Subsection 2.2 (0M38) · ScalingStacks

2.2 Pointed Unital and Reduced $\infty$ -operads[0M38]

Recall from [Lur] the following definitions:

Definition 2.2.1. (A.2.3.1.1, A.2.3.4.1) An $\infty$ -operad $\mathcal{O}$ is called:

(1)

Unital if for every object $X$ of $\underline{\mathcal{O}}$ , the space of constants $\text{Mul}_{\mathcal{O}}\left(\varnothing,X\right)$ is contractible. We denote the full $\infty$ -category spanned by the unital $\infty$ -operads by $\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{un}}}$ .
(2)

Reduced if it is unital and the underlying $\infty$ -category is a contractible space. We denote the full $\infty$ -category spanned by the reduced $\infty$ -operads by $\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{red}}}$ .

Example 2.2.2. A symmetric monoidal $\infty$ -category is unital if and only if the unit object is initial.

We proceed by listing the various adjunctions between the different $\infty$ -categories of $\infty$ -operads and $\infty$ -categories. First, recall from A.2.1.4.10 that there is an underlying $\infty$ -category functor $\underline{\left(-\right)}\colon\mathbf{Op}_{\infty}\to\mathbf{Cat}_{\infty}$ and that this functor has a left adjoint $\iota\colon\mathbf{Cat}_{\infty}\hookrightarrow\mathbf{Op}_{\infty}$ , which is a fully faithful embedding. Informally, $\iota$ regards an $\infty$ -category as an $\infty$ -operad with empty higher (and nullary) multi-mapping spaces. On the other hand,

[05X1]

Lemma 2.2.3. The restriction of the forgetful functor $\underline{\left(-\right)}\colon\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{un}}}\to\mathbf{Cat}_{\infty}$ admits a right adjoint that takes every $\infty$ -category $\mathcal{C}$ to the coCartesian $\infty$ -operad $\mathcal{C}_{\sqcup}$ and the unit map of the adjunction $\underline{\mathcal{C}_{\sqcup}}\to\mathcal{C}$ is an equivalence (ie $\left(-\right)_{\sqcup}$ is fully faithful).

[05X2]

Proof. The first claim follows from A.2.4.3.9 by passing to maximal $\infty$ -subgroupoids. The second claim follows from A.2.4.3.11. ∎

By A.2.3.1.9, the fully faithful embedding $\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{un}}}\hookrightarrow\mathbf{Op}_{\infty}$ has a left adjoint given by tensoring with $\mathbb{E}_{0}$ (which is a localization functor). From this follows,

[05X3]

Lemma 2.2.4. $\mathbb{E}_{0}$ is the initial object of $\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{red}}}$ .

[05X4]

Proof. The composition of forgetful functors

\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{un}}}\to\mathbf{Op}_{\infty}\to\mathbf{Cat}_{\infty}

has a left adjoint given as the composition of the corresponding left adjoints. The first one takes $\Delta^{0}$ to $\mathbf{Triv}$ (by A.2.1.4.8) and the second takes $\mathbf{Triv}$ to $\mathbf{Triv}\otimes\mathbb{E}_{0}\simeq\mathbb{E}_{0}$ (by A.2.3.1.9). Hence, for every reduced operad $\mathcal{P}$ (which is in particular unital), we get

\operatorname{Map}\left(\mathbb{E}_{0},\mathcal{P}\right)\simeq\operatorname{Map}\left(\Delta^{0},\underline{\mathcal{P}}\right)\simeq\underline{\mathcal{P}}^{\simeq}\simeq\Delta^{0}.

∎

One source of unital symmetric monoidal $\infty$ -categories is

[05X5]

Lemma 2.2.5. Let $\mathcal{Q}$ be a unital $\infty$ -operad and let $\mathcal{C}$ be a symmetric monoidal $\infty$ -category. The symmetric monoidal $\infty$ -category $\operatorname{Alg}_{\mathcal{Q}}\left(\mathcal{C}\right)$ is also unital.

[05X6]

Proof. By A.3.2.4.4, the $\infty$ -operad $\operatorname{Alg}_{\mathcal{Q}}\left(\mathcal{C}\right)^{\otimes}\to\mathbf{Fin}_{*}$ is also a symmetric monoidal $\infty$ -category and so we only need to show that the unit object of $\operatorname{Alg}_{\mathcal{Q}}\left(\mathcal{C}\right)$ is initial. Since $\mathcal{Q}$ is unital, the canonical map $\mathcal{Q}\to\mathbb{E}_{0}\otimes\mathcal{Q}$ is an equivalence of $\infty$ -operads (by A.2.3.1.9) and therefore the forgetful functor

