Georgian Mathematical Journal Volume 9 (2002), Number 1, 71–74

ON KAN FIBRATIONS FOR MALTSEV ALGEBRAS

M. JIBLADZE AND T. PIRASHVILI

Abstract. We prove that any surjective homomorphism of Maltsev algebras is a Kan fibration.

2000 Mathematics Subject Classification: 55U10, 18C10. Key words and phrases: Kan fibration, Maltsev algebras.

It is well known that simplicial groups are Kan complexes and, more generally, any surjective homomorphism of simplicial groups is a Kan fibration. Among other things, from the results in [1] the following statements follow:

Theorem 1. Any simplicial model of a Maltsev theory is a Kan complex.

Theorem 2. If 𝕋 is an algebraic theory such that any simplicial 𝕋-model is a Kan complex, then 𝕋 is a Maltsev theory.

This note has two aims: firstly, to give direct proofs of these facts without using the category theory machinery and, secondly, to get sharper results. In particular we prove that any surjective homomorphism of Maltsev algebras is a Kan fibration.

A Maltsev operation in an algebraic theory 𝕋 is a ternary operation [ , , ] in 𝕋 satisfying the identities

[a, a, b] = b and [a, b, b] = a.

An algebraic theory is called Maltsev if it possesses a Maltsev operation. Clearly, the theory of groups is a Maltsev theory by taking [a, b, c] = ab^{−1}c. More generally, the theory of loops is a Maltsev theory; this latter fact has already been used in the homotopy theory (see [3]). An example of another sort of a Maltsev theory is the theory of Heyting algebras.

We start by proving a stronger version of Theorem 2. Let

S^{1} = Δ^{1}/∂Δ^{1}

be the smallest simplicial model of the circle. Moreover, let S_{𝕋}^{1} be the simplicial 𝕋-model which is obtained by applying degreewise the free 𝕋-model functor to S^{1}.

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

Proposition 1. Let 𝕋 be an algebraic theory such that S_{𝕋}^{1} satisfies the (1, 2)- th Kan condition in dimension 2. That is, for any 1-simplices x_{1}, x_{2} with d_{1}x_{1} = d_{1}x_{2} there is a 2-simplex x with d_{1}x = x_{1} and d_{2}x = x_{2}. Then 𝕋 is a Maltsev theory.

Proof. Denote the unique nondegenerate 1-simplex of S^{1} by σ and the unique vertex d_{0}σ = d_{1}σ by ∗. So S_{𝕋}^{1} in dimension zero is the free 𝕋-model generated by ∗. Similarly, S_{𝕋}^{1} in dimension one is the free 𝕋-model generated by s_{0}∗ and σ, and in dimension two it is the free 𝕋-model generated by s_{1}s_{0}∗, s_{0}σ, and s_{1}σ. Since d_{1}s_{0}∗ = d_{1}σ = ∗, the (1, 2)-th Kan condition implies the existence of a 2-simplex x of S_{𝕋}^{1} with d_{1}x = s_{0}∗ and d_{2}x = σ. This means there is an element x(s_{1}s_{0}∗, s_{0}σ, s_{1}σ) in the free 𝕋-model with three generators s_{1}s_{0}∗, s_{0}σ, s_{1}σ such that the equalities x(d_{1}s_{1}s_{0}∗, d_{1}s_{0}σ, d_{1}s_{1}σ) = s_{0}∗ and x(d_{2}s_{1}s_{0}∗, d_{2}s_{0}σ, d_{2}s_{1}σ) = σ hold in the free 𝕋-algebra with two generators s_{0}∗, σ. Applying standard simplicial identities we see that this means x(s_{0}∗, σ, σ) = s_{0}∗ and x(s_{0}∗, s_{0}∗, σ) = σ, i. e., that x is a Maltsev operation.

The following Theorem shows that if 𝕋 is a Maltsev theory, then all surjective homomorphisms of simplicial 𝕋-models are Kan fibrations, which obviously implies Theorem 1. Our proof uses exactly the same inductive argument as the one given in [2] for simplicial groups (see page 130 in [2]) except that we put a new input for w_{0}.

Theorem 3. Any surjective homomorphism f : X → Y of simplicial models of a Maltsev theory is a Kan fibration.

Proof. For n > 0 and 0 ⩽ k ⩽ n, given y ∈ Y_{n} with d_{i}y = f(x_{i}) for i ≠ k, 0 ⩽ i ⩽ n, where x_{i} are elements of X_{n−1} with matching faces, we have to find x ∈ X_{n} with f(x) = y and d_{i}x = x_{i} for i ≠ k. Take x′ ∈ f^{−1}(y) and then put

\begin{aligned}w_{0} & = [s_{0}x_{0}, s_{0}d_{0}x′, x′], \\ w_{j} & = [w_{j−1}, s_{j}d_{j}w_{j−1}, s_{j}x_{j}] \end{aligned}

for 0 < j < k; in case k < n put

\begin{aligned}w_{n} & = [w_{k−1}, s_{n−1}d_{n}w_{k−1}, s_{n−1}x_{n}], \\ w_{j} & = [w_{j+1}, s_{j−1}d_{j}w_{j+1}, s_{j−1}x_{j}] \end{aligned}

for n > j > k.

