
f0��
L\prime
\partial \prime
2
// E\prime
\partial \prime
1
// R\prime
commutative and preserving the actions of R, R\prime , and the Peiffer lifting, namely
f1(r \triangleleft e) = f0(r) \triangleleft f1(e), for all e \in E and r \in R,
f2(r \triangleleft l) = f0(r) \triangleleft f2(l), for all l \in L and r \in R,
f2\{ e, e\prime \} = \{ f1(e), f1(e\prime )\} , for all e, e\prime \in E.
Thus we get the category of 2-crossed modules of algebras that is denoted by X2Mod.
Definition 2.6. The category of 2-crossed modules with fixed tail (E \rightarrow R) is the full subcategory
of X2Mod that is denoted by X2Mod/(E \rightarrow R). This type of 2-crossed modules will be called 2-
crossed (R \rightarrow E)-modules.
3. 2-quasi crossed modules of algebras.
Definition 3.1. A 2-quasi crossed module of algebras is a chain complex of R-algebra homo-
morphisms L
\partial 2 - \rightarrow E
\partial 1 - \rightarrow R, together with an R-equivariant bilinear map
\{ - , - \} : E \otimes R E - \rightarrow L,
satisfying the following axioms:
(2QX1) \partial 2\{ e, e\prime \} = ee\prime - \partial 1(e
\prime ) \triangleleft e,
(2QX2) \{ e, e\prime e\prime \prime \} = \{ ee\prime , e\prime \prime \} + \partial 1(e
\prime \prime ) \triangleleft \{ e, e\prime \} ,
(2QX3) \{ e\prime , e\} \partial 1(e\prime ) \triangleleft \{ e, \partial 2(l)\} = \{ ee\prime e, \partial 2(\partial 1(e\prime ) \triangleleft l)\} - (\partial 1(e
\prime ) \triangleleft e)\{ e, \partial 2(\partial 1(e\prime ) \triangleleft l)\}
for all l \in L and e, e\prime , e\prime \prime \in E.
Definition 3.2. 2-quasi crossed module morphisms can be defined in a similar way. Therefore,
we get the category of 2-quasi crossed modules of algebras denoted by QX2Mod.
The category of 2-quasi (E \rightarrow R)-modules, namely QX2Mod/(E \rightarrow R) can also be defined
according to Definition 2.6.
3.1. 2-crossed modules vs 2-quasi crossed modules.
Lemma 3.1. Any 2-crossed module is a 2-quasi crossed module. This leads an inclusion functor
X2Mod - \rightarrow QX2Mod. (1)
Proof. Let L
\partial 2 - \rightarrow E
\partial 1 - \rightarrow R be a 2-crossed module. We only have to prove that axiom 2QX3 is
verified. So we obtain
\{ e\prime , e\} \partial 1(e\prime ) \triangleleft \{ e, \partial 2(l)\} = (e\prime e - \partial 1(e) \triangleleft e
\prime )\{ e, \partial 2(\partial 1(e\prime ) \triangleleft l)\} =
= e\prime e \{ e, \partial 2(\partial 1(e\prime ) \triangleleft l)\} - (\partial 1(e) \triangleleft e
\prime )\{ e, \partial 2(\partial 1(e\prime ) \triangleleft l)\} =
= e\prime e (e \cdot (\partial 1(e\prime ) \triangleleft l)) - (\partial 1(e
\prime ) \triangleleft e) \cdot (e \cdot (\partial 1(e\prime ) \triangleleft l)) =
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
2-QUASI CROSSED MODULES OF COMMUTATIVE ALGEBRAS 327
= e
\bigl(
e\prime e (\partial 1(e
\prime ) \triangleleft l) - e((\partial 1(e
\prime ) \triangleleft e) \cdot (\partial 1(e\prime ) \triangleleft l)
\bigr)
=
= e
\bigl(
\{ e\prime e, \partial 2(\partial 1(e\prime ) \triangleleft l)\} - \{ \partial 1(e\prime ) \triangleleft e, \partial 2(\partial 1(e\prime ) \triangleleft l)\}
\bigr)
=
= \{ ee\prime e, \partial 2(\partial 1(e\prime ) \triangleleft l)\} - \{ e(\partial 1(e\prime ) \triangleleft e), \partial 2(\partial 1(e\prime ) \triangleleft l)\} =
= \{ ee\prime e, \partial 2(\partial 1(e\prime ) \triangleleft l)\} - (\partial 1(e
\prime ) \triangleleft e)\{ e, \partial 2(\partial 1(e\prime ) \triangleleft l)\} ,
for all l \in L and e, e\prime \in E, that completes the proof.
