Subsection 2.4 (0M3A) · ScalingStacks

2.4 Free Algebras[0M3A]

The symmetric sequence underlying a reduced $\infty$ -operad $\mathcal{P}$ features in the construction of free $\mathcal{P}$ -algebras. In what follows we briefly recall and summarize the material of A.3.1.3 specialized to the setting that is of interest to us. That is, let $\mathcal{P}$ be a reduced $\infty$ -operad and let $p\colon\mathcal{C}^{\otimes}\to\mathbf{Fin}_{*}$ be a presentably symmetric monoidal $\infty$ -category. By A.3.1.3.5 the forgetful functor

U_{\mathcal{P}}\colon\operatorname{Alg}_{\mathcal{P}}\left(\mathcal{C}\right)\to\operatorname{Alg}_{\mathbf{Triv}}\left(\mathcal{C}\right)\simeq\mathcal{C}

admits a left adjoint $F_{\mathcal{P}}$ (the free $\mathcal{P}$ -algebra functor) that can be characterized as follows. By definition A.3.1.3.1, for every object $X\in\mathcal{C}$ we get a diagram $\mathcal{P}_{\mathbf{SSeq}}\left(X\right)\colon\mathcal{P}_{\mathbf{SSeq}}^{\otimes}\to\mathcal{\mathcal{C}}_{\operatorname{\scriptsize{act}}}^{\otimes}$ that, loosely speaking, corresponds to a sequence of maps $\mathcal{P}_{\mathbf{SSeq}}\left(n\right)\to\mathcal{\mathcal{C}}_{\operatorname{\scriptsize{act}}}^{\otimes}$ , such that each map lands in the connected component of $X^{\otimes n}$ and is $\Sigma_{n}$ -equivariant in the evident way. Furthermore, a map $f\colon X\to U_{\mathcal{P}}\left(A\right)$ in $\mathcal{C}$ , gives a lift of $\mathcal{P}_{\mathbf{SSeq}}^{\otimes}\left(X\right)$ to a cone diagram

\mathcal{P}_{\mathbf{SSeq}}^{\otimes}\left(f\right)\colon\mathcal{P}_{\mathbf{SSeq}}^{\otimes}\to\left(\mathcal{\mathcal{C}}_{\operatorname{\scriptsize{act}}}^{\otimes}\right)_{/U_{\mathcal{P}}\left(A\right)}.

We say that $f$ exhibits $A$ as the free $\mathcal{P}$ -algebra on $X$ , if $\mathcal{P}_{\mathbf{SSeq}}^{\otimes}\left(f\right)$ is an operadic $p$ -colimit diagram. By A.3.1.3.2 and A.3.1.3.5, such a map $f$ exists for every $X$ and can be taken as the $X$ -component of a unit natural transformation for an adjunction $F_{\mathcal{P}}\dashv U_{\mathcal{P}}$ .

Using our assumption on $\mathcal{C}$ , we can reduce the operadic colimit in the above discussion to an ordinary colimit in $\mathcal{C}$ . Consider the following commutative diagram

where $\alpha$ is a natural transformation from $p$ to the constant diagram on $\left\langle 1\right\rangle$ that consists of active morphisms. Let $\overline{\alpha}$ be a coCartesian natural transformation that lifts $\alpha$ . The restricted functor $F=\overline{\alpha}|_{\Delta^{\left\{1\right\}}\times\mathcal{C}_{\operatorname{\scriptsize{act}}}^{\otimes}}$ lands in the fiber over $\left\langle 1\right\rangle$ and is therefore a functor $F\colon\mathcal{C}_{\operatorname{\scriptsize{act}}}^{\otimes}\to\mathcal{C}$ .

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Remark 2.4.1. Informally speaking, $F$ takes each multi-object $X_{1}\oplus\cdots\oplus X_{n}$ to the tensor product $X_{1}\otimes\cdots\otimes X_{n}$ . There are two abstract characterizations of $F$ (which we shall not use):

(1)

It is the left adjoint of the inclusion $\underline{\mathcal{C}}\hookrightarrow\mathcal{C}_{\operatorname{\scriptsize{act}}}^{\otimes}$ .
(2)

The symmetric monoidal envelope is a left adjoint to the inclusion of symmetric monoidal $\infty$ -categories into $\infty$ -operads. The functor $F$ is the induced functor on the underlying $\infty$ -categories of the unit of this adjunction at the object $\mathcal{C}$ .