We then have

f(w_{0}) = [s_{0}f(x_{0}), s_{0}d_{0}f(x′), f(x′)] = [s_{0}d_{0}y, s_{0}d_{0}y, y] = y,
f(w_{j}) = [f(w_{j−1}), s_{j}d_{j}f(w_{j−1}), s_{j}f(x_{j})]
= [y, s_{j}d_{j}y, s_{j}d_{j}y] = y

for 0 < j < k and, if k < n, then

\begin{aligned}f(w_{n}) & = [f(w_{k−1}), s_{n−1}d_{n}f(w_{k−1}), s_{n−1}f(x_{n})] = [y, s_{n−1}d_{n}y, s_{n−1}d_{n}y] = y, \\ f(w_{j}) & = [f(w_{j+1}), s_{j−1}d_{j}f(w_{j+1}), s_{j−1}f(x_{j})] = [y, s_{j−1}d_{j}y, s_{j−1}d_{j}y] = y \end{aligned}

for n > j > k. Furthermore,

d_{0}w_{0} = [d_{0}s_{0}x_{0}, d_{0}s_{0}d_{0}x′, d_{0}x′] = [x_{0}, d_{0}x′, d_{0}x′] = x_{0},
d_{i}w_{j} = [d_{i}w_{j−1}, d_{i}s_{j}d_{j}w_{j−1}, d_{i}s_{j}x_{j}]

= \begin{cases}[x_{i}, s_{j−1}d_{j−1}d_{i}w_{j−1}, s_{j−1}d_{i}x_{j}] \\ = [x_{i}, s_{j−1}d_{j−1}x_{i}, s_{j−1}d_{j−1}x_{i}] = x_{i}, & i < j, \\ [d_{j}w_{j−1}, d_{j}w_{j−1}, x_{j}] = x_{j}, & i = j \end{cases}

for 0 < j < k and, if k < n, then

\begin{aligned}d_{i}w_{n} & = [d_{i}w_{k−1}, d_{i}s_{n−1}d_{n}w_{k−1}, d_{i}s_{n−1}x_{n}] \\ & = \begin{cases}[x_{i}, s_{n−2}d_{n−1}d_{i}w_{k−1}, s_{n−2}d_{i}x_{n}] \\ = [x_{i}, s_{n−2}d_{n−1}x_{i}, s_{n−2}d_{n−1}x_{i}] = x_{i}, & 0 ⩽ i < k, \\ [d_{n}w_{k−1}, d_{n}w_{k−1}, x_{n}] = x_{n}, & i = n, \end{cases} \end{aligned}

\begin{aligned}d_{i}w_{j} & = [d_{i}w_{j+1}, d_{i}s_{j−1}d_{j}w_{j+1}, d_{i}s_{j−1}x_{j}] \\ & = \begin{cases}[x_{i}, s_{j−2}d_{j−1}d_{i}w_{j+1}, s_{j−2}d_{i}x_{j}] \\ = [x_{i}, s_{j−2}d_{j−1}x_{i}, s_{j−2}d_{j−1}x_{i}] = x_{i}, & 0 ⩽ i < k, \\ [x_{i}, s_{j−1}d_{j}d_{i}w_{j+1}, s_{j−1}d_{i−1}x_{j}] \\ = [x_{i}, s_{j−1}d_{j}x_{i}, s_{j−1}d_{j}x_{i}] = x_{i}, & j < i ⩽ n, \\ [d_{j}w_{j+1}, d_{j}w_{j+1}, x_{j}] = x_{j}, & i = j \end{cases} \end{aligned}

for n > j > k.

Thus if one takes x = w_{k−1} for k = n and x = w_{k+1} for k < n, one obtains f(x) = y and d_{i}x = x_{i} for i ≠ k, 0 ⩽ i ⩽ n, as desired.

Acknowledgements

The research was supported by the TMR research network ERB FMRX CT-97-0107 and the INTAS grant #99-00817. The authors are grateful to the Max Planck Institute for Mathematics (Bonn), where they were given an opportunity to work jointly on the paper.

References

1. A. Carboni, G. M. Kelly, and M. C. Pedicchio, Some remarks on Maltsev and Goursat categories. Appl. Categ. Structures 1(1993), No. 4, 385–421.

2. E. B. Curtis, Simplicial homotopy theory. Adv. Math. 6(1971), 107–209.

3. S. Klaus, On simplicial loops and H-spaces. Topology Appl. 112(2001), No. 3, 337–348.

(Received 15.08.2001)

Authors’ address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia E-mail: jib@rmi.acnet.ge

pira@rmi.acnet.ge