Lemma 3.2. Let L
\partial 2 - \rightarrow E
\partial 1 - \rightarrow R be a 2-quasi crossed module and let [L,L] be the ideal of L
generated by the elements of the form
e \star l = \partial 1(e) \triangleleft l - \{ e, \partial 2(l)\} + \{ \partial 2(l), e\} ,
l0 \sharp l1 = l0 l1 - \{ \partial 2(l0), \partial 2(l1)\} ,
for all l, l0, l1 \in L and e \in E. Then [L,L] is an R-invariant ideal of L.
Proof. For all r \in R, e \in E and l0, l1, l2 \in L, we get
r \triangleleft (e \star l) = r \triangleleft (\partial 1(e) \triangleleft l - \{ e, \partial 2(l)\} + \{ \partial 2(l), e\} ) =
= r \triangleleft (\partial 1(e) \triangleleft l) - r \triangleleft (\{ e, \partial 2(l)\} + \{ \partial 2(l), e\} ) =
= r\partial 1(e) \triangleleft l - r \triangleleft \{ e, \partial 2(l)\} + r \triangleleft \{ \partial 2(l), e\} =
= \partial 1(r \triangleleft e) \triangleleft l - \{ r \triangleleft e, \partial 2(l)\} + \{ \partial 2(l), r \triangleleft e\} ,
Fix r \triangleleft e = e\prime \in E; it follows
\partial 1(e
\prime ) \triangleleft l - \{ e\prime , \partial 2(l)\} + \{ \partial 2(l), e\prime \} = e\prime \ast l \in [L,L] .
If we handle the second type of elements, we get
r \triangleleft (l0 \sharp l1) = r \triangleleft (l0 l1 - \{ \partial 2(l0), \partial 2(l1)\} ) =
= r \triangleleft (l0 l1) - r \triangleleft (\{ \partial 2(l0), \partial 2(l1)\} ) =
= (r \triangleleft l0)l1 - \{ r \triangleleft \partial 2(l0), \partial 2(l1)\} =
= (r \triangleleft l0)l1 - \{ \partial 2(r \triangleleft l0), \partial 2(l1)\} ,
Fix r \triangleleft l0 = l2 \in L; then it follows
l2 l1 - \{ \partial 2(l2), \partial 2(l1)\} = l2\sharp l1 \in [L,L]
and proves that [L,L] is an R-invariant ideal.
Proposition 3.1. Hence we get the quotient R-algebra
Lcr = L/[L,L].
Lemma 3.3. The quotient map \phi : L \rightarrow L/[L,L] provides an induced functor
( )\ast cr : QX2Mod \rightarrow X2Mod
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
328 KADIR EMIR
which maps any 2-quasi crossed module L
\partial 2 - \rightarrow E
\partial 1 - \rightarrow R to a 2-crossed module
Lcr \partial cr
2 - \rightarrow E
\partial 1 - \rightarrow R (2)
with the new Peiffer lifting \{ - , - \} cr : E \otimes R E - \rightarrow Lcr given by the composition
E \otimes R E
\{ - , - \} - \rightarrow L
\phi - \rightarrow Lcr .
Proof. Axioms 2XM2 and 2XM4 are satisfied as the elements e \star l and l0\sharp l1 are already
quotiented out in the definition of Lcr.
Since \partial 2(e \ast l) = \partial 2(l0\sharp l1) = 0, we also have
\partial 2([L,L]) = 0,
for all e \in E and l, l0, l1 \in L.
Moreover we get
(2XM1) \partial cr
2 \{ e, e\prime \} cr = \partial cr
2
\bigl(
\phi \{ e, e\prime \}
\bigr)
=
= \partial 2
\bigl(
\{ e, e\prime \} + [L,L]
\bigr)
=
= \partial 2\{ e, e\prime \} + \partial 2([L,L]) =
= ee\prime - \partial 1(e
\prime ) \triangleleft e (\because 2QX1)
and
(2XM3) \{ e, e\prime e\prime \prime \} cr = \phi \{ e, e\prime e\prime \prime \} = \{ e, e\prime e\prime \prime \} + [L,L] =
=
\Bigl(
\{ ee\prime , e\prime \prime \} + \partial 1(e
\prime \prime ) \triangleleft \{ e, e\prime \}
\Bigr)
+ [L,L] (\because 2QX2) =
=
\Bigl(
\{ ee\prime , e\prime \prime \} + [L,L]
\Bigr)
+
\Bigl(
\partial 1(e
\prime \prime ) \triangleleft \{ e, e\prime \} + [L,L]
\Bigr)
=
=
\Bigl(
\{ ee\prime , e\prime \prime \} + [L,L]
\Bigr)
+ \partial 1(e
\prime \prime ) \triangleleft
\Bigl(
\{ e, e\prime \} + [L,L]
\Bigr)
=
= \phi \{ ee\prime , e\prime \prime \} + \partial 1(e
\prime \prime ) \triangleleft \phi \{ e, e\prime \} =
= \{ ee\prime , e\prime \prime \} cr + \partial 1(e
\prime \prime ) \triangleleft \{ e, e\prime \} cr
for all e, e\prime , e\prime \prime \in E, that completes the proof.
