The coarse shape groups

The Hurewicz theorem in Sh◦* and Sh◦*²

“…Proof. According to Remark 4.2 in [2], pro-π k (X, X 0 , x 0 ) is a zero-object in pro-Grp (pro-Set in case k = 1) when π * k (X, X 0 , x 0 ) = 0, for k ∈ N, and then, by Lemma 5.5 in [4], pro-π * k (X, X 0 , x 0 ) is a zero-object in pro-Grp. The conditions of Theorem 3.3 are fulfilled, so ( H1), ( H2) and ( H3) hold.…”

“…Proof. The space (X, x 0 ) is connected since π * 0 (X, x 0 ) = 0 if and only if pro-π 0 (X, x 0 ) = 0 (see [2], Theorem 4.4), which is equivalent to connectedness of the space (X, x 0 ). Furthermore, by Corollary 5.6 in [4], for every k ∈ N, π * k (X, x 0 ) = 0 ⇐⇒ pro-π * k (X, x 0 ) is a zero-object in pro-Grp.…”

“…According to Theorem I.6.8 in [5], every pointed pair of spaces has a HP ol 2 • -expansion, so we can consider the homotopy and homology pro-groups of a pointed pair of spaces and their relationships.…”

“…Let us briefly list main categories (given by its objects and morphisms) and functors (given by its acting on the objects and morphisms of the domain category) we deal with in the article. For more details on these categories and functors see [2] and [4]. Let C be an arbitrary category and D its full and dense subcategory.…”