ScalingStacks

Let \(M\) be a monoid (i.e. a set equipped with an associative and unital binary operation). Then, the data of \(M\) is entirely recorded by its bar construction, a simplicial set \({\textup{Bar}}(M)\) with \({\textup{Bar}}(M)_n \coloneqq M^{\times n}\) whose face and degeneracy maps respectively record the product and unit of \(M\). Indeed, \(M\) is already completely specified by the restriction \(\Delta^\mathrm{op}_{\leq 3} \hookrightarrow\Delta^\mathrm{op}\xrightarrow{{\textup{Bar}}(M)} \mathrm{Set}\); note that the associativity of its multiplication is guaranteed by the commutativity of a certain square of face maps in \(\Delta^\mathrm{op}\) (whose morphisms all correspond to endpoint-preserving injections in \(\Delta\)). Altogether, we can identify monoids as a full subcategory either of \(\mathrm{Fun}(\Delta^\mathrm{op},\mathrm{Set})\) or of \(\mathrm{Fun}(\Delta^\mathrm{op}_{\leq 3},\mathrm{Set})\).