Remark that, we used the fact that [L,L] is R-invariant, in the above calculations.
Corollary 3.1. We get the following adjunction:
Moreover we get
2XM1) @cr
2 {e, e0}cr = @cr
2
�
�{e, e0}
�
= @2
�
{e, e0} + [L, L]
�
= @2{e, e0} + @2([L, L])
= ee0 � @1(e
0) . e (* 2QX1)
and
2XM3) {e, e0e00}cr = �{e, e0e00}
= {e, e0e00} + [L, L]
=
⇣
{ee0, e00} + @1(e
00) . {e, e0}
⌘
+ [L, L] (* 2QX2)
=
⇣
{ee0, e00} + [L, L]
⌘
+
⇣
@1(e
00) . {e, e0} + [L, L]
⌘
=
⇣
{ee0, e00} + [L, L]
⌘
+ @1(e
00) .
⇣
{e, e0} + [L, L]
⌘
= �{ee0, e00} + @1(e
00) . �{e, e0}
= {ee0, e00}cr + @1(e
00) . {e, e0}cr
for all e, e0, e00 2 E, that completes the proof.
Remark that, we used the fact that [L, L] is R-invariant, in the above calculations.
Corollary 3.7 We get the following adjunction:
QX2Mod
( )⇤cr
**
jj G g
X2Mod? (3)
3.2 Simplicial algebras vs 2-quasi crossed modules of algebras
We know that the category of 2-crossed modules of algebras is equivalent to the category
of simplicial algebras with Moore complex of length two. This equivalence is proven by higher
dimensional Pei↵er elements in [3] with the method introduced in [5]. Briefly, for a given simplicial
algebra A = (An, di
n, si
n) with Moore complex of length two, the subalgebras generated by
(ker d0)(ker d1 \ ker d2), (ker d1)(ker d0 \ ker d2), (ker d2)(ker d0 \ ker d1)
of A2 are all trivial. However, the notion of 2-quasi crossed modules arises by weakening some of
these conditions as follows:
• Define A0
1 be the subalgebra of ker(d1 : A2 ! A1) generated by the elements in the form
s1(x) � s0(x), for all x 2 A1,
• Define A0
2 be the subalgebra of ker(d2 : A2 ! A1) generated by the elements in the form
s0(x) � s1s0d1(x), for all x 2 A1.
Then we get the definition of 2-quasi crossed modules corresponding to the 2-truncated simplicial
algebras with the following trivial subalgebras of A2:
(A0
1) (ker d0 \ ker d2), (A0
2) (ker d0 \ ker d1).
7
.
(3)
3.2. Simplicial algebras vs 2-quasi crossed modules of algebras. We know that the category
of 2-crossed modules of algebras is equivalent to the category of simplicial algebras with Moore
complex of length two. This equivalence is proven by higher dimensional Peiffer elements in [3] with
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
2-QUASI CROSSED MODULES OF COMMUTATIVE ALGEBRAS 329
the method introduced in [5]. Briefly, for a given simplicial algebra A = (An, d
i
n, s
i
n) with Moore
complex of length two, the subalgebras generated by
(ker d0)(ker d1 \cap ker d2), (ker d1)(ker d0 \cap ker d2), (ker d2)(ker d0 \cap ker d1)
of A2 are all trivial. However, the notion of 2-quasi crossed modules arises by weakening some of
these conditions as follows:
– define A\prime
1 be the subalgebra of ker(d1 : A2 \rightarrow A1) generated by the elements in the form
s1(x) - s0(x), for all x \in A1,
– define A\prime
2 be the subalgebra of ker(d2 : A2 \rightarrow A1) generated by the elements in the form
s0(x) - s1s0d1(x), for all x \in A1.
Then we get the definition of 2-quasi crossed modules corresponding to the 2-truncated simplicial
algebras with the following trivial subalgebras of A2 :
(A\prime
1) (ker d0 \cap ker d2), (A\prime
2) (ker d0 \cap ker d1).
