用户: Scavenger/Infinity Category and Higher Algebra/Modules in the context of Infinity Operads

The objective of this notes is to define the notion of modules in the context of $\infty$-operads.

The presentation of these materials in [2] is confusing (at least to me). There abounds typos and misleading (or, in some sense, wrong) notations and the technical details are vague. I try to remedy this in the notes below, by giving the right notations and statements of the theorem, arrange them in the correct order, and write down some remarks. Hope this can bring ease to those reading this part of [2].

1Coherent $\infty$-Operads

Definition 1.1 ([2] Definition B.3.1). We will say that a functor $\mathcal{C}\to {\Delta }^2$ of $\infty$-categories is flat if the inclusion $\mathcal{C}{\times }_{{\Delta }^2}{\Lambda }_1^2\to \mathcal{C}$ is a categorical equivalence.

Definition 1.2 ([2] Definition B.3.8). Let $X\to S$ be an inner fibration of simplicial sets. We say that it is flat if for every 2-simplex $\sigma$ of $S$, the pullback of $X\to S$ along $\Delta$, $X{\times }_S{\Delta }^2\to {\Delta }^2$ is flat in the sense of Definition 1.1.

Definition 1.3 ([2] Definition 2.3.1.1). Let ${\mathcal{O}}^{\otimes }$ be an $\infty$-operad. We will say that ${\mathcal{O}}^{\otimes }$ is unital if for every object $X\in \mathcal{O}={\mathcal{O}}_{⟨1⟩}^{\otimes }$, the Kan complex ${\mathrm{Mul}}_{{\mathcal{O}}^{\otimes }}(\varnothing ,X)$ is contractible.

Definition 1.4 ([2] Definition 3.3.1.1). We will say that a morphism $\alpha :⟨m⟩\to ⟨n⟩$ in $N({\mathrm{Fin}}_{\ast })$ is semi-inert if ${\alpha }^{-1}(i)$ has at most one element for every $i\in ⟨n{⟩}^{\circ }$. We will say that $\alpha$ is null if ${\alpha }^{-1}(⟨n{⟩}^{\circ })=\varnothing$.

Let $p:{\mathcal{O}}^{\otimes }\to N({\mathrm{Fin}}_{\ast })$ be an $\infty$-operad, and let $f:X\to Y$ be a morphism in ${\mathcal{O}}^{\otimes }$. We will say that $f$ is semi-inert if it satisfies the following conditions:

(i) $p(f)$ is semi-inert in $N({\mathrm{Fin}}_{\ast })$.

(ii) For every inert morphism $g:Y\to Z$ in ${\mathcal{O}}^{\otimes }$, if $p(g)\circ p(f)$ is an inert morphism in $N({\mathrm{Fin}}_{\ast })$, then $g\circ f$ is an inert morphism in ${\mathcal{O}}^{\otimes }$.

We will say that $f$ is null if $p(f)$ is null.

1.5. A null morphism in an $\infty$-operad is semi-inert (see [2] Remark 3.3.1.2)

Notation 1.6 ([2] Notation 3.3.2.1). Let ${\mathcal{O}}^{\otimes }$ be a unital $\infty$-operad. We use ${\mathcal{K}}_{{\mathcal{O}}^{\otimes }}$ (or simply ${\mathcal{K}}_{\mathcal{O}}$) to denote the full subcategory of $\mathrm{Fun}({\Delta }^1,{\mathcal{O}}^{\otimes })$ spanned by the semi-inert morphisms. For $i\in \{0,1\}$ we let $e_i:{\mathcal{K}}_{\mathcal{O}}\to {\mathcal{O}}^{\otimes }$ be the map given by evaluation at the vertex $\{i\}$ of ${\Delta }^1$. We will say that a morphism in ${\mathcal{K}}_{\mathcal{O}}$ is inert if its images under $e_0$ and $e_1$ are inert morphisms in ${\mathcal{O}}^{\otimes }$.

