refinement of ${\mathcal{K}_{\diamond}}$ and any element of ${\mathcal{K}}$ has an arcwise connected causal complement. Therefore adopting ${\mathcal{K}}$ as index set the cited problems are overcome.\
In Section \[C\], we consider an irreducible net ${\mathscr{A}}_{\mathcal{K}}$ satisfying the Borchers property and punctured Haag duality. The key for studying superselection sectors of the net ${\mathscr{A}}_{\mathcal{K}}$, namely the $\mathrm{C}^*-$category ${\mathcal{Z}^1_t({\mathscr{A}}_{\mathcal{K}})}$, is provided by the following fact. We introduce the causal puncture ${\mathcal{K}}_x$ of ${\mathcal{K}}$ induced by a point $x$ of ${\mathcal{M}}$ (\[Bba:1\]) and consider the categories ${\mathcal{Z}^1_t({\mathscr{A}}_{{\mathcal{K}}_x})}$ of 1-cocycles of ${\mathcal{K}}_x$, trivial in ${\mathfrak{B}(\mathcal{H}_o)}$, with values in the net ${\mathscr{A}}_{{\mathcal{K}}_x}$. We show that a family $z_x\in{\mathcal{Z}^1_t({\mathscr{A}}_{{\mathcal{K}}_x})}$ for any $x\in{\mathcal{M}}$ admits an extension to a 1-cocycle $z\in{\mathcal{Z}^1_t({\mathscr{A}}_{\mathcal{K}})}$ if, and only if, a suitable gluing condition is verified (Proposition \[Ca:2\]). A similar result holds for arrows (Proposition \[Ca:4\]), and can be easily generalized to functors. These results suggest that one could proceed as follows: first, prove that the categories ${\mathcal{Z}^1_t({\mathscr{A}}_{{\mathcal{K}}_x})}$ have the right structure to describe the superselection theory (*local theory*, Section \[Cb\]); secondly, check that the constructions we have made on ${\mathcal{Z}^1_t({\mathscr{A}}_{{\mathcal{K}}_x})}$ satisfy the mentioned gluing condition, and consequently can be extended to ${\mathcal{Z}^1_t({\mathscr{A}}_{\mathcal{K}})}$ (*global theory*, Section \[Cc\]). This argument works. We will prove that 