4. Coproduct of 2-quasi crossed modules. Let us recall the coproduct of crossed modules of
algebras from [13].
4.1. Coproduct of crossed modules. Let (A,R, \partial 1) and (B,R, \partial 2) be two crossed R-modules.
There exists an action of B on A with
b \triangleleft a = \partial 2(b) \triangleleft a . (4)
Then, we have the semidirect product B \rtimes A. Define \partial : B \rtimes A \rightarrow R by
\partial (b, a) = \partial 2(b) + \partial 1(a).
for all (b, a) \in B \rtimes A. Here \partial becomes a precrossed module where R acts on B \rtimes A in a natural
way, since
\partial (r \triangleleft (b, a)) = \partial (r \triangleleft b, r \triangleleft a) = \partial 2(r \triangleleft b) + \partial 1(r \triangleleft a) =
= r \partial 2(b) + r \partial 1(a) = r
\bigl(
\partial 2(b) + \partial 1(a)
\bigr)
= r \partial (b, a),
for all r \in R and (b, a) \in B \rtimes A.
Let P be the ideal of B \rtimes A generated by the elements of the form
(b, a)(b\prime , a\prime ) - \partial (b, a) \triangleleft (b\prime , a\prime )
for all (b, a), (b\prime , a\prime ) \in B \rtimes A. On the other hand, we have (by using (4))
(b, a)(b\prime , a\prime ) - \partial (b, a) \triangleleft (b\prime , a\prime ) = (b, a) \cdot (b\prime , a\prime ) - (\partial 2(b) + \partial 1(a)) \triangleleft (b
\prime , a\prime ) =
= (bb\prime , \partial 2(b) \triangleleft a
\prime + \partial 2(b
\prime ) \triangleleft a+ aa\prime ) -
\bigl(
(\partial 2(b) + \partial 1(a)) \triangleleft b
\prime , (\partial 2(b) + \partial 1(a)) \triangleleft a
\prime \bigr) =
= (bb\prime , \partial 2(b) \triangleleft a
\prime + \partial 2(b
\prime ) \triangleleft a+ aa\prime ) -
\bigl(
\partial 2(b) \triangleleft b
\prime + \partial 1(a) \triangleleft b
\prime , \partial 2(b) \triangleleft a
\prime + \partial 1(a) \triangleleft a
\prime \bigr) =
= (bb\prime , \partial 2(b) \triangleleft a
\prime + \partial 2(b
\prime ) \triangleleft a+ aa\prime ) -
\bigl(
bb\prime + \partial 1(a) \triangleleft b
\prime , \partial 2(b) \triangleleft a
\prime + aa\prime
\bigr)
=
= ( - \partial 1(a) \triangleleft b
\prime , \partial 2(b
\prime ) \triangleleft a).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
330 KADIR EMIR
That means P is generated by the elements
( - \partial 1(a) \triangleleft b
\prime , \partial 2(b
\prime ) \triangleleft a).
Remark that \partial (P ) = 0. Hence we have the induced morphism
\=\partial : (B \rtimes A)/P \rightarrow R,
defined by
\=\partial
\bigl(
(b, a) + P
\bigr)
= \partial 2(b) + \partial 1(a),
gives us a crossed module since, for all (b, a), (b\prime , a\prime ) \in B \rtimes A we have
\=\partial ((b, a) + P ) \triangleleft ((b\prime , a\prime ) + P ) = (\partial 2(b) + \partial 1(a)) \triangleleft ((b
\prime , a\prime ) + P ) =
=
\bigl(
\partial 2(b) \triangleleft b
\prime + \partial 1(a) \triangleleft b
\prime , \partial 2(b) \triangleleft a
\prime + \partial 1(a) \triangleleft a
\prime \bigr) =
= (bb\prime + \partial 1(a) \triangleleft b
\prime , \partial 2(b) \triangleleft a
\prime + aa\prime ).
We know that ( - \partial 1(a) \triangleleft b
\prime , \partial 2(b
\prime ) \triangleleft a) \in P, it follows
(bb\prime + \partial 1(a) \triangleleft b
\prime , \partial 2(b) \triangleleft a
\prime + aa\prime ) = (bb\prime , \partial 2(b
\prime ) \triangleleft a+ \partial 2(b) \triangleleft a
\prime + aa\prime ) =
= ((b, a)(b\prime , a\prime )) + P = ((b, a) + P )((b\prime , a\prime ) + P ).
Therefore we get the crossed module ((B \rtimes A)/P,R, \partial ) which is the coproduct in XMod/R.
