  Craig Squier proved that, if a monoid can be presented by a finite convergent
string rewriting system, then it satisfies the homological finiteness condition
left-FP3. Using this result, he constructed finitely presentable monoids with a
decidable word problem, but that cannot be presented by finite convergent
rewriting systems. Later, he introduced the condition of finite derivation
type, which is a homotopical finiteness property on the presentation complex
associated to a monoid presentation. He showed that this condition is an
invariant of finite presentations and he gave a constructive way to prove this
finiteness property based on the computation of the critical branchings: being
of finite derivation type is a necessary condition for a finitely presented
monoid to admit a finite convergent presentation. This survey presents Squier's
results in the contemporary language of polygraphs and higher-dimensional
categories, with new proofs and relations between them.

Category Theory