\underline{\operatorname{Alg}}_{\mathbb{E}_{0}\otimes\mathcal{Q}}\left(\mathcal{C}\right)\simeq\underline{\operatorname{Alg}}_{\mathbb{E}_{0}}\left(\operatorname{Alg}_{\mathcal{Q}}\left(\mathcal{C}\right)\right)\to\underline{\operatorname{Alg}}_{\mathcal{Q}}\left(\mathcal{C}\right)

is an equivalence of $\infty$ -categories. On the other hand, by A.2.1.3.10 we have

\underline{\operatorname{Alg}}_{\mathbb{E}_{0}}\left(\operatorname{Alg}_{\mathcal{Q}}\left(\mathcal{C}\right)\right)\simeq\underline{\operatorname{Alg}}_{\mathcal{Q}}\left(\mathcal{C}\right)_{1/},

where $1\in\underline{\operatorname{Alg}}_{\mathcal{Q}}\left(\mathcal{C}\right)$ is the unit object and the projection $\underline{\operatorname{Alg}}_{\mathcal{Q}}\left(\mathcal{C}\right)_{1/}\to\underline{\operatorname{Alg}}_{\mathcal{Q}}\left(\mathcal{C}\right)$ is an equivalence of $\infty$ -categories if and only if $1$ is initial (T.1.2.12.5). ∎

[0M2P]

Definition 2.2.6. The $\infty$ -category of pointed $\infty$ -categories is denoted by $\mathbf{Cat}_{\infty,*}=\left(\mathbf{Cat}_{\infty}\right)_{\Delta^{0}/}$ . The $\infty$ -category of pointed $\infty$ -operads is denoted by $\mathbf{Op}_{\infty,*}=\mathbf{Op}_{\infty}\times_{\mathbf{Cat}_{\infty}}\mathbf{Cat}_{\infty,*}$ . We also denote by $\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{un}}}$ and $\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{red}}}$ the corresponding $\infty$ -categories of pointed unital (resp. reduced) $\infty$ -operads.

[05X7]

Remark 2.2.7. Using 2.1.1, we have an equivalence of $\infty$ -categories $\mathbf{Op}_{\infty,*}\simeq\left(\mathbf{Op}_{\infty}\right)_{\mathbf{Triv}/}$ . Since $\mathbf{Triv}\to\mathbb{E}_{0}$ is an equivalence after tensoring with $\mathbb{E}_{0}$ , we get $\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{un}}}\simeq\left(\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{un}}}\right)_{\mathbb{E}_{0}/}$ and therefore also $\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{red}}}\simeq(\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{red}}})_{\mathbb{E}_{0}/}$ . We allow ourselves to pass freely between the two points of view on (unital, reduced) pointed $\infty$ -operads.

[05X8]

Remark 2.2.8. Observe that by 2.2.4, the projection $\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{red}}}\to\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{red}}}$ is an equivalence. Hence, the inclusion $\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{red}}}\hookrightarrow\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{un}}}$ induces a functor

\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{red}}}\simeq\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{red}}}\hookrightarrow\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{un}}}.

Moreover, it exhibits $\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{red}}}$ as the full subcategory of $\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{un}}}$ spanned by the reduced objects with respect to the underlying $\infty$ -category functor $\underline{\left(-\right)}\colon\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{un}}}\to\mathbf{Cat}_{\infty}$ in the sense of 2.1.4. Thus, the two notions of “reduced $\infty$ -operad” coincide.

We now apply the general observations from the previous subsection to deduce the following:

[05X9]

Proposition 2.2.9. The inclusion

\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{red}}}\simeq\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{red}}}\hookrightarrow\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{un}}}

has a right adjoint $\left(-\right)^{\operatorname{\scriptsize{red}}}$ . Moreover, for a pointed unital $\infty$ -operad $\mathcal{Q}$ the value of the right adjoint is given by the pullback

in the $\infty$ -category $\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{un}}}$ . Furthermore, the top map can be taken to be the counit of the adjunction at $\mathcal{Q}$ .

[05XA]

Proof. We need to verify the hypothesis of 2.1.5. The underlying $\infty$ -category functor $L\colon\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{un}}}\to\mathbf{Cat}_{\infty}$ is a composition of two functors $\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{un}}}\to\mathbf{Op}_{\infty}\to\mathbf{Cat}_{\infty}$ . The first is a right adjoint by A.2.3.1.9 and the second is a right adjoint by A.2.1.4.10. Hence, the composition is a right adjoint as well and therefore preserves limits. Moreover, by 2.2.3, $L$ is also a left adjoint and its right adjoint is fully faithful. Hence, the functor $\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{un}}}\to\mathbf{Cat}_{\infty,*}$ also has a fully faithful right adjoint and we have $\left(\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{un}}}\right)^{\operatorname{\scriptsize{red}}}\simeq\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{red}}}$ . Finally, by 2.1.5, the inclusion $\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{red}}}\simeq\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{red}}}\hookrightarrow\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{un}}}$ admits a right adjoint with the stated description. ∎

[0M2Q]

Definition 2.2.10. A unital $\infty$ -operad $\mathcal{Q}$ and an object $X\in\underline{\mathcal{Q}}$ determine a pointed unital $\infty$ -operad $\mathcal{Q}_{X}\in\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{un}}}$ . We denote $\operatorname{End}_{\mathcal{Q}}^{\operatorname{\scriptsize{red}}}\left(X\right):=\left(\mathcal{Q}_{X}\right)^{\operatorname{\scriptsize{red}}}$ and call it the reduced endomorphism $\infty$ -operad of $X$ in $\mathcal{Q}$ .