4.2. Coproduct of 2-quasi crossed modules. Let us fix two 2-quasi crossed (E \rightarrow R)-modules
\scrA = L1
\partial 2 - \rightarrow E
\partial 1 - \rightarrow R, \{ - , - \} 1 : E \times E \rightarrow L1,
\scrA \prime = L2
\partial \prime
2 - \rightarrow E
\partial 1 - \rightarrow R, \{ - , - \} 2 : E \times E \rightarrow L2
throughout the entire section.
Remark that, we have
\partial 2\{ e, e\prime \} 1 = \partial \prime
2\{ e, e\prime \} 2 = ee\prime - \partial 1(e
\prime ) \triangleleft e,
when we consider \scrA and \scrA \prime .
Construct L1 \rtimes L2 in the sense of subsection 4.1. Then let P be the ideal of L1 \rtimes L2 generated
by the elements
(\epsilon 1\{ e, e\prime \} 1, \epsilon 2\{ e, e\prime \} 2),
where \epsilon i = \pm 1 and \epsilon 1 \not = \epsilon 2.
Define the Peiffer lifting
\{ - , - \} : E \times E \rightarrow (L1 \rtimes L2)/P,
with
\{ e, e\prime \} = (\{ e, e\prime \} 1, 0) + P = (0, \{ e, e\prime \} 2) + P,
by considering (\{ e, e\prime \} 1, - \{ e, e\prime \} 2) \in P.
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2-QUASI CROSSED MODULES OF COMMUTATIVE ALGEBRAS 331
E acts on (L1 \rtimes L2)/P in a natural way, namely
e \triangleleft ((l, l\prime ) + P ) = (e \triangleleft l, e \triangleleft l\prime ) + P,
for all e \in E and (l, l\prime ) \in L1 \rtimes L2.
There exists an induced morphism
\=\partial : (L1 \rtimes L2)/P \rightarrow E,
where
\=\partial
\bigl(
(l1, l2) + P
\bigr)
= \partial 2(l1) + \partial \prime
2(l2),
by using the fact that \partial (P ) = 0.
Thus we get a 2-quasi crossed module
(L1 \rtimes L2)/P
\=\partial - \rightarrow E
\partial 1 - \rightarrow R
since
(2QX1) \=\partial \{ e, e\prime \} = \=\partial (\{ e, e\prime \} 1, 0) + P =
= \partial 2\{ e, e\prime \} 1 = ee\prime - \partial 1(e
\prime ) \triangleleft e,
(2QX2) \{ e, e\prime e\prime \prime \} = (\{ e, e\prime e\prime \prime \} 1, 0) + P =
=
\bigl(
\{ ee\prime , e\prime \prime \} 1 + \partial 1(e
\prime \prime ) \triangleleft \{ e, e\prime \} 1, 0
\bigr)
+ P =
= (\{ ee\prime , e\prime \prime \} 1, 0) + P + (\partial 1(e
\prime \prime ) \triangleleft \{ e, e\prime \} 1, 0) + P =
= \{ ee\prime , e\prime \prime \} + \partial 1(e
\prime \prime ) \triangleleft \{ e, e\prime \}
and also
2QX3) \{ e, e\prime \} \partial 1(e) \triangleleft \{ e\prime , \=\partial ((l, l\prime ) + P )\} = \{ e, e\prime \} \partial 1(e) \triangleleft \{ e\prime , \partial 2(l) + \partial \prime
2(l
\prime )\} =
= \{ e, e\prime \} \partial 1(e) \triangleleft (\{ e\prime , \partial 2(l)\} + \{ e\prime , \partial \prime
2(l
\prime )\} ) =
= \{ e, e\prime \}
\bigl(
\partial 1(e) \triangleleft \{ e\prime , \partial 2(l)\} + \partial 1(e) \triangleleft \{ e\prime , \partial \prime
2(l
\prime )\}
\bigr)
=
= \{ e, e\prime \} \partial 1(e) \triangleleft \{ e\prime , \partial 2(l)\} + \{ e, e\prime \} \partial 1(e) \triangleleft \{ e\prime , \partial \prime
2(l
\prime )\} =
= ((\{ e, e\prime \} 1, 0) + P )\partial 1(e) \triangleleft ((\{ e\prime , \partial 2(l)\} 1, 0) + P )+
+((0, \{ e, e\prime \} 2) + P )\partial 1(e) \triangleleft ((0, \{ e\prime , \partial \prime
2(l
\prime )\} ) + P ) =
= ((\{ e, e\prime \} 1, 0)\partial 1(e) \triangleleft (\{ e\prime , \partial 2(l)\} 1, 0) + P )+
+((0, \{ e, e\prime \} 2)\partial 1(e) \triangleleft (0, \{ e\prime , \partial \prime
2(l
\prime )\} ) + P ) =
