The coarse shape

Abstract

Abstract. Given a category C, a certain category pro * -C on inverse systems in C is constructed, such that the usual pro-category pro-C may be considered as a subcategory of pro * -C. By simulating the (abstract) shape category construction, Sh (C,D) , an (abstract) coarse shape categoryis obtained. An appropriate functor of the shape category to the coarse shape category exists. In the case of topological spaces, C = HT op and D = HP ol or D = HAN R, the corresponding realizing category for Sh * is pro * -HP ol or pro * -HAN R respectively. Concerning an operative characterization of a coarse shape isomorphism, a full analogue of the well known Morita lemma is proved, while in the case of inverse sequences, a useful sufficient condition is established. It is proved by examples that for C = Grp (groups) and C = HT op, the classification of inverse systems in pro * -C is strictly coarser than in pro-C. Therefore, the underlying coarse shape theory for topological spaces makes sense.

Enriched Pro-Categories and Shapes

“…Finally, one can readily see that if an F : P → Q is induced by an ( f j ) and a G : Q → R is induced by a (g j ), then the composition GF : P → R is induced by (g j f j ). The following theorem generalizes Claim 3 of [26].…”

“…We assume that the notion of a pro-category is well known as well as the basics of the (abstract) shape theory, especially, via the inverse systems approach due to Mardešić and Segal [13]. For the sake of completeness, we shall briefly recall the needed notions and main facts concerning a pro * -category and the coarse shape obtained in [26]. The category language follows [29].…”

“…An enriched pro-category pro J -C is interesting and useful by itself because, in general, it divides (classifies) the objects into larger classes (isomorphisms types) than the underlying pro-category pro-C (see Examples 7.1 and 7.2 of [26]). Moreover, in many important cases one can go on much further, i.e., to develop the corresponding J-shape theory.…”

“…becomes a functor, called the (abstract) J-shape functor. Comparing to the (abstract) shape functor, we know that the restriction of S J to D into the full subcategory of Sh J (C, D) , determined by ObD, is not a category isomorphism (Example 3 of [26]). Nevertheless, we shall prove that P and Q are isomorphic objects of D if and only if they are isomorphic in Sh J (C, D) , i.e.…”

“…which, in general, is not a surjection (Example 3 of [26]). Finally, the functor Sh J (C, D) factorizes as Sh J (C, D) = I (C, D) S (C, D) , where S (C, D) : C → Sh (C, D) is the (abstract) shape functor, while…”