We can describe the reduced $\infty$ -operad $\operatorname{End}_{\mathcal{Q}}^{\operatorname{\scriptsize{red}}}\left(X\right)$ informally as follows. For every $m\in\mathbb{N}$ , denote by $X^{\left(m\right)}$ the $m$ -tuple $\left(X,\dots,X\right)$ . The space of $m$ -ary operations is the “subspace” of $\text{Mul}_{\mathcal{Q}}\left(X^{\left(m\right)},X\right)$ of those maps that are reduced in the sense that plugging the unique constant in all arguments but one results in an identity morphism $X\to X$ . We end this subsection by making the above description precise in a special case of a symmetric monoidal $\infty$ -category. For this, we first need to analyze the way multi-mapping spaces interact with limits of $\infty$ -operads.

For every integer $m$ , there is a functor $hG^{(m)}\colon h\mathbf{Op}_{\infty,*}\to h\mathcal{S}$ , that takes each $\infty$ -operad $\mathcal{P}$ pointed by an object $X$ to the space $\operatorname{Mul}_{\mathcal{P}}\left(X^{(m)},X\right)$ and a map of pointed $\infty$ -operads $f\colon\mathcal{P}\to\mathcal{Q}$ to the homotopy class of the induced map on multi-mapping spaces $\operatorname{Mul}_{\mathcal{P}}(X^{(m)},X)\to\operatorname{Mul}_{\mathcal{Q}}(f(X)^{(m)},f(X))$ .

[05XB]

Lemma 2.2.11. For every integer $m$ , there is a limit-preserving functor $G^{(m)}\colon\mathbf{Op}_{\infty,*}\to\mathcal{S}$ , that lifts the functor $hG^{(m)}\colon h\mathbf{Op}_{\infty,*}\to h\mathcal{S}$ .

[05XC]

Proof. Recall the combinatorial simplicial model category $\mathbf{POp}_{\infty}$ of $\infty$ -preoperads, whose underlying $\infty$ -category is $\mathbf{Op}_{\infty}$ (see A.2.1.4). Let $\overline{\mathcal{Z}}_{0}\subseteq\overline{\mathcal{Z}}_{1}\subseteq\mathbf{Fin}_{*}$ be the following subcategories:

(1)

The category $\overline{\mathcal{Z}}_{0}$ is discrete and contains only the objects $\left\langle 1\right\rangle$ and $\left\langle m\right\rangle$ .
(2)

The category $\overline{\mathcal{Z}}_{1}$ contains $\overline{\mathcal{Z}}_{0}$ together with a unique non-identity morphism, which is the active map $\alpha\colon\left\langle m\right\rangle\to\left\langle 1\right\rangle$ .

We endow $\overline{\mathcal{Z}}_{0}$ and $\overline{\mathcal{Z}}_{1}$ with the induced (trivial) marking. Unwinding the definition, for any $\infty$ -operad $\mathcal{P}$ , the simplicial set $\operatorname{Map}_{\mathbf{POp}_{\infty}}\left(\overline{\mathcal{Z}}_{0},\mathcal{P}^{\natural}\right)$ is isomorphic to $\mathcal{P}_{\left\langle m\right\rangle}^{\simeq}\times\mathcal{P}_{\left\langle 1\right\rangle}^{\simeq}$ . Moreover, given

\underline{X}=\left(X_{1}\oplus\cdots\oplus X_{m},Y\right)\in\mathcal{P}_{\left\langle m\right\rangle}^{\simeq}\times\mathcal{P}_{\left\langle 1\right\rangle}^{\simeq},

the fiber of the fibration (hence also the homotopy fiber)