By A.3.1.1.15 and A.3.1.1.16, $\mathcal{P}_{\mathbf{SSeq}}^{\otimes}\left(f\right)$ is an operadic $p$ -colimit diagram if and only if the diagram $\mathcal{P}_{\mathbf{SSeq}}\left(f\right)=F\circ\mathcal{P}_{\mathbf{SSeq}}^{\otimes}\left(f\right)$ is a colimit diagram in $\mathcal{C}$ . In particular, we get

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Lemma 2.4.2. Let $\mathcal{P}$ be a reduced $\infty$ -operad and let $\mathcal{C}$ be a presentably symmetric monoidal $\infty$ -category. The forgetful functor

U_{\mathcal{P}}\colon\underline{\operatorname{Alg}}_{\mathcal{P}}\left(\mathcal{C}\right)\to\underline{\operatorname{Alg}}_{\mathbf{Triv}}\left(\mathcal{C}\right)\simeq\mathcal{C}

admits a left adjoint $F_{\mathcal{P}}$ and the associated monad $T_{\mathcal{P}}=U_{\mathcal{P}}\circ F_{\mathcal{P}}$ acts on an object $X\in\mathcal{C}$ as follows:

T_{\mathcal{P}}\left(X\right)=U_{\mathcal{P}}F_{\mathcal{P}}\left(X\right)=\operatorname*{colim}\mathcal{P}_{\boldsymbol{\mathbf{SSeq}}}\left(X\right)=\coprod_{n\geq 0}\left(\mathcal{P}\left(n\right)\otimes X^{\otimes n}\right)_{h\Sigma_{n}}

(where we let $\otimes$ denote the canonical enrichment of $\underline{\mathcal{C}}$ over $\mathcal{S}$ as well).

Our next goal is to articulate the functoriality of $T_{\mathcal{P}}$ in the $\infty$ -operad $\mathcal{P}$ .

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Construction 2.4.3. Given a map of reduced $\infty$ -operads $\mathcal{P}\to\mathcal{Q}$ we get a forgetful functor $G\colon\underline{\operatorname{Alg}}_{\mathcal{Q}}\left(\mathcal{C}\right)\to\underline{\operatorname{Alg}}_{\mathcal{P}}\left(\mathcal{C}\right)$ , such that $U_{\mathcal{P}}G=U_{\mathcal{Q}}$ . The unit map

\operatorname{Id}\to U_{\mathcal{Q}}F_{\mathcal{Q}}=U_{\mathcal{P}}GF_{\mathcal{Q}}

has an adjunct $F_{\mathcal{P}}\to GF_{\mathcal{Q}}$ and by applying $U_{\mathcal{P}}$ we obtain an induced map of the associated monads (as endofunctors of $\mathcal{C}$ ):

\alpha_{G}\colon T_{\mathcal{P}}=U_{\mathcal{P}}F_{\mathcal{P}}\to U_{\mathcal{P}}GF_{\mathcal{Q}}=U_{\mathcal{Q}}F_{\mathcal{Q}}=T_{\mathcal{Q}},

which is well defined up to homotopy.

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Lemma 2.4.4. In the setting of Construction 2.4.3, if $G$ is an equivalence of $\infty$ -categories, then the map $\alpha_{G}\colon T_{\mathcal{P}}\to T_{\mathcal{Q}}$ is a natural equivalence of functors.

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Proof. Since all the steps in the construction are invariant, we may assume without loss of generality that $G$ is the identity functor and $U_{\mathcal{P}}=U_{\mathcal{Q}}$ . In this case, the map $\alpha_{G}$ is given by applying $U_{\mathcal{Q}}$ to the composition

F_{\mathcal{Q}}\xrightarrow{F_{\mathcal{Q}}u}F_{\mathcal{Q}}U_{\mathcal{Q}}F_{\mathcal{Q}}\xrightarrow{cF_{\mathcal{Q}}}F_{\mathcal{Q}}

where $u$ and $c$ are the unit and counit of the adjunction $F_{\mathcal{Q}}\dashv U_{\mathcal{Q}}$ . This composition is homotopic to the identity by the zig-zag identities. ∎

Our last task is to show that the map from Construction 2.4.3 is induced from the map of symmetric sequences $\mathcal{P}_{\mathbf{SSeq}}\to\mathcal{Q}_{\mathbf{SSeq}}$ by the functoriality of the explicit formula given in 2.4.2.

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Lemma 2.4.5. Given a map $f\colon X\to U_{\mathcal{P}}\left(A\right)$ , the map $\operatorname*{colim}\mathcal{P}_{\mathbf{SSeq}}\left(X\right)\to U_{\mathcal{P}}\left(A\right)$ induced by the diagram $\mathcal{P}_{\mathbf{SSeq}}\left(f\right)$ is equivalent to the canonical map $\tilde{f}\colon U_{\mathcal{P}}F_{\mathcal{P}}\left(X\right)\to U_{\mathcal{P}}\left(A\right)$ (ie $U_{\mathcal{P}}$ of the adjunct of $f$ ).

