The coshape invariant and continuous extensions of functors
Let \(\mathfrak T\) be a category and \(\mathfrak P \subset \mathfrak T\) be a full subcategory. Using the category of direct systems \(\pmb{dir}-\mathfrak T\) resp. a quotient category \(\pmb {dir}-\mathfrak T/\sim =\pmb{inj}-\mathfrak T\), the author defines the notion of a \(\mathfrak P\)-coexpansion \(\pmb p: \pmb X \to (X)\) of an object \(X \in \mathfrak T\) (\((X)\) denoting the trivial direct system of one single object), analogous to the concept of an \(ANR\)- resolution of a space, and he calls the category \(\mathfrak P\) to be codense in \(\mathfrak T\), whenever every object of \(\mathfrak T\) admits such a \(\mathfrak P\)-coexpansion. He constructs in this case a coshape category \(\pmb {CSH}_{(\mathfrak T, \mathfrak P)}\). Since he is primarily interested in the case that \(\mathfrak T\) is the homotopy category of (based or unbased) pairs of topological spaces and \(\mathfrak P \) the subcategory of finite CW-spaces, his main theorem asserts, that in this case the category \(\mathfrak P\) is codense in the category \(\mathfrak T\). In the last section the paper deals with the problem of extending functors \(T: \mathfrak P \to \pmb {Gr}\) =\,category of groups over \(\mathfrak T\), using the coshape functor.
scientific article; zbMATH DE number 5935887
scientific article