ScalingStacks

5.2 Topoi and the Reduced Endomorphism Operad[0M3L]

In this subsection we describe a simple application of 5.1.4. Let 𝒞\mathcal{C} be an ∞\infty-topos and let 𝒞∗\mathcal{C}_{*} be the ∞\infty-category of pointed objects in 𝒞\mathcal{C} with the Cartesian symmetric monoidal structure.

[0M34]

Definition 5.2.1. For a pair of integers m,k≥−2m,k\geq-2, we denote by 𝒞∗[k,m]⊆𝒞∗\mathcal{C}_{*}^{\left[k,m\right]}\subseteq\mathcal{C}_{*} the full subcategory spanned by objects which are simultaneously (k−1)(k-1)-connected (ie kk-connective) and mm-truncated.

[05ZF]

Theorem 5.2.2. Let 𝒞\mathcal{C} be an ∞\infty-topos and let k,d≥−2k,d\geq-2. For every X∈𝒞∗[k,2​k+d]X\in\mathcal{C}_{*}^{\left[k,2k+d\right]} the ∞\infty-operad End𝒞∗red⁡(X)\operatorname{End}_{\mathcal{C}_{*}}^{\operatorname{\scriptsize{red}}}\left(X\right) is an essentially (d+1)\left(d+1\right)-operad. In particular, for d=−1d=-1, the ∞\infty-operad End𝒞∗red⁡(X)\operatorname{End}_{\mathcal{C}_{*}}^{\operatorname{\scriptsize{red}}}\left(X\right) is either 𝔼0\mathbb{E}_{0} or 𝔼∞\mathbb{E}_{\infty} and for d=−2d=-2, it is 𝔼∞\mathbb{E}_{\infty}.

[05ZG]

Proof. By A.5.2.6.10 and A.5.2.6.12, we have a commutative diagram of ∞\infty-categories

𝒞∗≥k\textstyle{\mathcal{C}_{*}^{\geq k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Ωk\scriptstyle{\Omega^{k}}∼\scriptstyle{\sim}Alg¯𝔼kgrp​(𝒞∗)\textstyle{\underline{\operatorname{Alg}}_{\mathbb{E}_{k}}^{\operatorname{\scriptsize{grp}}}\left(\mathcal{C}_{*}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}U\scriptstyle{U}𝒞∗,\textstyle{\mathcal{C}_{*},}

in which UU is the forgetful functor. Since the kk-fold loop space functor restricts to a functor 𝒞∗[k,2​k+d]→τ≤k+d​𝒞∗\mathcal{C}_{*}^{\left[k,2k+d\right]}\to\tau_{\leq k+d}\mathcal{C}_{*}, we can restrict the above diagram to

𝒞∗[k,2​k+d]\textstyle{\mathcal{C}_{*}^{\left[k,2k+d\right]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Ωk\scriptstyle{\Omega^{k}}∼\scriptstyle{\sim}Alg¯𝔼kgrp​(τ≤k+d​𝒞∗)\textstyle{\underline{\operatorname{Alg}}_{\mathbb{E}_{k}}^{\operatorname{\scriptsize{grp}}}\left(\tau_{\leq k+d}\mathcal{C}_{*}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}U\scriptstyle{U}τ≤k+d​𝒞∗.\textstyle{\tau_{\leq k+d}\mathcal{C}_{*}.}

The ∞\infty-category Alg¯𝔼kgrp​(τ≤k+d​𝒞∗)\underline{\operatorname{Alg}}_{\mathbb{E}_{k}}^{\operatorname{\scriptsize{grp}}}\left(\tau_{\leq k+d}\mathcal{C}_{*}\right) is a full subcategory of Alg¯𝔼k​(τ≤k+d​𝒞∗)\underline{\operatorname{Alg}}_{\mathbb{E}_{k}}\left(\tau_{\leq k+d}\mathcal{C}_{*}\right), which is itself equivalent to Alg¯𝔼k​(τ≤k+d​𝒞)\underline{\operatorname{Alg}}_{\mathbb{E}_{k}}\left(\tau_{\leq k+d}\mathcal{C}\right). The ∞\infty-category τ≤k+d​𝒞\tau_{\leq k+d}\mathcal{C} is a (k+d+1)\left(k+d+1\right)-topos (with the Cartesian symmetric monoidal structure) and 𝔼k\mathbb{E}_{k} is (k−2)\left(k-2\right)-connected. Thus, 5.1.4 implies that for every XX in Alg¯𝔼kgrp​(τ≤k+d​𝒞∗)\underline{\operatorname{Alg}}_{\mathbb{E}_{k}}^{\operatorname{\scriptsize{grp}}}\left(\tau_{\leq k+d}\mathcal{C}_{*}\right), the reduced endomorphism operad of XX is an essentially (d+1)\left(d+1\right)-operad.

