GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 6, 1998, 575-581

CROSSED SEMIMODULES AND SCHREIER INTERNAL CATEGORIES IN THE CATEGORY OF MONOIDS

A. PATCHKORIA

Abstract. We introduce the notion of a Schreier internal category in the category of monoids and prove that the category of Schreier internal categories in the category of monoids is equivalent to the category of crossed semimodules. This extends a well-known equivalence of categories between the category of internal categories in the category of groups and the category of crossed modules.

1. Introduction and Statement of Results

The description of internal categories in the category of groups as crossed modules is well known (see, e.g., [1]). There exist the same descriptions of internal categories in Rings, Lie algebras, etc. One can find different descriptions (not as crossed modules) of internal categories and groupoids in groups, Mal’tsev varieties of universal algebras, congruence modular varieties, and Mal’tsev categories in [2], [3], [4] and [5] respectively.

Let Mon denote the category of monoids. In this note we show that there exists an equivalence between the category of internal categories in Mon (note that monoids are not congruence modular) satisfying the socalled Schreier condition and the category of crossed semimodules. This answers a question posed by G. Janelidze (oral communication).

By a crossed semimodule we mean a pair

Φ = (μ : A ⟶ X, φ : X ⟶ End(A)),

where A, X are monoids and μ, φ are homomorphisms of monoids satisfying (CSM 1) μ(φ(x)(a)) + x = x + μ(a), x ∈ X, a ∈ A, (CSM 2) φ(μ(a))(a′) + a = a + a′, a, a′ ∈ A.

575 A crossed semimodule morphism is a commutative diagram

1991 Mathematics Subject Classification. 18B40, 20M50.

Key words and phrases. Monoid, crossed semimodule, internal category, internal groupoid, Schreier condition.

1072-947X/98/1100-0575$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

τ = (β, λ) \begin{matrix}Φ : A & ⟶^{μ} & X \\ ↓ & ^{β}↓ & ^{λ}↓ \\ Φ′ : A′ & ⟶^{μ′} & X′ \end{matrix}

with β and λ monoid homomorphisms, which satisfies

β(φ(x)(a)) = φ′(λ(x))(β(a))

for all x ∈ X and a ∈ A.

Recall that an internal category in the category of monoids is a diagram

[figure: π_{1}; m; π_{2}; s_{0}; d_{0}; d_{1}; M_{1}; M_{2}; M :; M_{0}; ,]

where M_{0} is the monoid of objects, M_{1} the monoid of morphisms, and d_{0}, d_{1}, s_{0} are homomorphisms of monoids called the domain, the codomain, and the identity respectively; M_{2} together with π_{1} and π_{2} forms the pullback

\begin{matrix}M_{2} & ⟶^{π_{2}} & M_{1} \\ ^{π_{1}}↓ & & ↓^{d_{1}} & ; \\ M_{1} & ⟶_{d_{0}} & M_{0} \end{matrix}

m is a homomorphism of monoids called the composition of M and usually written as m(g, f) = g ∘ f; and the following conditions hold:

(i) d_{0}s_{0} = d_{1}s_{0} = 1_{M_{0}}, (ii) d_{0}m = d_{0}π_{1}, d_{1}m = d_{1}π_{2}, (iii) m(m(h, g), f) = m(h, m(g, f)), (iv) m(s_{0}d_{1}(f), f) = f = m(f, s_{0}d_{0}(f)). A morphism of internal categories in Mon is a commutative diagram

[figure: π′_{1}; m′; π′_{2}; s′_{0}; d′_{0}; d′_{1}; M′_{1}; M′_{2}; M′ :; M′_{0}; π_{1}; m; π_{2}; s_{0}; d_{0}; d_{1}; M_{1}; M_{2}; M :; M_{0}; γ = (γ_{1}, γ_{0}); γ_{1} × γ_{1}; γ_{1}; γ_{0}; ,] where γ_{0} and γ_{1} are monoid homomorphisms.

An internal groupoid in Mon is an internal category M in Mon in which every morphism has an inverse, i.e., for every f ∈ M_{1} there exists a unique f^{−1} ∈ M_{1} such that m(f, f^{−1}) = s_{0}d_{1}(f) and m(f^{−1}, f) = s_{0}d_{0}(f).

Definition. A Schreier internal category in Mon is an internal category in Mon which satisfies the Schreier condition: for any f ∈ M_{1} there exists a unique g ∈ Ker(d_{0}) such that

f = g + s_{0}d_{0}(f).

We prove

Theorem. The category of Schreier internal categories in the category of monoids is equivalent to the category of crossed semimodules.

Restricting this equivalence gives

Corollary 1. The category of Schreier internal groupoids in the category of monoids is equivalent to the category of crossed semimodules Φ = (μ : A ⟶ X, φ : X ⟶ End(A)) such that A is a group.