= ((\{ e, e\prime \} 1\partial 1(e) \triangleleft \{ e\prime , \partial 2(l)\} 1, 0) + P )+
+((0, \{ e, e\prime \} 2\partial 1(e) \triangleleft \{ e\prime , \partial \prime
2(l
\prime )\} 2) + P ) =
= ((\{ e\prime ee\prime , \partial 2(\partial 1(e) \triangleleft l)\} 1 - (\partial 1(e) \triangleleft e
\prime )\{ e\prime , \partial 2(\partial 1(e) \triangleleft l)\} 1, 0) + P )+
+(
\bigl(
0, \{ e\prime ee\prime , \partial \prime
2(\partial 1(e) \triangleleft l
\prime )\} 2 - (\partial 1(e) \triangleleft e
\prime )\{ e\prime , \partial \prime
2(\partial 1(e) \triangleleft l
\prime )\} 2
\bigr)
+ P ) =
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
332 KADIR EMIR
= ((\{ e\prime ee\prime , \partial 2(\partial 1(e) \triangleleft l)\} 1, 0) + P ) - (\partial 1(e) \triangleleft e
\prime )((\{ e\prime , \partial 2(\partial 1(e) \triangleleft l)\} 1, 0) + P )+
+((0, \{ e\prime ee\prime , \partial \prime
2(\partial 1(e) \triangleleft l
\prime )\} 2) + P ) - (\partial 1(e) \triangleleft e
\prime )((0, \{ e\prime , \partial \prime
2(\partial 1(e) \triangleleft l
\prime )\} 2)
\bigr)
+ P ) =
= \{ e\prime ee\prime , \partial 2(\partial 1(e) \triangleleft l)\} - (\partial 1(e) \triangleleft e
\prime )\{ e\prime , \partial 2(\partial 1(e) \triangleleft l)\} +
+\{ e\prime ee\prime , \partial \prime
2(\partial 1(e) \triangleleft l
\prime )\} - (\partial 1(e) \triangleleft e
\prime )\{ e\prime , \partial \prime
2(\partial 1(e) \triangleleft l
\prime )\} =
= \{ e\prime ee\prime , \partial 2(\partial 1(e) \triangleleft l) + \partial \prime
2(\partial 1(e) \triangleleft l
\prime )\} -
- (\partial 1(e) \triangleleft e
\prime )\{ e\prime , \partial 2(\partial 1(e) \triangleleft l) + \partial \prime
2(\partial 1(e) \triangleleft l
\prime )\} =
= \{ e\prime ee\prime , \=\partial ((\partial 1(e) \triangleleft l, \partial 1(e) \triangleleft l\prime ) + P )\} -
- (\partial 1(e) \triangleleft e
\prime )\{ e\prime , \=\partial (\partial 1(e) \triangleleft l, \partial 1(e) \triangleleft l\prime ) + P )\} =
= \{ e\prime ee\prime , \=\partial (\partial 1(e) \triangleleft ((l, l\prime ) + P ))\} - (\partial 1(e) \triangleleft e
\prime )\{ e\prime , \=\partial (\partial 1(e) \triangleleft ((l, l\prime ) + P ))\} ,
for all e, e\prime , e\prime \prime \in E and ((l, l\prime ) + P ) \in (L1 \rtimes L2)/P.
Theorem 4.1. Given two 2-quasi crossed (E \rightarrow R)-modules \scrA and \scrA \prime , we have the coproduct
\scrA \amalg
QX2Mod
\scrA \prime = (L1 \rtimes L2)/P
\=\partial - \rightarrow E
\partial 1 - \rightarrow R
in the category QX2Mod/(E \rightarrow R).
Proof. Let
L1
\partial 2 //
\alpha
��
E
\partial 1 //
id
��
R
id
��
D
\partial \prime \prime
2
// E
\partial 1
// R
and
L2
\partial \prime
2 //
\beta
��
E
\partial 1 //
id
��
R
id
��
D
\partial \prime \prime
2
// E
\partial 1
// R
be two 2-quasi crossed module morphisms. Then there exists a unique 2-quasi crossed module
morphism
(L1 \rtimes L2)/P
\=\partial //
\phi
��
E
\partial 1 //
id
��
R
id
��
D
\partial \prime \prime
2
// E
\partial 1
// R
where the morphism
\phi : (L1 \rtimes L2)/P - \rightarrow D,
is given by
\phi ((l1, l2) + P ) = \alpha (l1) + \beta (l2),
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
2-QUASI CROSSED MODULES OF COMMUTATIVE ALGEBRAS 333
which satisfies the universal property of the coproduct object with the following diagram, and
completes the proof.
