Theory and Applications of Categories, Vol. 25, No. 16, 2011, pp. 426–435.

SEMIDIRECT PRODUCTS AND CROSSED MODULES IN VARIETIES OF RIGHT Ω-LOOPS

EDWARD B. INYANGALA

Abstract. We present a new explicit construction of categorical semidirect products in an arbitrary variety V of right Ω-loops and use it to obtain simplified descriptions of internal precrossed and crossed modules in V.

1. Introduction

Categorical semidirect products were introduced by D. Bourn and G. Janelidze in [6], and, as follows from the results of [6], they exist in all semi-abelian categories in the sense of G. Janelidze, L. Márki and W. Tholen [13], and in particular in all semi-abelian varieties of universal algebras. They have also been studied in several contexts by various authors; see e.g. F. Borceux, G. Janelidze, and G.M. Kelly [4], S. Mantovani and G. Metere [14], G. Metere and A. Montoli [15], and references therein. This paper is devoted to their construction in an arbitrary variety V of right Ω-loops, and to an accordingly simplified description of internal precrossed and crossed modules in V in the sense of G. Janelidze [12].

While Ω-groups, also called groups with multiple operators are well known from P.J. Higgins [9], the Ω-loops are defined similarly, just replacing the group structure with a loop structure. As it was observed already in [9], they share many basic properties of Ω-groups; it is also known that some of such properties, and in particular Bourn protomodularity, hold for right (and left) Ω-loops too. The following definition of right Ω-loop should be considered as well known; according to the definition of left closed magma in [3], the term right closed Ω-magma would also be appropriate.

1.1. Definition. A variety V of right Ω-loops is a pointed variety of universal algebras that has, among its terms, a binary + and a binary − satisfying the identities

x + 0 = x,  (1)
0 + x = x,  (2)
(x − y) + y = x,  (3)
(x + y) − y = x,  (4)

Received by the editors 2010-02-18 and, in revised form, 2011-07-05.

Transmitted by Walter Tholen. Published on 2011-10-30.

2000 Mathematics Subject Classification: 08C05, 18D35, 18G50, 18C10.

Key words and phrases: semidirect products, variety of right loops, crossed module, precrossed module.

\begin{matrix}c \\ ◯ \end{matrix} Edward B. Inyangala, 2011. Permission to copy for private use granted.

where 0 is the unique constant of V.

1.2. Remark. The identities above easily imply that x − x = 0. Indeed, x − x = (0 + x) − x = 0.

Briefly, our descriptions of semidirect products and (pre)crossed modules are obtained using the fact that the right Ω-loops are, in some sense, exactly the algebraic structures whose semidirect products have the corresponding cartesian products as their underlying sets.

2. Preliminaries

A pointed finitely cocomplete category C is said to be semi-abelian (in the sense of [13]) if it is exact in the sense of M. Barr [1] and protomodular in the sense of D. Bourn [5] (see also F. Borceux [2] and F. Borceux and D. Bourn [3]). Since every variety of universal algebras is (small-)cocomplete and exact, it is semi-abelian if and only if it is pointed and protomodular. As shown by D. Bourn and G. Janelidze [7], a variety of universal algebras is protomodular if and only if it admits nullary terms e_{1}, ..., e_{n}, binary terms t_{1}, ..., t_{n}, and (n + 1)-ary term t, satisfying the identities

t(t_{1}(x, y), ..., t_{n}(x, y), y) = x and t_{i}(x, x) = e_{i} (i = 1, ..., n).

In particular every variety of right Ω-loops is protomodular, and therefore semi-abelian: just take n = 1, e_{1} = 0, t_{1}(x, y) = x − y, and t(x, y) = x + y. Moreover, the pointed case is well known in universal algebra from the work of A. Ursini and his collaborators (see [8], [18], [19]).