Note that Corollary 1 can be obtained as a special case of the theorem of [6].

Restricting this equivalence further, we obtain

Corollary 2 (R. Brown and C. B. Spencer [1]). . The category of internal categories (= groupoids) in the category of groups is equivalent to the category of crossed modules.

2. The Proofs

Let

x ⟶^{f} y ⟶^{f′} z

be a composable pair in a Schreier internal category M in Mon. Then there exist uniquely defined g, h ∈ Ker(d_{0}) such that

f = g + s_{0}d_{0}(f) and f′ = h + s_{0}(d_{1}(g) + d_{0}(f)).

Since m is a monoid homomorphism and (iv) holds, we obtain

m(f′, f) = m(h + s_{0}(d_{1}(g) + d_{0}(f)), g + s_{0}d_{0}(f)) =
= m(h, 0) + m(s_{0}(d_{1}(g) + d_{0}(f)), g + s_{0}d_{0}(f)) =
= m(h, s_{0}d_{0}(h)) + m(s_{0}d_{1}(g + s_{0}d_{0}(f)), g + s_{0}d_{0}(f)) =
= h + g + s_{0}d_{0}(f).

So the composition m is uniquely defined,

m(h + s_{0}(d_{1}(g) + d_{0}(f)), g + s_{0}d_{0}(f)) = h + g + s_{0}d_{0}(f).  (1)

Proof of the theorem. Let

[figure: π_{1}; m; π_{2}; s_{0}; d_{0}; d_{1}; M_{1}; M_{2}; M :; M_{0}]

be a Schreier internal category in Mon. By the Schreier condition, for any x ∈ M_{0} and any g ∈ Ker(d_{0}) there exists a unique α(x, g) ∈ Ker(d_{0}) such that

s_{0}(x) + g = α(x, g) + s_{0}(x).

We have

α(x, g_{1} + g_{2}) + s_{0}(x) = s_{0}(x) + g_{1} + g_{2} =
= α(x, g_{1}) + s_{0}(x) + g_{2} = α(x, g_{1}) + α(x, g_{2}) + s_{0}(x),
α(x, 0) + s_{0}(x) = s_{0}(x),
α(x + y, g) + s_{0}(x + y) = s_{0}(x + y) + g = s_{0}(x) + s_{0}(y) + g =
= s_{0}(x) + α(y, g) + s_{0}(y) = α(x, α(y, g)) + s_{0}(x) + s_{0}(y) =
= α(x, α(y, g)) + s_{0}(x + y),
α(0, g) + s(0) = s(0) + g.

Hence, in view of the Schreier condition, we obtain

α(x, g_{1} + g_{2}) = α(x, g_{1}) + α(x, g_{2}), α(x, 0) = 0,

CROSSED SEMIMODULES AND SCHREIER CATEGORIES 579

α(x + y, g) = α(x, α(y, g)), α(0, g) = g.

This means that we have a homomorphism of monoids ψ : M_{0} ⟶ End ( Ker(d_{0})), ψ(x)(g) = α(x, g), x ∈ M_{0}, g ∈ Ker(d_{0}). Consider the pair of monoid homomorphisms

Ψ = (d_{1} : Ker(d_{0}) ⟶ M_{0}, ψ : M_{0} ⟶ End ( Ker(d_{0}))).

It is clear that

d_{1}(ψ(x)(g)) + x = x + d_{1}(g), x ∈ M_{0}, g ∈ Ker(d_{0})

since s_{0}(x) + g = ψ(x)(g) + s_{0}(x) and d_{1}s_{0} = 1_{M_{0}}. On the other hand,

s_{0}d_{1}(g) + g′ = ψ(d_{1}(g))(g′) + s_{0}d_{1}(g), g, g′ ∈ Ker(d_{0}).

This and (1) give ψ(d_{1}(g))(g′) + g = m(ψ(d_{1}(g))(g′) + s_{0}d_{1}(g), g) =

= m(s_{0}d_{1}(g) + g′, g) = m(s_{0}d_{1}(g), g) + m(g′, 0) = g + g′,

i.e.,

ψ(d_{1}(g))(g′) + g = g + g′, g, g′ ∈ Ker(d_{0}).

So Ψ is a crossed semimodule.

Let γ = (γ_{1}, γ_{0}) : M ⟶ M′ be a morphism of Schreier internal categories in Mon. The Schreier condition allows us to write

γ_{1}(ψ(x)(g)) + s′_{0}γ_{0}(x) = γ_{1}(ψ(x)(g)) + γ_{1}s_{0}(x) =

= γ_{1}(ψ(x)(g) + s_{0}(x)) = γ_{1}(s_{0}(x) + g) = s′_{0}γ_{0}(x) + γ_{1}(g) =
= ψ′(γ_{0}(x))(γ_{1}(g)) + s′_{0}γ_{0}(x),

i.e.,

γ_{1}(ψ(x)(g)) + s′_{0}γ_{0}(x) = ψ′(γ_{0}(x))(γ_{1}(g)) + s′_{0}γ_{0}(x).