D
\partial \prime \prime
2 - \rightarrow E
\partial 1 - \rightarrow R
L1
\partial 2 - \rightarrow E
\partial 1 - \rightarrow R
(\alpha ,id,id)
66
(i1,id,id)
// (L1 \rtimes L2)/P
\=\partial - \rightarrow E
\partial 1 - \rightarrow R
(\phi ,id,id)
OO
L2
\partial \prime
2 - \rightarrow E
\partial 1 - \rightarrow R .
(\beta ,id,id)
hh
(i2,id,id)
oo
5. Coproduct of 2-crossed modules. In this section, we construct the coproduct object in the
category X2Mod/(E \rightarrow R) through 2-quasi crossed modules and their functorial relationship with
2-crossed modules.
First of all, let us denote two fixed 2-crossed (E \rightarrow R)-modules
\scrA = L1
\partial 2 - \rightarrow E
\partial 1 - \rightarrow R, \{ - , - \} 1 : E \times E \rightarrow L1,
\scrA \prime = L2
\partial \prime
2 - \rightarrow E
\partial 1 - \rightarrow R, \{ - , - \} 2 : E \times E \rightarrow L2.
Theorem 5.1. Given two 2-crossed (E \rightarrow R)-modules \scrA and \scrA \prime , we have the coproduct
\scrA \amalg
X2Mod
\scrA \prime =
\Bigl(
\scrA \amalg
QX2Mod
\scrA \prime
\Bigr) \ast
cr
in the category X2Mod/(E \rightarrow R).
Proof. Suppose that we have 2-crossed (E \rightarrow R)-modules \scrA , \scrA \prime . Considering the inclusion
functor X2Mod \rightarrow QX2Mod given in (1), we naturally have 2-quasi crossed (E \rightarrow R)-modules \scrA
and \scrA \prime . Thus, from Theorem 4.1, we obtain
\scrA \amalg
QX2Mod
\scrA \prime = (L1 \rtimes L2)/P
\=\partial - \rightarrow E
\partial 1 - \rightarrow R (5)
which is the coproduct object in the category QX2Mod/(E \rightarrow R).
Recall from (3) that the functor
( )\ast cr : QX2Mod - \rightarrow X2Mod (6)
is left adjoint to the inclusion functor (1).
From the categorical point of view, it is a well-known property that left adjoints preserve colimits;
moreover, the coproduct object is defined as a colimit over the diagram that consists of just two objects
[1]. Consequently, the functor ( )\ast cr maps coproducts to coproducts.
Then, when we apply the functor (6) to (5) we get the 2-crossed module
\Bigl(
\scrA \amalg
QX2Mod
\scrA \prime
\Bigr) \ast
cr
=
\Bigl(
(L1 \rtimes L2)/P
\Bigr) cr \=\partial cr
- - - \rightarrow E
\partial 1 - - \rightarrow R (7)
that follows from (2); which gives the coproduct \scrA \amalg
X2Mod
\scrA \prime in the category X2Mod/(E \rightarrow R).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
334 KADIR EMIR
Remark 5.1. In fact, (7) defines the coproduct of (\scrA )\ast cr and (\scrA \prime )\ast cr in general. However, the
inclusion map in our adjunction (3) provides that: when we have a 2-crossed module and apply the
inclusion functor and ( )\ast cr respectively, we obtain the same 2-crossed module up to isomorphism.
Summarily, \scrA \in X2Mod
5 Coproduct of 2-crossed modules
In this section, we construct the coproduct object in the category X2Mod/(E ! R) through
2-quasi crossed modules and their functorial relationship with 2-crossed modules.
First of all, let us denote two fixed 2-crossed (E ! R)-modules
A = L1
@2�! E
@1�! R , {�,�}1 : E ⇥ E ! L1
A0 = L2
@0
2�! E
@1�! R , {�,�}2 : E ⇥ E ! L2
Theorem 5.1 Given two 2-crossed (E ! R)-modules A and A0, we have the coproduct
A q
X2Mod
A0 =
⇣
A q
QX2Mod
A0
⌘⇤
cr
in the category X2Mod/(E ! R).
