  Modeling distributed computing in a way enabling the use of formal methods is
a challenge that has been approached from different angles, among which two
techniques emerged at the turn of the century: protocol complexes, and directed
algebraic topology. In both cases, the considered computational model generally
assumes communication via shared objects, typically a shared memory consisting
of a collection of read-write registers. Our paper is concerned with network
computing, where the processes are located at the nodes of a network, and
communicate by exchanging messages along the edges of that network. Applying
the topological approach for verification in network computing is a
considerable challenge, mainly because the presence of identifiers assigned to
the nodes yields protocol complexes whose size grows exponentially with the
size of the underlying network. However, many of the problems studied in this
context are of local nature, and their definitions do not depend on the
identifiers or on the size of the network. We leverage this independence in
order to meet the above challenge, and present $\textit{local}$ protocol
complexes, whose sizes do not depend on the size of the network. As an
application of the design of "compact" protocol complexes, we reformulate the
celebrated lower bound of $\Omega(\log^*n)$ rounds for 3-coloring the $n$-node
ring, in the algebraic topology framework.
