Envelopes for algebras and modules

Example 17.4.12 (Envelopes for algebras and modules). All the following categories are symmetric monoidal under disjoint union.

(1)

The envelope \(\Env (\Assoc )\) is the category \(\Fin ^{\Alg }\) whose objects are finite sets and whose morphisms \(S\to T\) are maps of finite sets together with a linear order on every fiber over \(t\in T\).

(2)

The envelope \(\Env (\oLMod )\) is the category \(\Fin ^{\LMod }\) whose objects are pairs \((S_A,S_M)\) of finite sets. A morphism \((S_A,S_M)\to (T_A,T_M)\) is a map \[ f\colon S_A\sqcup S_M\longrightarrow T_A\sqcup T_M \] which restricts to a bijection \(S_M\to T_M\), together with a linear order on \(f^{-1}(t)\cap S_A\) for every \(t\in T_A\sqcup T_M\).

(3)

The envelope \(\Env (\oMod )\) is the category \(\Fin ^{\Mod }\) with the same objects and morphisms as in (2), but without the choices of linear orders.

These descriptions follow immediately from Lemma 17.3.17 and the multimorphism sets of \(\Assoc \), \(\oLMod \), and \(\oMod \).