Notation 1.7 ([2] Notation 3.3.3.2). Let ${\mathcal{O}}^{\otimes }$ be a unital $\infty$-operad. We use ${\mathcal{K}}_{\mathcal{O}}^0$ to denote the full subcategory of ${\mathcal{K}}_{\mathcal{O}}$ spanned by the null morphisms in ${\mathcal{O}}^{\otimes }$. For $i=0,1$, we let $e_i^0$ denote the restriction to ${\mathcal{K}}_{\mathcal{O}}^0$ of the evaluation map $e_i:{\mathcal{K}}_{\mathcal{O}}\to {\mathcal{O}}^{\otimes }$.

Definition 1.8 ([1] Definition 3.1.10). We will say that a unital $\infty$-operad ${\mathcal{O}}^{\otimes }$ is coherent if the evaluation map $e_0:{\mathcal{K}}_O\to {\mathcal{O}}^{\otimes }$ is a flat isofibration.

Remark 1.9. This definition of coherent $\infty$-operads (probably) disagrees with the one in [2] (i.e. Definition 3.3.1.9). The criterion given in [2] (i.e. Theorem 3.3.2.2) only guarantees the conincidence of the two definitions in the case $\mathcal{O}$ is a Kan complex. I believe the definition on [1] is the right one since $e_0$ being a flat isofibration is the condition that is essentially used in proving various properties regarding modules. However, a concern of me is that this means if ${\mathcal{O}}^{\otimes }$ is a coherent $\infty$-operad in the sense of [2], then $e_0$ is not necessarily a flat isofibration, leading to possible incorrectness of [2].

Example 1.10. ${\mathrm{Assoc}}^{\otimes }$ is coherent by [2] Proposition 4.1.1.20 or [1] Proposition 3.5.1.

${\mathrm{Comm}}^{\otimes }$ is coherent by [2] Example 3.3.1.12 or [1] Proposition 3.6.1.

2Modules

Construction 2.1 ([2] Construction 3.3.3.1). Let ${\mathcal{O}}^{\otimes }$ be a unital $\infty$-operad and let ${\mathcal{C}}^{\otimes }\to {\mathcal{O}}^{\otimes }$ be a fibration of generalized $\infty$-operads. We define a simplicial set ${\widetilde{\mathrm{Mod}}}^{{\mathcal{O}}^{\otimes }}({\mathcal{C}}^{\otimes }{)}^{\otimes }$ (or simply ${\widetilde{\mathrm{Mod}}}^{\mathcal{O}}(\mathcal{C}{)}^{\otimes }$) over the simplicial set ${\mathcal{O}}^{\otimes }$ so that the following universal property is satisfied: for every simplicial set $X$ over ${\mathcal{O}}^{\otimes }$, there is a canonical bijection of sets ${\mathrm{Map}}_{/{\mathcal{O}}^{\otimes }}(X,{\widetilde{\mathrm{Mod}}}^{\mathcal{O}}(\mathcal{C}{)}^{\otimes })\simeq {\mathrm{Map}}_{\mathrm{Fun}(\{1\},{\mathcal{O}}^{\otimes })}(X{\times }_{\mathrm{Fun}(\{0\},{\mathcal{O}}^{\otimes })}{\mathcal{K}}_{\mathcal{O}},{\mathcal{C}}^{\otimes })$.

We let ${\overline{\mathrm{Mod}}}^{\mathcal{O}}(\mathcal{C}{)}^{\otimes }$ denote the full simplicial subset of ${\widetilde{\mathrm{Mod}}}^{\mathcal{O}}(\mathcal{C}{)}^{\otimes }$ spanned by those vertices $\overline{v}$ with the following property: let $v\in {\mathcal{O}}^{\otimes }$ be the image of $\overline{v}$ in ${\mathcal{O}}^{\otimes }$, then the functor $\{v\}{\times }_{\mathrm{Fun}(\{0\},{\mathcal{O}}^{\otimes })}{\mathcal{K}}_{\mathcal{O}}\to {\mathcal{C}}^{\otimes }$ corresponding to $\overline{v}$ brings inert morphisms to inert morphisms (here we view $\{v\}{\times }_{\mathrm{Fun}(\{0\},{\mathcal{O}}^{\otimes })}{\mathcal{K}}_{\mathcal{O}}$ as a simplicial subset of ${\mathcal{K}}_{\mathcal{O}}$).

