Homology theory in the alternative set theory. I: Algebraic preliminaries. (English) Zbl 0735.03032

Guričan, Jaroslav

Commentat. Math. Univ. Carol. 32, No. 1, 75-93 (1991).

Nonstandard methods make it possible to create chains of simplexes such that their numbers of elements are infinitely large natural numbers. On the other hand, the members of chains may be infinitely small simplexes. A successive approximation of a topological object by closer and closer chains of simplexes may be substituted by an approximation using a suitable nonstandard chain. A new mathematical structure may be originated as it was with Loeb measure in the case of measure theory.
The author builds a technical, algebraic apparatus for a nonstandard treatment of homology theory in alternative set theory (AST). In the continuation of the paper [”Homology theory in the AST. II. Basic concepts, Eilenberg-Steenrod’s axioms”, Comment. Math. Univ. Carol. (to appear)] the author presents the homology theory in the framework of AST and proves that the Eilenberg-Steenrod axioms are satisfied.

Reviewer: K.Čuda (Praha)

MSC:

03H05	Nonstandard models in mathematics
03E70	Nonclassical and second-order set theories
18G99	Homological algebra in category theory, derived categories and functors
55N99	Homology and cohomology theories in algebraic topology
20F99	Special aspects of infinite or finite groups

Keywords:

commutative \(\pi\)-group; inverse system of \(Sd\)-classes and \(Sd\)-maps; tensor product; prolongation; homology theory; alternative set theory