${\mathcal{Z}^1_t({\mathscr{A}}_{\mathcal{K}})}$ has a tensor product, a symmetry and that any object has left inverses. The full subcategory ${\mathcal{Z}^1_t({\mathscr{A}}_{\mathcal{K}})}_{\mathrm{f}}$ of ${\mathcal{Z}^1_t({\mathscr{A}}_{\mathcal{K}})}$ whose objects have finite statistics has conjugates (Theorem \[Cbc:11\]).\
In Appendix \[X\] we give some basics definitions and results on tensor $\mathrm{C}^*-$categories.

Homotopy and net-cohomology of posets {#A}
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After some preliminaries, the main topics are discussed in full generality in the first three sections: the first homotopy group of a poset; the connection between homotopy and net-cohomology; the behaviour of net-cohomology under a change of the index set. The remaining two sections are devoted to study the case that the poset is a basis for the topology of a topological space. We stress that the results obtained in the first three sections in terms of abstract posets can be applied, not only to sharply localized charges which are the subject of the present investigation, but also to charges like those studied in [@BF; @AS].

Preliminaries: the simplicial set and net-cohomology {#Aa}
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In the present section we recall the definition of simplicial set of a poset and the notion of net-cohomology of a poset, thereby establishing our notations. References for this section are [@Rob2; @GLRV; @Rob3].\
**The simplicial set 