We briefly record alternative simplicial models for associative algebras and their modules. We will use their cut diagrams in the construction of endomorphism categories below. Lurie’s proof of the colimit statement in Corollary 19.1.17 also uses these models to lift an operadic colimit of the underlying diagram to a colimit of modules.
19.3.1 Associative algebras as simplicial objects
For a nonempty finite linear order \(P\), let \(\Cut (P)\) be its set of Dedekind cuts, namely the nontrivial partitions \((P_0,P_1)\) such that \(p_0\leq p_1\) for all \(p_0\in P_0\) and \(p_1\in P_1\). Pullback of cuts along a map of finite linear orders defines a span of finite sets. The canonical ordering of the cuts refines this construction to a functor \[ \Cut \colon \simp \catop \longrightarrow \Assoc ^{\otimes }. \]
Definition 19.3.1. Let \(C\) be a monoidal \(\infty \)-category. An \(\bbA _{\infty }\)-algebra in \(C\) is a functor \[ A\colon \simp \catop \longrightarrow C^{\otimes } \] over \(\Cut \colon \simp \catop \to \Assoc ^{\otimes }\) which sends the dual of every interval inclusion \[ e_i\colon [1]\simeq \{i-1\leq i\}\hookrightarrow [n] \] to an inert morphism. We write \[ \Alg _{\bbA _{\infty }}(C) \subseteq \Fun _{/\Assoc ^{\otimes }}(\simp \catop ,C^{\otimes }) \] for the full subcategory of such functors.
The inert condition identifies the components of \(A_n\in C^{\otimes }_n\simeq C^n\) with \(A_1\). The resulting simplicial structure records the multiplication and all its coherences.
Theorem 19.3.2 ([Lurie (2017), Proposition 4.1.3.19]). Restriction along \(\Cut \) induces an equivalence \[ \Cut ^*\colon \Alg (C)\iso \Alg _{\bbA _{\infty }}(C). \]
Applied to the cartesian monoidal \(\infty \)-category \(\Cat _{\infty }\), this identifies \(\bbA _{\infty }\)-monoidal \(\infty \)-categories with monoidal \(\infty \)-categories.
19.3.2 Left modules as simplicial objects
Construction 19.3.3. Define \[ \LCut \colon \simp \catop \times [1]\longrightarrow \oLMod ^{\otimes } \] as the composite \[ \simp \catop \times [1] \xrightarrow {\Cut \times [1]} \Assoc ^{\otimes }\times [1] \longrightarrow \oLMod ^{\otimes }, \] where the final functor sends \((I,1)\) to \((I,\emptyset )\) and \((I,0)\) to \((I,\{m\})\). The morphism from \((I,0)\) to \((I,1)\) is represented by the span \[ (I,\{m\})\hookleftarrow (I,\emptyset )\xrightarrow {=}(I,\emptyset ). \]
Definition 19.3.4. An \(\bbA _{\infty }\)-module in \(C\) is a functor \[ F\colon \simp \catop \times [1]\longrightarrow C^{\otimes } \] over \(\LCut \) satisfying the following conditions:
- (1)
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The restriction to \(\simp \catop \times \{1\}\) is an \(\bbA _{\infty }\)-algebra.
- (2)
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Each morphism \(F([n],0)\to F([n],1)\) is inert.
- (3)
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If \(\alpha \colon [n]\to [m]\) is an interval inclusion with \(\alpha (n)=m\), then \(F(\alpha \catop ,\id _0)\) is inert.
We write \(\LMod ^{\bbA _{\infty }}(C)\) for the resulting full subcategory of \(\Fun _{/\oLMod ^{\otimes }}(\simp \catop \times [1],C^{\otimes })\).
Theorem 19.3.5 ([Lurie (2017), Proposition 4.2.2.12]). Restriction along \(\LCut \) induces an equivalence \[ \LCut ^*\colon \LMod (C)\iso \LMod ^{\bbA _{\infty }}(C). \]
For a module represented by \(F\), write \(A=F([1],1)\) and \(M=F([0],0)\). The conditions identify \[ F([n],0)\simeq (A,\dots ,A,M), \] so the simplicial structure records all the action maps \(A^{\otimes n}\otimes M\to A^{\otimes m}\otimes M\). Applied to a diagram of \(A\)-modules, this model explains why a colimit of the underlying objects inherits an \(A\)-action when \(A\otimes -\) preserves that colimit. In the proof cited in Corollary 19.1.17, this is made precise by taking the colimit in the functor category defining the simplicial model.
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