From this, by the same condition, we obtain

γ_{1}(ψ(x)(g)) = ψ′(γ_{0}(x))(γ_{1}(g)), x ∈ M_{0}, g ∈ Ker(d_{0}).

This means that the commutative diagram

Ker(d_{0}) ⟶^{d_{1}} M_{0}
γ_{1}↓  ^{γ_{0}}↓
Ker(d′_{0}) ⟶^{d′_{1}} M′_{0}

is a crossed semimodules morphism.

So we have the functor

S : Schreier internal categories in Mon ⟶ Crossed semimodules,

defined by S(M) = Ψ and S(γ_{1}, γ_{0}) = (γ_{1}| Ker(d_{0}), γ_{0}) : Ψ ⟶ Ψ′.

There is a standard way to show that S is an equivalence of categories. Let

Φ = (μ : A ⟶ X, φ : X ⟶ End(A))

be a crossed semomodule. Using the semidirect product A ×_{φ} X, we obtain the diagram

[figure: π_{1}; m; π_{2}; s_{0}; d_{0}; d_{1}; A ×_{φ} X; (A ×_{φ} X)_{d_{0}} ×_{d_{1}} (A ×_{φ} X); C :; X; ,]

where d_{0}(a, x) = x, d_{1}(a, x) = μ(a) + x, s_{0}(x) = (0, x), m((a′, μ(a) + x), (a, x)) = (a′ + a, x) and π_{1}, π_{2} are the projections. It is plain to see that C is a Schreier internal category in Mon. On the other hand, any crossed semimodule morphism τ = (β, λ) : Φ ⟶ Φ′ defines the morphism of Schreier internal categories γ = (γ_{1}, λ) : C ⟶ C′, where γ_{1}(a, x) = (β(a), λ(x)).

Thus we have the functor

T : Crossed semimodules ⟶ Schreier internal categories in Mon,

defined by T (Φ) = C and T (τ) = γ : C ⟶ C′.

It is straightforward to see that 1 ≅ ST. Due to the Schreier condition, 1 ≅ T S holds, too.

Proof of Corollary 1. Suppose M is a Schreier internal groupoid in Mon. Consider the crossed semimodule

S(M) = (d_{1} : Ker(d_{0}) ⟶ M_{0}, ψ : M_{0} ⟶ End ( Ker(d_{0}))).

For any g ∈ Ker(d_{0}) there exists a unique h ∈ Ker(d_{0}) such that

m(h + s_{0}d_{1}(g), g) = 0 and m(g, h + s_{0}d_{1}(g)) = s_{0}d_{1}(g).

Using (1), we obtain

h + g = 0 and g + h + s_{0}d_{1}(g) = s_{0}d_{1}(g).

Then the Schreier condition gives

h + g = 0 and g + h = 0,

i.e., Ker(d_{0}) is a group.

Conversely, suppose Φ = (μ : A ⟶ X, φ : X ⟶ End(A)) is a semimodule such that A is a group. Then it is clear that for any morphism (a, x) of T (Φ) we have (a, x)^{−1} = (−a, μ(a) + x), i.e., T (Φ) is a Schreier internal groupoid in Mon.

Acknowledgment

This research was supported by INTAS Grant 93-436 “Algebraic K-theory, groups and categories.”

References

1. R. Brown and C. B. Spencer, G-groupoids, crossed modules and the fundamental groupoid of a topological group. Proc. Konink. Nederl. Akad. Wetensch. 79(1976) 296–302.

2. J.-L. Loday, Spaces with finitely many non-trivial homotopy groups. J. Pure Appl. Algebra 24(1982) 179–202.

3. G. Janelidze, Internal categories in Mal’cev varieties. Preprint, The University of North York, Toronto, 1990.

4. G. Janelidze and M. C. Pedicchio, Internal categories and groupoids in congruence modular varieties. Preprint, McGill University, Montreal, 1995.

5. A. Carboni, M. C. Pedicchio, and N. Pirovano, Internal graphs and internal groupoids in Mal’cev categories. Category Theory 1991 (Montreal, PQ, 1991), 97–109, CMS Conf. Proc. 13, Amer. Math. Soc., Providence, RI, 1992.

6. T. Porter, Crossed modules in Cat and a Brown–Spencer theorem for 2-categories. Cahiers Topologie Géom. Différentielle Catégoriques 26 (1985), 381–388.

(Received 29.07.1996)

Author’s address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, M. Aleksidze St., Tbilisi 380093

Georgia