Construction 2.2 ([2] Notation 3.3.3.5). Let ${\mathcal{O}}^{\otimes }$ be a unital $\infty$-operad, and let ${\mathcal{C}}^{\otimes }\to {\mathcal{O}}^{\otimes }$ be a fibration of generalized $\infty$-operads. We define a simplicial set ${\widetilde{\mathrm{Alg}}}_{/\mathcal{O}}(\mathcal{C})$ over ${\mathcal{O}}^{\otimes }$ by the following universal property: for every simplicial set $X$ over ${\mathcal{O}}^{\otimes }$, we have a canonical bijection ${\mathrm{Map}}_{/{\mathcal{O}}^{\otimes }}(X,{\widetilde{\mathrm{Alg}}}_{/\mathcal{O}}(\mathcal{C}))\simeq {\mathrm{Map}}_{/\mathrm{Fun}(\{1\},{\mathcal{O}}^{\otimes })}(X{\times }_{\mathrm{Fun}(\{0\},{\mathcal{O}}^{\otimes })}{\mathcal{K}}_{\mathcal{O}}^0,{\mathcal{C}}^{\otimes })$. We let ${}^p{\mathrm{Alg}}_{/\mathcal{O}}(\mathcal{C})$ denote the full simplicial subset of ${\widetilde{\mathrm{Alg}}}_{/\mathcal{O}}(\mathcal{C})$ spanned by those vertices whose corresponding functor ${\Delta }^0{\times }_{\mathrm{Fun}(\{0\},{\mathcal{O}}^{\otimes })}{\mathcal{K}}_{\mathcal{O}}\to {\mathcal{C}}^{\otimes }$ brings inert morphisms to inert morphisms.

2.3. The natural inclusion ${\mathcal{K}}_{\mathcal{O}}^0\to {\mathcal{K}}_{\mathcal{O}}$ induces a natural map ${\widetilde{\mathrm{Mod}}}^{\mathcal{O}}(\mathcal{C}{)}^{\otimes }\to {\widetilde{\mathrm{Alg}}}_{/\mathcal{O}}(\mathcal{C})$ by the defining universal properties. It is easy to see that ${\overline{\mathrm{Mod}}}^{\mathcal{O}}(\mathcal{C}{)}^{\otimes }$ is mapped into ${}^p{\mathrm{Alg}}_{/\mathcal{O}}(\mathcal{C})$.

There is a map ${\mathcal{K}}_{\mathcal{O}}^0\times {\mathrm{Fun}}_{/{\mathcal{O}}^{\otimes }}({\mathcal{O}}^{\otimes },{\mathcal{C}}^{\otimes })\to {\mathcal{C}}^{\otimes }$ of simplicial sets over ${\mathcal{O}}^{\otimes }$ constructed as the composition ${\mathcal{K}}_{\mathcal{O}}^0\times {\mathrm{Fun}}_{/{\mathcal{O}}^{\otimes }}({\mathcal{O}}^{\otimes },{\mathcal{C}}^{\otimes })\to \mathrm{Fun}({\Delta }^1,{\mathcal{O}}^{\otimes })\times \mathrm{Fun}({\mathcal{O}}^{\otimes },{\mathcal{C}}^{\otimes })\to \mathrm{Fun}({\Delta }^1,{\mathcal{C}}^{\otimes })\stackrel{e{v}_1}{\to }{\mathcal{C}}^{\otimes }$, which induces a map ${\mathcal{O}}^{\otimes }\times {\mathrm{Fun}}_{/{\mathcal{O}}^{\otimes }}({\mathcal{O}}^{\otimes },{\mathcal{C}}^{\otimes })\to {\widetilde{\mathrm{Alg}}}_{/\mathcal{O}}(\mathcal{C})$. It is easy to see that the preimage of ${}^p{\mathrm{Alg}}_{/\mathcal{O}}(\mathcal{C})$ under this map can be identified with ${\mathcal{O}}^{\otimes }\times {\mathrm{Alg}}_{/\mathcal{O}}(\mathcal{C})$. Thus we have a natural map ${\mathcal{O}}^{\otimes }\times {\mathrm{Alg}}_{/\mathcal{O}}(\mathcal{C}){\to }^p{\mathrm{Alg}}_{/\mathcal{O}}(\mathcal{C})$.

