in which the front square is a pull-back. Then P is often called the fibre-product o f f and p, and it is also said that ~: P ~ X is induced by f from p. The map ~: Q~ P is determined by ~01 and q)2. Our object is to give conditions on the front and back squares which ensure that if q~l, q~2 and ~o are homotopy equivalences, then so also is ~. First of all we shall assume throughout that the ba...

In this paper we show that it is possible to model observable behaviour of coalgebras independently from their internal dynamics, but within the general framework of representing behaviour by a map into a “final” coalgebra. In the first part of the paper we characterise Set-endofunctors F with the property that bisimilarity of elements of F -coalgebras coincides with having the same observable ...

In this paper we define a weak (n+1,ε)−strainer on an Alexandrov space with curvature≥ 1, and prove an almost isometric sphere theorem in the setting of a weak strainer, making use of a rigidity theorem for round spheres. To prove the rigidity theorem we investigate several properties of weak strainers, e.g. the maximality property, the covering property of the balls centered at strainer points...

Laan in (Ph.D Thesis, Tartu. 1999) introduced the principal weak form of Condition $(P)$ as Condition $(PWP)$ and gave some characterization of monoids by this condition of their acts. In this paper first we introduce Condition (G-PWP), a generalization of Condition $(PWP)$ of acts over monoids and then will give a characterization of monoids when all right acts satisfy this condition. We also ...

We introduce a very weak version of the square principle which may hold even under failure of the generalized continuum hypothesis. Under this weak square principle, we give a new characterization (Theorem 10) of partial orderings with κ-Freese–Nation property (see below for the definition). The characterization is not a ZFC theorem: assuming Chang’s Conjecture for אω , we can find a counter-ex...

Let X and Y be Banach spaces. We give a “non-separable” proof of the Kalton-Werner-Lima-Oja theorem that the subspaceK(X, X) of compact operators forms an M -ideal in the space L(X, X) of all continuous linear operators from X to X if and only if X has Kalton’s property (M∗) and the metric compact approximation property. Our proof is a quick consequence of two main results. First, we describe h...

We formulate a weak Gorenstein property for the Eisenstein component of the p-adic Hecke algebra associated to modular forms. We show that this weak Gorenstein property holds if and only if a weak form of Sharifi’s conjecture and a weak form of Greenberg’s conjecture hold.