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Proof. One only has to observe that the map $\tilde{f}\colon U_{\mathcal{P}}F_{\mathcal{P}}\left(X\right)\to U_{\mathcal{P}}\left(A\right)$ is a map of cones on $\mathcal{P}_{\mathbf{SSeq}}\left(X\right)$ . Let $u_{X}\colon X\to U_{\mathcal{P}}F_{\mathcal{P}}\left(X\right)$ be the unit map of the free-forgetful adjunction at $X$ . The adjunct map $F_{\mathcal{P}}\left(X\right)\to A$ induces a map $\tilde{f}^{\triangleright}\colon\left(\mathcal{\mathcal{C}}_{\operatorname{\scriptsize{act}}}^{\otimes}\right)_{/U_{\mathcal{P}}F_{\mathcal{P}}\left(X\right)}\to\left(\mathcal{\mathcal{C}}_{\operatorname{\scriptsize{act}}}^{\otimes}\right)_{/U_{\mathcal{P}}\left(A\right)}$ . Inspecting Construction A.3.1.3.1, it can be seen that the cone diagram $\mathcal{P}_{\mathbf{SSeq}}\left(f\right)$ is equivalent to the composition of the universal cone diagram $\mathcal{P}_{\mathbf{SSeq}}\left(u_{X}\right)$ and $\tilde{f}^{\triangleright}$ . ∎

From this we get

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Proposition 2.4.6. Let $g\colon\mathcal{P}\to\mathcal{Q}$ be a map of reduced $\infty$ -operads and let $\mathcal{C}$ be a presentably symmetric monoidal $\infty$ -category. For every object $X\in\mathcal{C}$ , the induced map of the associated monads

T_{\mathcal{P}}\left(X\right)=\operatorname*{colim}\mathcal{P}_{\mathbf{SSeq}}\left(X\right)\to\operatorname*{colim}\mathcal{Q}_{\mathbf{SSeq}}\left(X\right)=T_{\mathcal{Q}}\left(X\right)

is equivalent to the canonical map on colimits that is induced by pre-composition with

g_{\mathbf{SSeq}}\colon\mathcal{P}_{\mathbf{SSeq}}\to\mathcal{Q}_{\mathbf{SSeq}}.

[05XW]

Proof. We denote by $G\colon\underline{\operatorname{Alg}}_{\mathcal{Q}}\left(\mathcal{C}\right)\to\underline{\operatorname{Alg}}_{\mathcal{P}}\left(\mathcal{C}\right)$ the forgetful functor induced by the map $g$ . Let

f\colon X\to U_{\mathcal{Q}}F_{\mathcal{Q}}\left(X\right)\simeq U_{\mathcal{P}}GF_{\mathcal{Q}}\left(X\right)

be the unit map. It induces a cone diagram

\mathcal{P}_{\mathbf{SSeq}}\left(f\right)\colon\mathcal{P}_{\mathbf{SSeq}}\to\mathcal{\mathcal{C}}_{/U_{\mathcal{Q}}F_{\mathcal{Q}}\left(X\right)}

and, by 2.4.5, the associated map $\tilde{f}\colon U_{\mathcal{P}}F_{\mathcal{P}}\left(X\right)\to U_{\mathcal{Q}}F_{\mathcal{Q}}\left(X\right)$ is equivalent to the map $\operatorname*{colim}\mathcal{P}_{\mathbf{SSeq}}\left(X\right)\to U_{\mathcal{Q}}F_{\mathcal{Q}}\left(X\right)$ specified by the cone diagram $\mathcal{P}_{\mathbf{SSeq}}\left(f\right)$ . On the other hand, inspecting Construction A.3.1.3.1, it can be seen that the diagram $\mathcal{P}_{\mathbf{SSeq}}\left(f\right)$ is obtained from the diagram

\mathcal{Q}_{\mathbf{SSeq}}\left(f\right)\colon\mathcal{Q}_{\mathbf{SSeq}}\to\mathcal{\mathcal{C}}_{/U_{\mathcal{Q}}F_{\mathcal{Q}}\left(X\right)}

by pre-composition with $g_{\mathbf{SSeq}}\colon\mathcal{P}_{\mathbf{SSeq}}\to\mathcal{Q}_{\mathbf{SSeq}}$ and that $\mathcal{Q}_{\mathbf{SSeq}}\left(f\right)$ exhibits $U_{\mathcal{Q}}F_{\mathcal{Q}}\left(X\right)$ as the colimit of $\mathcal{Q}_{\mathbf{SSeq}}\left(X\right)$ . Thus, we get the desired equivalence. ∎

Tomer Schlank, Lior Yanovski

Original source: arXiv:1808.06006v3

Original source · 1808.06006v3