Proof : Suppose that we have 2-crossed (E ! R)-modules A, A0. Considering the inclusion
functor X2Mod ! QX2Mod given in (1), we naturally have 2-quasi crossed (E ! R)-modules A
and A0. Thus, from Theorem 4.1, we obtain
A q
QX2Mod
A0 = (L1 o L2)/P
@̄�! E
@1�! R (5)
which is the coproduct object in the category QX2Mod/(E ! R).
Recall from (3) that the functor
( )⇤cr : QX2Mod �! X2Mod (6)
is left adjoint to the inclusion functor (1).
From the categorical point of view, it is a well-known property that left adjoints preserve
colimits; moreover, the coproduct object is defined as a colimit over the diagram that consists of
just two objects [1]. Consequently, the functor ( )⇤cr maps coproducts to coproducts.
Then, when we apply the functor (6) to (5) we get the 2-crossed module
⇣
A q
QX2Mod
A0
⌘⇤
cr
=
⇣
(L1 o L2)/P
⌘cr @̄cr
���! E
@1��! R (7)
that follows from (2); which gives the coproduct A q
X2Mod
A0 in the category X2Mod/(E ! R).
Remark 5.2 In fact, (7) defines the coproduct of (A)⇤cr and (A0)⇤cr in general. However, the
inclusion map in our adjunction (3) provides that: when we have a 2-crossed module and apply the
inclusion functor and ( )⇤cr respectively, we obtain the same 2-crossed module up to isomorphism.
Summarily, A 2 X2Mod ,���! A 2 QX2Mod
( )⇤cr���! (A)⇤cr
⇠= A 2 X2Mod.
12
\scrA \in QX2Mod
( )\ast cr - - - \rightarrow (\scrA )\ast cr
\sim = A \in X2Mod.
Acknowledgment. The author thanks the anonymous referee for his/her comments to
improve the paper. This research was supported by the projects Mathematical Structures
9 (MUNI/A/0885/2019), and Group Techniques and Quantum Information (MUNI/G/1211/2017)
by Masaryk University Grant Agency (GAMU).
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Topology, 23, 337 – 345 (1984).
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Received 05.11.18,
after revision — 08.04.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
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| id | umjimathkievua-article-467 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:52Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/58/4adf67c8c16eb8ffce4cd02ddf706658.pdf |
| spelling | umjimathkievua-article-4672025-03-31T08:44:52Z 2-Quasi crossed modules of commutative algebras 2-Quasi Crossed Modules of Commutative Algebras 2-Quasi crossed modules of commutative algebras Kadir, Emir Kadir, Emir Kadir, Emir 2-quasi crossed module commutative algebra coproduct 2-quasi crossed module commutative algebra coproduct UDC 512.6 We define 2-quasi crossed modules of commutative algebras obtained by relaxing some 2-crossed module conditions. Moreover, we prove that there exists a functorial relationship between these two structures which enables us to construct the coproduct object in the category of 2-crossed modules of commutative algebras. UDC 512.6 2-квазі схрещені модулі комутативних алгебр Дано визначення 2-квазi схрещених модулiв комутативних алгебр на базi послаблення деяких умов для 2-схрещених модулiв. Крiм того, доведено, що iснує функторне спiввiдношення мiж цими двома структурами, яке дозволяє збудувати об’єкт ко-добутку у категорiї 2-схрещених модулiв комутативних алгебр. Institute of Mathematics, NAS of Ukraine 2022-04-26 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/467 10.37863/umzh.v74i3.467 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 3 (2022); 323-334 Український математичний журнал; Том 74 № 3 (2022); 323-334 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/467/9200 Copyright (c) 2022 kadir emir |
| spellingShingle | Kadir, Emir Kadir, Emir Kadir, Emir 2-Quasi crossed modules of commutative algebras |
| title | 2-Quasi crossed modules of commutative algebras |
| title_alt | 2-Quasi Crossed Modules of Commutative Algebras 2-Quasi crossed modules of commutative algebras |
| title_full | 2-Quasi crossed modules of commutative algebras |
| title_fullStr | 2-Quasi crossed modules of commutative algebras |
| title_full_unstemmed | 2-Quasi crossed modules of commutative algebras |
| title_short | 2-Quasi crossed modules of commutative algebras |
| title_sort | 2-quasi crossed modules of commutative algebras |
| topic_facet | 2-quasi crossed module commutative algebra coproduct 2-quasi crossed module commutative algebra coproduct |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/467 |
| work_keys_str_mv | AT kadiremir 2quasicrossedmodulesofcommutativealgebras AT kadiremir 2quasicrossedmodulesofcommutativealgebras AT kadiremir 2quasicrossedmodulesofcommutativealgebras |