For an object B in a semi-abelian category C consider the diagram

\xymatrix{ & P t_{C}(B) \ar[dr]^{K} \ar[dl]^{U} \\ C \ar[ur]^{F} \ar[rr]_{F^{T}} &  & _{C_{T}} \ar[ll]_{U^{T}} \ar[ul]^{L}}  (5)

in which:

• P t_{C}(B) is the category of points over B as used in various above-mentioned papers (originally [5]). That is, the objects in P t_{C}(B) are triples (A, α, β), where α : A ⟶ B and β : B ⟶ A are morphisms in C in with αβ = 1_{B}. A morphism f : (A, α, β) ⟶ (A′, α′, β′) in P t_{C}(B) is a morphism f : A ⟶ A′ in C with α′f = α and fβ = β′.

• U is a functor defined by U(A, α, β) = Ker(α), and F is its left adjoint, defined therefore by F (X) = (B + X, [1, 0], ι_{1}), in the obvious notation.

• T is the monad on C determined by the adjoint pair (F, U), C^{T} the category of T^{B}-algebras, and U^{T} , F^{T} , and K are the corresponding forgetful functor, free functor, and comparison functor respectively. L is the left adjoint of K. Recall that, for X in C, we have T (X) = B♭X = ker([1, 0] : B + X ⟶ B).

As shown in [6], the functor U is monadic; and, according to [6], for a T^{B}-algebra (X, ξ) (or a B-action (X, ξ) in the sense of [4]), the semidirect product (B ⋉ (X, ξ), π_{ξ}, ι_{ξ}) is defined as the object in P t_{C}(B) corresponding to (X, ξ) under the equivalence (K, L). Equivalently,

(B ⋉ (X, ξ), π_{ξ}, ι_{ξ}) = L(X, ξ).  (6)

As usually, by the semidirect product of B and (X, ξ) we will sometimes mean (just) the object B ⋉ (X, ξ). When C is a pointed protomodular variety of universal algebras, we have

B♭X = {t(b_{1}, ..., b_{p}, x_{1}, ..., x_{q}) ∈ B + X| t(b_{1}, ..., b_{p}, 0, ..., 0) = 0},  (7)

and B ⋉(X, ξ) can be presented as B ⋉ (X, ξ) = (B +X)/E, where E is the congruence on the coproduct B + X generated by {(t, ξ(t))|t ∈ B♭X}, considering both B♭X and X as subalgebras of B + X.

Let us also recall the explicit description of the comparison functor K. For (A, α, β) in P t_{C}(B), consider the diagram

[figure: B♭X; ξ; κ_{B,X}; B + X; [1,0]; [β,κ]; B; ι_{1}; X; κ; A; α; B; β]  (8)

where (B♭X, κ_{B,X}) is the kernel of [1, 0], (X, κ) is the kernel of α, and ξ is the induced morphism between these kernels. We can write K(A, α, β) = (X, ξ).

3. Simplified description of semidirect products

The main result of this paper is the following:

3.1. Theorem. Let V variety of right Ω-loops. Given an object B and a T^{B}-algebra (X, ξ), the semidirect product B ⋉ (X, ξ) is the set-theoretical product B × X equipped with the Ω-algebra structure defined by:

ω((b_{1}, x_{1}), ..., (b_{n}, x_{n})) = (ω(b_{1}, ..., b_{n}), ξ(ω(x_{1} + b_{1}, ..., x_{n} + b_{n}) − ω(b_{1}, ..., b_{n}))), (9)

for each n-ary operation ω ∈ Ω and for all b_{1}, ..., b_{n} ∈ B, x_{1}, ..., x_{n} ∈ X. The corresponding π_{ξ} : B × X ⟶ B and ι_{ξ} : B ⟶ B × X are given by π_{ξ}(b, x) = b and ι_{ξ}(b) = (b, 0) respectively.

