## Journal of Homotopy and Related Structures, vol. 1(1), 2006, pp.1–5 SEPARABLE MORPHISMS OF SIMPLICIAL SETS (2006)

### BibTeX

@MISC{Chikhladze06journalof,

author = {Dimitri Chikhladze},

title = {Journal of Homotopy and Related Structures, vol. 1(1), 2006, pp.1–5 SEPARABLE MORPHISMS OF SIMPLICIAL SETS},

year = {2006}

}

### OpenURL

### Abstract

(communicated by George Janelidze) We show that the class of separable morphisms in the sense of G. Janelidze and W. Tholen in the case of Galois structure

### Citations

94 |
Calculus of Fractions and Homotopy Theory
- Gabriel, Zisman
- 1967
(Show Context)
Citation Context ...A → B of the category C is a trivial covering or a cartesian morphism with respect to the Galois structure Γ if it is a fibration, and the square A ηA �� HIA (*) f B ηB � HIf � HIB is a pullback (see =-=[5]-=-, 3.1). Definition 1.1. A fibration h : A → B from F is called a separable morphism with respect to the Galois structure Γ = (C, X, H, I, η, F, F ′ ) if the diagonal ∆ = 〈1A, 1A〉 : A → A ×B A is a car... |

23 |
Pure Galois theory in categories
- Janelidze
- 1990
(Show Context)
Citation Context ... a pair of adjoint functors I, H : X ⇆ C consists of specified classes of morphisms F and F ′ , called fibrations, in C and X respectively (see G. Janelidze [7], the earlier reference is G. Janelidze =-=[6]-=-), and we require separable morphisms to be fibrations. Our purpose is to describe the class of separable morphisms for the Galois structure introduced by R. Brown and G. Janelidze in [2] (Γ2 below). ... |

18 |
Galois Theories. Cambridge
- Borceux, Janelidze
- 2001
(Show Context)
Citation Context ...(a, a) to (d1, d2) in A ×B A. Here we observe, if (e1, e2) is a 1-simplex in A ×B A “connecting” (a ′ , a ′ ) with (d1 ′ , d2 ′ ), then there is a commutative diagram ∆[0] � A �� ���� �� �� �� �� �� ∆=-=[1]-=- � B wherein the upper horizontal morphism is determined by a ′ and the diagonals by e1 and e2. Since these diagonals are equal e1 = e2, yielding also d1 ′ = d2 ′ . Applying this argument consecutivel... |

13 | Functorial factorization, well-pointedness and separability
- Janelidze, Tholen
- 1999
(Show Context)
Citation Context ...objects in a category A. What we call Γ1 below can be seen as a special case of this situation, a characterization of separable morphisms for which is given by Theorem 2.1. G. Janelidze and W. Tholen =-=[8]-=- defined separable morphisms in a category C for a given pointed endofunctor of C. Given an adjunction I, H : X ⇆ C one can consider separable morphisms with respect to the induced monad. Then, in a s... |

11 |
Decidable (=separable) objects and morphisms in lextensive categories
- Carboni, Janelidze
- 1996
(Show Context)
Citation Context ... of Galois structure of second order coverings of simplicial sets due to R. Brown and G. Janelidze coincides with the class of covering maps of simplicial sets. Separable morphisms were introduced in =-=[3]-=- by A. Carboni and G. Janelidze for lextensive categories. In the way of [3] one can consider separable morphism in a lextensive category Fam(A), the category of families of objects in a category A. W... |

6 | Galois theory of second order covering maps of simplicial sets
- Brown, Janelidze
- 1999
(Show Context)
Citation Context ...G. Janelidze [6]), and we require separable morphisms to be fibrations. Our purpose is to describe the class of separable morphisms for the Galois structure introduced by R. Brown and G. Janelidze in =-=[2]-=- (Γ2 below). Theorem 2.4 states that for this Galois structure separable morphisms are exactly the Kan fibrations which are covering maps of simplicity sets. 1. Separable morphisms In this section C i... |

6 | On localization and stabilization for factorization systems, Applied categorical structures 5 - Carboni, Janelidze, et al. - 1997 |

4 |
Categorical Galois theory: revision and some recent developments, Galois connections and applications
- Janelidze
(Show Context)
Citation Context ...der a Galois structure, which together with a pair of adjoint functors I, H : X ⇆ C consists of specified classes of morphisms F and F ′ , called fibrations, in C and X respectively (see G. Janelidze =-=[7]-=-, the earlier reference is G. Janelidze [6]), and we require separable morphisms to be fibrations. Our purpose is to describe the class of separable morphisms for the Galois structure introduced by R.... |