Definition 2.4 ([2] Definition 3.3.3.8). Let ${\mathcal{O}}^{\otimes }$ be a unital $\infty$-operad, and let ${\mathcal{C}}^{\otimes }\to {\mathcal{O}}^{\otimes }$ be a fibration of generalized $\infty$-operads. We let ${\mathrm{Mod}}^{\mathcal{O}}(\mathcal{C}{)}^{\otimes }$ denote the fiber product ${\overline{\mathrm{Mod}}}^{\mathcal{O}}(\mathcal{C}{)}^{\otimes }{\times }_{{}^p{\mathrm{Alg}}_{/\mathcal{O}}(\mathcal{C})}({\mathcal{O}}^{\otimes }\times {\mathrm{Alg}}_{/\mathcal{O}}(\mathcal{C}))$. For every algebra object $A\in {\mathrm{Alg}}_{/\mathcal{O}}(\mathcal{C})$, we let ${\mathrm{Mod}}_A^{\mathcal{O}}(\mathcal{C}{)}^{\otimes }$ denote the fiber product ${\mathrm{Mod}}^{\mathcal{O}}(\mathcal{C}{)}^{\otimes }{\times }_{{\mathrm{Alg}}_{/\mathcal{O}}(\mathcal{C})}\{A\}$. We will refer to this simplicial set as the $\infty$-operad of $\mathcal{O}$-module objects over $A$.

Below is the main theorem of this section, and justifies the name of ${\mathrm{Mod}}_A^{\mathcal{O}}(\mathcal{C}{)}^{\otimes }$.

Theorem 2.5 ([2] Theorem 3.3.3.9). Let ${\mathcal{C}}^{\otimes }\to {\mathcal{O}}^{\otimes }$ be a fibration of $\infty$-operads, where ${\mathcal{O}}^{\otimes }$ is coherent, and let $A\in {\mathrm{Alg}}_{/\mathcal{O}}(\mathcal{C})$. Then the natural map ${\mathrm{Mod}}_A^{\mathcal{O}}(\mathcal{C}{)}^{\otimes }\to {\mathcal{O}}^{\otimes }$ is a fibration of $\infty$-operads.

(Sketch of) Proof of the Theorem

Definition 2.6 ([2] Notation B.4.4). Suppose we are given a map of simplicial sets $X\stackrel{\varphi }{\to }Y\stackrel{\pi }{\to }Z$. We let ${\pi }_{\ast }(X)$ denote the simplicial set over $Z$ defined by the following universal property: for every map of simplicial sets $K\to Z$, we have a canonical bijection ${\mathrm{Map}}_{/Z}(K,{\pi }_{\ast }(X))\simeq {\mathrm{Map}}_{/Y}(K{\times }_{Z}Y,X)$.

Proposition 2.7 ([2] Proposition B.4.5). Let $\varkappa$ be a strongly inaccessible cardinal. Let $Y,Z$ be $\varkappa$-$\infty$-categories and let $\pi :Y\to Z$ be a flat isofibration of simplicial sets. Then the functor ${\pi }_{\ast }:{\mathrm{sSet}}_{/Y}^{\varkappa }\to {\mathrm{sSet}}_{/Z}^{\varkappa }$ is a right Quillen functor with respect to the Joyal model structures. In particular, if $X\to Y$ is an isofibration, then the induced map ${\pi }_{\ast }X\to Z$ is an isofibration.