Proof. Consider the diagram in the category of sets

[figure: X; κ; A; α; ψ; B; β; X; ⟨0,1⟩; B × X; φ; π_{1}; B; ⟨1,0⟩]  (10)

where αβ = 1_{B}, κ = ker(α) and the maps φ and ψ are defined as follows:

φ : B × X ⟶ A, (b, x) ⟼ κ(x) + β(b)
ψ : A ⟶ B × X, a ⟼ (α(a), κ^{−1}(a − βα(a))) .

We then have

ψφ(b, x) = ψ(κ(x) + β(b)) = (b, κ^{−1}((κ(x) + β(b)) − β(b)) = (b, x)
φψ(a) = φ(α(a), κ^{−1}(a − βα(a))) = (a − βα(a)) + βα(a) = a.

Therefore φ and ψ are bijections, inverse to each other. Together with (6) and (8) this allows us to construct the semidirect product B ⋉ (X, ξ) as the cartesian product B × X (in the category of sets) equipped with the unique algebraic structure making φ and ψ isomorphisms. For this, structure we have

ω((b_{1}, x_{1}), ...(b_{n}, x_{n})) = ψ(ω(φ(b_{1}, x_{1}), ..., φ(b_{n}, x_{n}))) (using ψφ = 1_{B×X})

= ψ(ω(κ(x_{1}) + β(b_{1}), ..., κ(x_{n}) + β(b_{n}))) = (α(ω(κ(x_{1}) + β(b_{1}), ..., κ(x_{n}) + β(b_{n}))), κ^{−1}[ω(κ(x_{1}) + β(b_{1}), ..., κ(x_{n}) + β(b_{n})) − βαω(κ(x_{1}) + β(b_{1}), ..., κ(x_{n}) + β(b_{n}))])

= (ω(b_{1}, ..., b_{n}), κ^{−1}[ω(κ(x_{1}) + β(b_{1}), ..., κ(x_{n}) + β(b_{n})) − ω(β(b_{1}), ..., β(b_{n}))])
= (ω(b_{1}, ..., b_{n}), κ^{−1}[β, κ](ω(x_{1} + b_{1}, ..., x_{n} + b_{n}) − ω(b_{1}, ..., b_{n})))

= (ω(b_{1}, ..., b_{n}), ξ(ω(x_{1} + b_{1}, ..., x_{n} + b_{n}) − ω(b_{1}, ..., b_{n}))) (11)

where the last equality follows from the commutativity of the left-hand square in diagram (8). Next, we have αφ = π_{1}, φ⟨1, 0⟩ = β and φ⟨0, 1⟩ = κ in (10). Indeed

αφ(b, x) = ακ(x) + αβ(x) = 0 + x,
φ⟨0, 1⟩(x) = φ(0, x) = κ(x) + 0 = κ(x),
φ⟨1, 0⟩(b) = φ(b, 0) = 0 + β(b) = β(b).

This allows us to identify π_{ξ} with π_{1} : B × X ⟶ B and ι_{ξ} with ⟨1, 0⟩ : B ⟶ B × X. Therefore

π_{ξ}(b, x) = π_{1}(b, x) = b, ι_{ξ}(b) = ⟨1, 0⟩(b) = (b, 0).

The identities (1), (2), (3) and (4) of Definition 1.1 are actually not only sufficient but also necessary for our definition of φ and ψ to determine bijections inverse to each other and to make the relevant parts of diagram (10) commutative. To show this consider the following three instances of diagram (10):

[figure: X; κ=1_{X}; X; α=0; ψ; 0; β=0; X; ⟨0,1⟩; 0 × X; φ; π_{1}; 0; ⟨0,0⟩]  (12)

[figure: 0; κ=0; X; α=1_{X}; ψ; X; β=1_{X}; 0; ⟨0,0⟩; X × 0; φ; π_{1}; X; ⟨1,0⟩]  (13)

[figure: X; κ=⟨0,1⟩; X × X; α=π_{1}; ψ; X; β=⟨1,1⟩; X; ⟨0,1⟩; X × X; φ; π_{1}; X; ⟨1,0⟩]  (14)

In (12), we have φ⟨0, 1⟩(x) = φ(0, x) = x + 0, κ(x) = x, and since the left-hand square commutes, we have x + 0 = x.

