  Given any task $\Pi$, Brandt's speedup theorem (PODC 2019) provides a
mechanical way to design another task~$\Pi'$ on the same input-set as $\Pi$
such that, for any $t\geq 1$, $\Pi$ is solvable in $t$ rounds if and only if
$\Pi'$ is solvable in $t-1$ rounds. The theorem applies to the anonymous
variant of the LOCAL model, in graphs with sufficiently large girth, and to
locally checkable labeling (LCL) tasks. In this paper, using combinatorial
topology applied to distributed computing, we dissect the construction in
Brandt's speedup theorem for expressing it in the broader framework of
round-based models supporting full information protocols, which includes models
as different as wait-free shared-memory computing with iterated immediate
snapshots, and synchronous failure-free network computing. In particular, we
provide general definitions for notions such as local checkability and local
independence, in our broader framework. In this way, we are able to identify
the hypotheses on the computing model, and on the tasks, that are sufficient
for Brandt's speedup theorem to apply. More precisely, we identify which
hypotheses are sufficient for the each direction of the if-and-only-if
condition. Interestingly, these hypotheses are of different natures. Our
general approach enables to extend Brandt's speedup theorem from LOCAL to
directed networks, to hypergraphs, to dynamic networks, and even to graphs
including short cyclic dependencies between processes (i.e., the large girth
condition is, to some extend, not necessary). The theorem can even be extended
to shared-memory wait-free computing. In particular, we provide new
impossibility proofs for consensus and perfect renaming in 2-process systems.
