Galois Theories

Abstract

In this paper by using the ring of real-valued continuous functions C(X), we prove a theorem in profinite spaces which states that for a compact Hausdorff space X, the set of its connected components X/∼ endowed with the quotient topology is a profinite space. Then we apply this result to give an alternative proof to the fact that the category of profinite spaces is a reflective subcategory in the category of compact Hausdorff spaces. Finally, under some circumstances on a space X, we compute the connected components of the space t(X) in terms of the ones of the space X.

Enriched and internal categories: an extensive relationship

“…Unauthenticated Download Date | 5/8/18 5:01 AM using the left adjoint in (4). So Int maps a matrix (M (i, j)) i∈I,j∈J to the span…”

“…We accordingly define the category Cat d (V) of categories in V with discrete object of objects to be the pullback Proof. For any V-category C, the assertion that the C-component of the unit of the adjunction is an isomorphism follows from extensivity of V where I = ob(C) × ob(C) as in (4), noting that (ob(C) × ob(C)) • 1 is canonically isomorphic to (ob(C) • 1) × (ob(C) • 1).…”

“…Specifically, we will require V to be cartesian closed and have finite limits, together with small coproducts satisfying a stengthening from a finitary condition to an infinitary condition of Lawvere's [11,12] notion of extensivity as developed by Carboni et al [6]. This infinitary version of extensivity has been studied by Borceux and Janelidze [4], and by Centazzo and Vitale [7], probably also by others, and is an instance of the notion of van Kampen colimit [8].…”

“…In Section 2, we follow Borceux and Janelidze [4] and Centazzo and Vitale [7] in extending Carboni et al's development [6] of Lawvere's notion of extensivity from a finitary condition to an infinitary condition, and in extending generic results about it. We also prove that, if V is extensive, so are V-Cat and Cat(V).…”

On factorization systems for surjective quandle homomorphisms

“…We describe this factorisation system in Section 2, by using an important property of permutability of a class of congruences in Qnd (Lemma 1.3), explicitly described in Section 1, that is of independent interest. This factorisation system (E, M) satisfies a characteristic property of the so-called reflective ones [7]: E is the class of surjective homomorphisms which are inverted (= sent to an isomorphism) by the reflector π 0 : Qnd → Qnd * , and for two composable surjective homomorphisms f and g, then g ∈ E whenever f • g ∈ E and f ∈ E. The class M is the class of trivial extensions (also called trivial coverings) in the sense of categorical Galois theory [1,14] (see also [9,10,11]). …”