Proof. The functor ${\pi }_{\ast }$ admits a left adjoint ${\pi }^{\ast }$ given by the formula ${\pi }^{\ast }A=A{\times }_{Z}Y$. To prove that ${\pi }_{\ast }$ is a right Quillen functor it suffices to show that ${\pi }^{\ast }$ preserves monomorphisms and categorical equivalences. The first case is clear, and the second case follows from [2] Corollary B.3.15.

$\square$

Remark 2.8. Proposition B.4.5 of [2] only requires $Y$ and $Z$ to be simplicial sets instead of $\infty$-categories. I think this is incorrect since the conditions of [2] Corollary B.3.15 require them to be $\infty$-categories.

Lemma 2.9 ([2] Lemma 3.3.3.12). Let ${\mathcal{O}}^{\otimes }$ be a unital $\infty$-operad and let ${\mathcal{C}}^{\otimes }\to {\mathcal{O}}^{\otimes }$ be a fibration of generalized $\infty$-operads. Then the map ${\widetilde{\mathrm{Alg}}}_{/\mathcal{O}}(C)\to {\mathcal{O}}^{\otimes }$ is an isofibration. In particular, ${\widetilde{\mathrm{Alg}}}_{/\mathcal{O}}(C)$ is an $\infty$-category.

Proof. The simplicial set ${\widetilde{\mathrm{Alg}}}_{/\mathcal{O}}(C)$ can be identified with $(e_0^0{)}_{\ast }({\mathcal{K}}_{\mathcal{O}}^0{\times }_{\mathrm{Fun}(\{1\},{\mathcal{O}}^{\otimes })}{\mathcal{C}}^{\otimes })$. The result follows from Proposition 2.7 and [2] Corollary 3.3.3.4.

$\square$

Lemma 2.10 ([2] Lemma 3.3.3.14). Let ${\mathcal{O}}^{\otimes }$ be a unital $\infty$-operad and let ${\mathcal{C}}^{\otimes }\to {\mathcal{O}}^{\otimes }$ be a fibration of generalized $\infty$-operads. Then the map ${\widetilde{\mathrm{Mod}}}^{\mathcal{O}}(\mathcal{C}{)}^{\otimes }\to {\widetilde{\mathrm{Alg}}}_{/\mathcal{O}}(\mathcal{C})$ is an isofibration.

Remark 2.11. (i): The statement of Lemma 2.10 is slightly stronger then [2] Lemma 3.3.3.14, but the proof of the latter is essentially the former.

(ii): One can prove that ${\overline{\mathrm{Mod}}}^{\mathcal{O}}(\mathcal{C}{)}^{\otimes }$ and ${}^p{\mathrm{Alg}}_{/\mathcal{O}}(\mathcal{C})$ are replete full subcategories of ${\widetilde{\mathrm{Mod}}}^{\mathcal{O}}(\mathcal{C}{)}^{\otimes }$ and ${\widetilde{\mathrm{Alg}}}_{/\mathcal{O}}(\mathcal{C})$, respectively.

Proposition 2.12 ([2] Remark 3.3.3.7). Let ${\mathcal{O}}^{\otimes }$ be a unital $\infty$-operad and let ${\mathcal{C}}^{\otimes }\to {\mathcal{O}}^{\otimes }$ be a fibration of generalized $\infty$-operads. The map ${\mathcal{O}}^{\otimes }\times {\mathrm{Fun}}_{/{\mathcal{O}}^{\otimes }}({\mathcal{O}}^{\otimes },{\mathcal{C}}^{\otimes })\to {\widetilde{\mathrm{Alg}}}_{/\mathcal{O}}(\mathcal{C})$ is an equivalence of $\infty$-categories. Consequently, ${\mathcal{O}}^{\otimes }\times {\mathrm{Alg}}_{/\mathcal{O}}(\mathcal{C}){\to }^p{\mathrm{Alg}}_{/\mathcal{O}}(\mathcal{C})$ is also an equivalence of $\infty$-categories.