\varphi_{\mathcal{P}}\colon\operatorname{Map}_{\mathbf{POp}_{\infty}}\left(\overline{\mathcal{Z}}_{1},\mathcal{P}^{\natural}\right)\to\operatorname{Map}_{\mathbf{POp}_{\infty}}\left(\overline{\mathcal{Z}}_{0},\mathcal{P}^{\natural}\right)\simeq\mathcal{P}_{\left\langle m\right\rangle}^{\simeq}\times\mathcal{P}_{\left\langle 1\right\rangle}^{\simeq}

over $\underline{X}$ is homotopy equivalent to the multi-mapping space $\operatorname{Mul}_{\mathcal{P}}\left(\left\{X_{1},\dots,X_{m}\right\};Y\right)$ . Let $\mathcal{Z}_{0}$ and $\mathcal{Z}_{1}$ be $\infty$ -operads that are fibrant replacements of $\overline{\mathcal{Z}}_{0}$ and $\overline{\mathcal{Z}}_{1}$ , respectively. Moreover, let $f\colon\mathcal{Z}_{0}\to\mathcal{Z}_{1}$ be a map corresponding to the inclusion $\overline{\mathcal{Z}}_{0}\hookrightarrow\overline{\mathcal{Z}}_{1}$ . The functor $F\colon\left(\mathbf{Op}_{\infty}\right)_{\mathcal{Z}_{0}/}\to\mathcal{S}$ , co-represented by $f\colon\mathcal{Z}_{0}\to\mathcal{Z}_{1}$ , preserves limits. Furthermore, its value on $g\colon\mathcal{Z}_{0}\to\mathcal{P}$ fits by T.5.5.5.12 into a fiber sequence

which therefore identifies $F\left(\mathcal{P}\right)$ with $\operatorname{Mul}_{\mathcal{P}}\left(\left\{X_{1},\dots,X_{m}\right\};Y\right)$ for the objects $X_{1},\dots,X_{m},Y\in\mathcal{P}$ determined by $g$ .

Let $U\colon\mathbf{Op}_{\infty,*}\to\left(\mathbf{Op}_{\infty}\right)_{\mathcal{Z}_{0}/}$ be the functor induced from the map $\mathcal{Z}_{0}\to\mathbf{Triv}$ corresponding to the inclusion $\overline{\mathcal{Z}}_{0}\hookrightarrow\mathbf{Triv}$ . By T.1.2.13.8, the functor $U$ preserves limits. We define $G^{(m)}\colon\mathbf{Op}_{\infty,*}\to\mathcal{S}$ to be the composition of $F$ and $U$ , which is limit-preserving as a composition of limit preserving-functors. Unwinding the definitions, $G^{(m)}$ indeed lifts $hG^{(m)}$ . ∎

Let $\mathcal{C}$ be a symmetric monoidal $\infty$ -category that is unital as an $\infty$ -operad (ie the unit is an initial object). For every $X\in\mathcal{C}$ and $m\in\mathbb{N}$ we have a canonical map $\sigma\colon X^{\sqcup m}\to X^{\otimes m}$ defined as follows. For $k=1,\dots,m$ , on the $k$ -th summand of $X^{\sqcup m}$ the map is the tensor product of $m$ maps, where the $k$ -th one is $X\overset{\operatorname{Id}}{\longrightarrow}X$ and the rest are the unique map $1_{\mathcal{C}}\to X$ .

[05XD]

Lemma 2.2.12. Let $\mathcal{C}$ be a symmetric monoidal $\infty$ -category that is unital as an $\infty$ -operad and that admits finite coproducts. For every $X\in\mathcal{C}$ and every $m\in\mathbb{N}$ , there is a fiber sequence

\operatorname{End}_{\mathcal{C}}^{\operatorname{\scriptsize{red}}}\left(X\right)\left(m\right)\to\operatorname{Map}_{\mathcal{C}}\left(X^{\otimes m},X\right)\xrightarrow{\sigma^{*}}\operatorname{Map}_{\mathcal{C}}\left(X^{\sqcup m},X\right),

where the fiber is taken over the fold map $\nabla\colon X^{\sqcup m}\to X$ .

[05XE]

Proof. By 2.2.9 we have a pullback square of pointed unital $\infty$ -operads

which, by 2.2.11, induces a pullback square of multi-mapping spaces

The bottom map is the map $\Delta^{0}\to\operatorname{Map}_{\mathcal{C}}\left(X^{\sqcup m},X\right)$ that chooses the fold map since it is induced from the map $\mathbb{E}_{\infty}=\left(\Delta^{0}\right)_{\sqcup}\to\underline{\mathcal{C}}_{\sqcup}$ . The right vertical map is induced by pre-composition with the map $\sigma\colon X^{\sqcup m}\to X^{\otimes m}$ , since it is induced by the adjunction

\left(-\right)_{\sqcup}\colon\mathbf{Cat}_{\infty}\leftrightarrows\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{un}}}\colon\underline{\left(-\right)}.

∎

Tomer Schlank, Lior Yanovski

Original source: arXiv:1808.06006v3

Original source · 1808.06006v3

2.2 Pointed Unital and Reduced ∞\infty-operads[0M38]

2.2 Pointed Unital and Reduced $\infty$ -operads[0M38]