The shapes in a concrete category

Abstract

Abstract.We show under what conditions, and how, one can obtain a shape theory (various shape theories) in a concrete category. The technique is, roughly speaking, reduced to the quotients by congruences providing the objects of lower cardinalities. The application yields the new (coarser) classifications in every concrete category which admits sufficiently many non-trivial quotients. Thus, the ordering, (ultra)pseudometric, uniform and topological structures, as well as many algebraic and mixed (multi-) structures, give rise to interesting results.

On the quotient shapes of vectorial spaces

“…Further, our category theory language follows [6], while all necessary facts concerning ordinals and cardinals one can find in [4], Chapter II. Nevertheless, we have to recall some indispensable notions and constructions which are introduced or exhibited in [10].…”

“…Given a field F , let V ect F denote the category of all vectorial spaces over F and all their linear functions. According to [10], Section 12, we propose that every finite-dimensional vectorial space X is "nice", and if X is infinite-κdimensional, then every vectorial space (over the same field) having dimension less than κ is "nice comparing to" X. As usually, dim X = |B|, where B is an algebraic (Hamel) basis of X.…”

“…As usually, dim X = |B|, where B is an algebraic (Hamel) basis of X. For the sake of completeness, let us briefly recall the canonical construction and the main result of [10], Section 12.…”

“…The quotient shape theory was recently introduced by the author, [10]. It is, of course, a kind of the general (abstract) shape theory, [9], I.…”

“…In [10], several well known concrete categories were considered and many examples are given which show that the quotient shape theory yields classifications strictly coarser than those by isomorphisms. In this paper we continue the consideration of quotient shapes of vectorial spaces ( [10], Section 12), especially, of the topological and normed vectorial spaces.…”