In (13), we have αφ(x, 0) = 0 + x, π_{1}(x, 0) = 0 and since αφ = π_{1} , we obtain 0 + x = x.

In (14), we have

φψ(y, x) = φ(y, κ^{−1}((y, x) − (y, y)))
= ((y, x) − (y, y)) + (y, y)
= ((y − y) + y, (x − y) + y),

ψφ(y, x) = ψ((0, x) + (y, y))
= ψ(0 + y, x + y) = ψ(y, x + y)
= (y, κ^{−1}((y, x + y) − (y, y)))
= (y, κ^{−1}(y − y, (x + y) − y))

and since φ and ψ must be inverse to each other, we obtain (x − y) + y = x and (x + y) − y = x.

3.2. Remarks. (a) As follows from (9), we have

(b_{1}, x_{1}) + (b_{2}, x_{2}) = (b_{1} + b_{2}, ξ(((x_{1} + b_{1}) + (x_{2} + b_{2})) − (b_{1} + b_{2}))).

(b) For any variety of right Ω-loops, we have

(0, x)+(b, 0) = (0+b, ξ(((x+0)+(0+b))−(0+b))) = (b, ξ((x+b)−b)) = (b, ξ(x)) = (b, x),

and in particular for Ω-groups this gives (b_{1}, x_{1}) + (b_{2}, x_{2}) = (b_{1} + b_{2}, x_{1} + ξ(b_{1} + x_{2} − b_{1})). for all b_{1}, b_{2} ∈ B and x_{1}, x_{2} ∈ X.

4. Crossed modules in a variety of right Ω-loops

In this section we apply the construction of the semidirect product to describe precrossed and crossed modules in varieties of right Ω-loops. Let us recall the following definitions, in which ι_{2} : X ⟶ B + X and κ_{B,X} : B♭X ⟶ B + X denote the second coproduct injection and the kernel of [1, 0] : B + X ⟶ B respectively.

4.1. Definition. [12] An internal precrossed module in a semi-abelian category C is a 4-tuple (B, X, ξ, δ) in which (X, ξ) is a B-action and δ : X ⟶ B a morphism in C such that the diagram

\xymatrix{B♭X \ar[d]_{ξ} \ar[r]^{κ_{B,X}} & B + X \ar[d]^{[1,δ]} \\ X \ar[r]^{δ} & B}  (15)

commutes.

4.2. Definition. [12] An internal crossed module in a semi-abelian category C is an internal precrossed module (B, X, ξ, δ) for which the diagram

\xymatrix{(B + X)♭X \ar[r]^{[1_{B},δ]♭1_{X}} \ar[d]_{_{[1_{B+X},ι_{2}]_{♯}}} & B♭X \ar[d]^{ξ} \\ B♭X \ar[r]_{ξ} & X}  (16)

commutes. Here, [1_{B+X}, ι_{2}]^{♯} is the unique morphism such that κ_{B,X}[1_{B+X}, ι_{2}]^{♯} = [1_{B+X}, ι_{2}]κ_{B+X,X}.

4.3. Theorem. A precrossed module in a variety V of right Ω-loops can equivalently be defined as a quadruple (B, X, ξ, δ) in which (X, ξ) is a B-action and δ : X ⟶ B is a morphism such that for an n-ary operation ω ∈ Ω,

ω(δ(x_{1}) + b_{1}, ..., δ(x_{n}) + b_{n}) − ω(b_{1}, ..., b_{n})

= δ(ξ(ω(x_{1} + b_{1}, ..., x_{n} + b_{n}) − ω(b_{1}, ..., b_{n}))) (17)

for all x_{1}, ..., x_{n} ∈ X, b_{1}, ..., b_{n} ∈ B.

