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In this paper we continue the investigation of some aspects of descent theory for schemes that was begun in Mesablishvili, Appl. Categ. Structures. Let 𝐒𝐂𝐇 be a category of schemes. We show that quasi-compact pure morphisms of schemes are effective descent morphisms with respect to...
It is proved that every regularly movable metric compactum has the shape of the inverse limit of an inverse sequence consisting of compact polyhedra and retractions.
Some geometric results are presented which can be derived from algebraic formulae for topological degree and Euler characteristic. In particular, it is shown that Euler characteristics of configuration spaces and work spaces of mechanical linkages can be computed in an algorithmic way. We also...
Border homology and cohomology groups of pairs of uniform spaces are defined and studied. These groups give an intrinsic characterization of Čech type homology and cohomology groups of the remainder of a uniform space.
Let 𝐻(𝑋) := (ℝ × 𝑋) ⋋ 𝑋* be the generalized Heisenberg group induced by a normed space 𝑋. We prove that 𝑋 and 𝑋* are relatively minimal subgroups of 𝐻(𝑋). We show that the group 𝐺 := 𝐻(𝐿 4 0, 1) is reflexively representable but weakly continuous unitary...
Let 𝔎 be a category of pairs of spaces, 𝔏 ⊂ 𝔎 the category of pairs of ANRs or CW-spaces, 𝐴∗ a chain functor (e.g., one associated with a spectrum). Then the derived homology 𝑠 ℎ∗ of the 𝔏-localization of 𝐴∗ is the strong homology theory on 𝔎 which is up to an...
A mapping is said to be confluent over locally connected continua if for each locally connected subcontinuum 𝑄 of the range each component of its preimage is mapped onto 𝑄. For mappings of compact spaces this class is a very natural generalization of locally confluent mappings. Various...
As observed by J. Beck, and as we know from M. Barr's and his joint work on triple cohomology, the classical isomorphism Opext ≅ 𝐻 2 that describes group extensions with abelian kernels, can be deduced from the equivalence between such extensions and torsors (in an appropriate sense). The...
For topological spaces 𝑋, 𝑌 with a fixed compatible quasi-uniformity 𝑄 in 𝑌 and for a family (𝑓 𝑖 ) 𝑖∈𝐼 of mappings from 𝑋 to 𝑌, the notions of even continuity in the sense of Kelley, topological equicontinuity in the sense of Royden and 𝑄-equicontinuity (i.e.,...
A subcategory of the category of groups with action is determined and it is proved that the functor defined in Datuashvili, Georgian Math. J. 9: 671–681, 2002 takes free objects from this category to free Leibniz algebras. This result gives a solution to a problem stated by J.-L. Loday Enseign....
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