Let d=−1d=-1. We recall from 3.2.3 that if 𝒫≄𝔼0\mathcal{P}\not\simeq\mathbb{E}_{0}, then it is (−1)\left(-1\right)-connected. Therefore, if 𝒫\mathcal{P} is an essentially 00-operad, then 𝒫≃𝔼∞\mathcal{P}\simeq\mathbb{E}_{\infty}. Hence, 𝒫\mathcal{P} is either 𝔼0\mathbb{E}_{0} or 𝔼∞\mathbb{E}_{\infty}.

Let d=−2d=-2. We get that 𝒫\mathcal{P} is an essentially (−1)\left(-1\right)-operad and hence equivalent to 𝔼∞\mathbb{E}_{\infty}. ∎

For every reduced ∞\infty-operad 𝒫\mathcal{P}, the structure of a 𝒫\mathcal{P}-algebra on an object X∈𝒞∗X\in\mathcal{C}_{*} is equivalent to the data of a map 𝒫→End𝒞∗red⁡(X)\mathcal{P}\to\operatorname{End}_{\mathcal{C}_{*}}^{\operatorname{\scriptsize{red}}}\left(X\right). Thus, if X∈𝒞∗[k,2​k−2]X\in\mathcal{C}_{*}^{\left[k,2k-2\right]}, then XX has a unique 𝒫\mathcal{P}-algebra structure for every reduced ∞\infty-operad 𝒫\mathcal{P}. Combining this with the fact that for a pointed connected object in an ∞\infty-topos, a structure of an 𝔼∞\mathbb{E}_{\infty}-algebra is equivalent to an ∞\infty-delooping, we get the following classical fact:

[05ZH]

Corollary 5.2.3. Let 𝒞\mathcal{C} be an ∞\infty-topos and let k≥1k\geq 1 be an integer. Every X∈𝒞∗[k,2​k−2]X\in\mathcal{C}_{*}^{\left[k,2k-2\right]} admits a unique ∞\infty-delooping.

In fact, we can get slightly more from 5.2.2. For example,

[05ZI]

Corollary 5.2.4. Let 𝒞\mathcal{C} be an ∞\infty-topos, let k≥1k\geq 1 be an integer, and let X∈𝒞∗[k,2​k−1]X\in\mathcal{C}_{*}^{\left[k,2k-1\right]}. If XX admits an HH-structure, then it admits a unique ∞\infty-delooping.

[05ZJ]

Proof. By 5.2.2, the ∞\infty-operad End𝒞∗red⁡(X)\operatorname{End}_{\mathcal{C}_{*}}^{\operatorname{\scriptsize{red}}}\left(X\right) is either 𝔼0\mathbb{E}_{0} or 𝔼∞\mathbb{E}_{\infty}. On the other hand, the existence of an HH-structure is equivalent to End𝒞∗red⁡(X)​(2)≠∅\operatorname{End}_{\mathcal{C}_{*}}^{\operatorname{\scriptsize{red}}}\left(X\right)\left(2\right)\neq\varnothing. Thus, XX admits an HH-structure if and only if End𝒞∗red⁡(X)≃𝔼∞\operatorname{End}_{\mathcal{C}_{*}}^{\operatorname{\scriptsize{red}}}\left(X\right)\simeq\mathbb{E}_{\infty} if and only if XX admits a unique ∞\infty-delooping. ∎

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

Tomer Schlank, Lior Yanovski

Original source: arXiv:1808.06006v3

Original source · 1808.06006v3