Proof. As explained in [12], a precrossed module in V corresponds to a reflexive graph

\xymatrix{B ⋉ X \ar[r]^{α} \ar[r]_{γ} & B \ar[l]_{β}}  (18)

with α(b, x) = b and β(b) = (b, 0). Since γ is a homomorphism and (b, x) = (0, x) + (b, 0) , we have γ(b, x) = γ(0, x) + γ(b, 0) = γ(0, x) + b. This shows that γ is completely determined by γ(0, x), for all x ∈ X. We introduce the morphism δ : X ⟶ B defined by δ(x) = γ(0, x) and then γ(b, x) = δ(x) + b for each b ∈ B and each x ∈ X. This means that a precrossed module in V can be defined as quadruple (B, X, ξ, δ), in which (X, ξ) is a B-action and δ : X ⟶ B a morphism in V such that γ : B ⋉ X ⟶ B defined by γ(b, x) = δ(x) + b is also a morphism in V. For an n-ary operation ω ∈ Ω,

ω(γ(b_{1}, x_{1}), ..., γ(b_{n}, x_{n})) = ω(δ(x_{1}) + b_{1}, ..., δ(x_{n}) + b_{n}),  (19)

γ(ω((b_{1}, x_{1}), ..., (b_{n}, x_{n}))) = γ(ω(b_{1}, ..., b_{n}), ξ(ω(x_{1} + b_{1}, ..., x_{n} + b_{n}) − ω(b_{1}, ...b_{n}))) = δ(ξ(ω(x_{1} + b_{1}, ..., x_{n} + b_{n}) − ω(b_{1}, ..., b_{n}))) + ω(b_{1}, ..., b_{n}) (20)

for x_{1}, ..., x_{n} ∈ X and b_{1}, ..., b_{n} ∈ B. Therefore, γ is a morphism if and only if (17) holds.

4.4. Theorem. A crossed module in a variety V of Ω-loops can be defined as a precrossed module (B, X, ξ, δ) with

ξ(ω(x′_{1} + (δ(x_{1}) + b_{1}), ..., x′_{n} + (δ(x_{n}) + b_{n})) − ω(δ(x_{1}) + b_{1}, ..., δ(x_{n}) + b_{n}))+

ξ(ω(x_{1} + b_{1}, ..., x_{n} + b_{n}) − ω(b_{1}, ..., b_{n})) = ξ(ω((x′_{1} + x_{1}) + b_{1}, ..., (x′_{n} + x_{n}) + b_{n}) − ω(b_{1}, ..., b_{n})) (21)

for all b′_{1}, ..., b′_{n} ∈ B, x_{1}, ..., x_{n} ∈ X and x′_{1}, ...x′_{n} ∈ X.

Proof. As follows from the results of [11] and [12], a precrossed module module (B, X, ξ, δ) is a crossed module if and only if the map

\xymatrix{m : (B ⋉ X) ×_{B} (B ⋉ X) \ar[r] & B ⋉ X}

defined by m(u, v) = p(u, βα(u), v) is a morphism in V; here p is any Mal’tsev term in V and α and β are as in (18). In our case, using Theorem 3.1, we can simplify the definition of m as follows:

m((b′, x′), (b, x)) = p((b′, x′), (b′, 0), (b, x)) = ((b′, x′) − (b′, 0)) + (b, x) = (b, x′ + x).

We then calculate, for any n-ary ω

m(ω((b′_{1}, x′_{1}), ..., (b′_{n}, x′_{n})), ω((b_{1}, x_{1}), ..., (b_{n}, x_{n})))

= m((ω(b′_{1}, ..., b′_{n}), ξ(ω(x′_{1} + b′_{1}, ..., x′_{n} + b′_{n}) − ω(b′_{1}, ..., b′_{n}))),