Proof. The idea is to use the fact that ${\mathcal{K}}_{\mathcal{O}}^0\to {\mathcal{O}}^{\otimes }\times {\mathcal{O}}^{\otimes }$ is a trivial Kan fibration ([2] Lemma 3.3.3.3) to show that ${\mathcal{O}}^{\otimes }\times {\mathrm{Fun}}_{/{\mathcal{O}}^{\otimes }}({\mathcal{O}}^{\otimes },{\mathcal{C}}^{\otimes })\to {\widetilde{\mathrm{Alg}}}_{/\mathcal{O}}(\mathcal{C})$ is an equivalence of inner fibrations over ${\mathcal{O}}^{\otimes }$.

$\square$

Proposition 2.13 ([2] Proposition 3.3.3.10). Let ${\mathcal{O}}^{\otimes }$ be a coherent $\infty$-operad, and let ${\mathcal{C}}^{\otimes }\to {\mathcal{O}}^{\otimes }$ be a fibration of generalized $\infty$-operads. Then the structure map ${\overline{\mathrm{Mod}}}^{\mathcal{O}}(\mathcal{C}{)}^{\otimes }\to {\mathcal{O}}^{\otimes }$ is again a fibration of generalized $\infty$-operads. Moreover, a morphism $f$ in ${\overline{\mathrm{Mod}}}^{\mathcal{O}}(\mathcal{C}{)}^{\otimes }$ is inert if and only if it satisfies the following conditions:

(i) The image of $f$ in ${\mathcal{O}}^{\otimes }$ under the structure map (which we denote by $f_0$) is inert.

(ii) Let $F:{\mathcal{K}}_{\mathcal{O}}{\times }_{\mathrm{Fun}(\{0\},{\mathcal{O}}^{\otimes })}{\Delta }^1\to {\mathcal{C}}^{\otimes }$ be the functor classified by $f$. For every $e_0$-Cartesian morphism $\tilde{f}$ of ${\mathcal{K}}_{\mathcal{O}}$ lifting $f_0$ (which is equivalent to $e_1(\tilde{f})$ being an equivalent, see part (4) of the proof of [2] Proposition 3.3.3.18), the morphism $F(\tilde{f})$ of ${\mathcal{C}}^{\otimes }$ is inert.

Remark 2.14. (i) Proposition 3.3.3.10 of [2] only requires ${\mathcal{C}}^{\otimes }\to {\mathcal{O}}^{\otimes }$ to be a map of generalized $\infty$-operads, instead of a fibration of generalized $\infty$-operads. This is incorrect.

(ii) One also needs a version of Proposition 2.13 concerning ${}^p{\mathrm{Alg}}_{/\mathcal{O}}(\mathcal{C})$ instead of ${\overline{\mathrm{Mod}}}^{\mathcal{O}}(\mathcal{C}{)}^{\otimes }$. The proof is simple since ${}^p{\mathrm{Alg}}_{/\mathcal{O}}(\mathcal{C})$ and ${\mathcal{O}}^{\otimes }\times {\mathrm{Alg}}_{/\mathcal{O}}(\mathcal{C})$ are equivalent inner fibrations over ${\mathcal{O}}^{\otimes }$. In particular, ${\overline{\mathrm{Mod}}}^{\mathcal{O}}(\mathcal{C}{)}^{\otimes }{\to }^p{\mathrm{Alg}}_{/\mathcal{O}}(\mathcal{C})$ is a fibration of generalized $\infty$-operads.

Now we can prove [2] Remark 3.3.3.16, using which we can finally prove 2.5.

References

[1]	Jacob Lurie, “Derived Algebraic Geometry III: Commutative Algebra”, arXiv:math/0703204v4, 2009
[2]	Jacob Lurie, “Higher Algebra”, Avalible for download at https://www.math.ias.edu/ lurie/
[3]	Jacob Lurie, “Kerodon”, https://kerodon.net