engleski

The Cartesian product of a compactum and a space is a bifunctor in shape

In 2003 the author has associated with every cofinite inverse system of compact Hausdorff space <b>X</b> with limit <i>X</i> and every simplicial complex <i>K</i> (possibly infinite) with geometric realization <i>P</i>=|<i>K</i>| a resolution R(<b>X</b>, <i>K</i>) of <i>X</i>×<i>P</i>, which consists of paracompact spaces. If <b>X</b> consists of compact polyhedra, then R(<b>X</b>, <i>K</i>) consists of spaces having the homotopy type of polyhedra. In two subsequent papers the author proved that R(<b>X</b>, <i>K</i>) is a covariant functor in each of its variables <b>X</b> and <i>K</i>. In the present paper it is proved that R(<b>X</b>, <i>K</i>) is also a covariant functor in the variable <i>K</i>. Moreover, R(<b>X</b>, <i>K</i>) is a bifunctor. This implies that the Cartesian product <i>X</i>×<i>P</i> is a bifunctor SSh(Cpt)×</>Sh(Top)→</>Sh(Top) from the product category SSh(Cpt)×</>H(Pol) of the strong shape category of compact Hausdorff spaces SSh(Cpt) and the homotopy category H(Pol) of polyhedra to the shape category Sh(Top) of topological spaces. This holds in spite of the fact that <i>X</i>×<i>Z</i> need not be a direct product in Sh(Top).

Inverse system; inverse limit; resolution; coherent mapping; Cartesian product; shape; strong shape; simplicial mapping; bifunctor.

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