(ω(b_{1}, ..., b_{n}), ξ(ω(x_{1} + b_{1}, ...x_{n} + b_{n}) − ω(b_{1}, ..., b_{n}))))
= (ω(b_{1}, ..., b_{n}), ξ(ω(x′_{1} + b′_{1}, ..., x′_{n} + b′_{n}) − ω(b′_{1}, ..., b′_{n}))+
ξ(ω(x_{1} + b_{1}, ..., x_{n} + b_{n}) − ω(b_{1}, ..., b_{n}))) (22)

and

ω(m((b′_{1}, x′_{1})(b_{1}, x_{1})), ..., m((b′_{n}, x′_{n})(b_{n}, x_{n})))

= ω((b_{1}, x′_{1} + x_{1}), ..., (b_{n}, x′_{n} + x_{n})) = (ω(b_{1}, ..., b_{n}), ξ(ω((x′_{1} + x_{1}) + b_{1}, ..., (x′_{n} + x_{1}) + b_{n}) − ω(b_{1}, ..., b_{n}))) (23)

for b_{1}, ..., b_{n} ∈ B, b′_{1}, ..., b′_{n} ∈ B, x_{1}, ..., x_{n} ∈ X, x′_{1}, ..., x′_{n} ∈ X with b′_{i} = δ(x_{i}) + b_{i}, i = 1, ..., n. Using the fact that

((b′, x′), (b, x)) ∈ (B ⋉ X) ×_{B} (B ⋉ X)

if and only if b′ = δ(x) + b, we conclude that m is a morphism in V if and only if (21) is satisfied.

4.5. Remarks and examples. When ω is the same as +, and it is associative, the formula bx = ξ(b+x−b) defined by a B-action on X (see [6]), and the equalities (17) and (21) become δ(b_{1}x_{2}) = b_{1} +δ(x_{2})−b_{1} and δ(x_{1})(b_{1}x′_{2}) = x_{1} +b_{1}x′_{2} −x_{1} respectively. They are obviously equivalent to the classical δ(bx) = b + δ(x) − b and δ(x)(x′) = x + x′ − x, which define crossed modules, originally by J.H.C. Whitehead [17]. More generally, (17) and (21) conveniently apply to the context of G. Orzech [16], and still, more generally, to the context of distributive Ω_{2}-loops in the sense of S. Mantovani and G. Metere [14]. On the other hand (17) and (21) are special cases of conditions imposed in [12]; they are expressed in [12] as commutativity of diagrams (2.1) and (3.14) respectively. However, we did not obtain (17) and (21) directly from those diagrams, and:

(a) We do not know how to deduce the commutativity of diagrams (2.1) and (3.14) in [12] directly from (17) and (21), that is, not using the reflexive graph (18) and the map m : (B ⋉ X) ×_{B} (B ⋉ X) ⟶ B ⋉ X (which in fact means: we do not know how to do it using Theorem 3.1).

(b) Applying (17) and (21) to groups, and to contexts of [14] and [16] (which include not only groups but many classical algebraic structures, e.g. associative and Lie, and general non-associative algebras over rings), gives the known descriptions of (pre)crossed modules much more easily than the results of [12].

Acknowledgements

The results of this paper formed part of the author’s PhD thesis [10] at the University of Cape Town. I would like to thank my supervisor Prof. George Janelidze for his encouragement and support. I would also like to thank the referee whose comments and suggestions helped improve the final version of the paper.

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Department of Mathematics and Applied Mathematics, University of Limpopo Private Bag X1106, Sovenga 0727, South Africa

Email: E.Inyangala@gmail.com

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Transmitting editors

Clemens Berger, Université de Nice-Sophia Antipolis, cberger@math.unice.fr Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: ronnie.profbrown(at)btinternet.com Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it

Valeria de Paiva: valeria.depaiva@gmail.com

Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, Macquarie University: steve.lack@mq.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@buffalo.edu Tom Leinster, University of Glasgow, Tom.Leinster@glasgow.ac.uk Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Brooke Shipley, University of Illinois at Chicago: bshipley@math.uic.edu James Stasheff, University of North Carolina: jds@math.upenn.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca