用户: Scavenger/Infinity Category and Higher Algebra/Sheaves

1Sheaves on $\infty$-Topoi

Definition 1.1 ([3] Definition 1.3.1.1). Let $\mathcal{T}$ be an $\infty$-category. A Grothendieck topology (or simply a topology) on $\mathcal{T}$ is a Grothendieck topology on the homotopy category $\text{h}\mathcal{T}$ (see [2] Subsection 6.2.2 for related discussions).

Let $\mathcal{T}$ be an $\infty$-category equipped with a topology. Let $\mathcal{C}$ be an $\infty$-category. We say that a functor $F:{\mathcal{T}}^{\mathrm{op}}\to \mathcal{C}$ is a $\mathcal{C}$-valued sheaf on $\mathcal{T}$ if the following condition is satisfied: For every object $U\in \mathcal{T}$ and every covering sieve ${\mathcal{T}}_0\subseteq {\mathcal{T}}_{/U}$ on $U$, the composite map ${\mathcal{T}}_0^{▹}\to ({\mathcal{T}}_{/U}{)}^{▹}\to \mathcal{T}\stackrel{F^{\mathrm{op}}}{\to }{\mathcal{C}}^{\mathrm{op}}$ is a colimit diagram in ${\mathcal{C}}^{\mathrm{op}}$. We let ${\mathrm{Sh}}_{\mathcal{C}}(\mathcal{T})$ denote the full subcategory of $\mathrm{Fun}({\mathcal{T}}^{\mathrm{op}},\mathcal{C})$ spanned by the $\mathcal{C}$-valued sheaves on $\mathcal{T}$.

Example 1.2 ([3] Example 1.3.1.3). Let $\mathcal{T}$ be a small $\infty$-category equipped with a topology, let $\mathcal{S}$ be the $\infty$-category of spaces. We will denote the $\infty$-category ${\mathrm{Sh}}_{\mathcal{S}}(\mathcal{T})$ simply by $\mathrm{Sh}(\mathcal{T})$, and refer to it as the $\infty$-category of sheaves on $\mathcal{T}$. One can check that the definition of sheaves here coincides with the one of [2] Definition 6.2.2.6. In particular $\mathrm{Sh}(\mathcal{T})$ is an $\infty$-topos.

Example 1.3. Let $\mathcal{C}$ and $\mathcal{D}$ be small $\infty$-categories equipped with Grothendieck topologies. A functor $F:\mathcal{C}\to \mathcal{D}$ is called cocontinuous if for every object $C\in \mathcal{C}$ and every covering sieve ${\mathcal{D}}_0\subseteq {\mathcal{D}}_{/F(C)}$ on $F(C)$, ${\mathcal{D}}_0{\times }_{{\mathcal{D}}_{/F(C)}}{\mathcal{C}}_{/C}\subseteq {\mathcal{C}}_{/C}$ is a covering sieve on $C$.

Let $F:\mathcal{C}\to \mathcal{D}$ be a cocontinuous functor. Let $L_{\mathcal{C}}$ be a left adjoints to the inclusion $\mathrm{Sh}(\mathcal{C})\to P(\mathcal{C})$. Let $f:P(\mathcal{C})\to P(\mathcal{D})$ be a right adjoint to the functor $P(\mathcal{D})\to P(\mathcal{C})$ given by composing with $F$. One can check that the functor $f$ takes sheaves on $\mathcal{C}$ to sheaves on $\mathcal{D}$, and therefore $g_{\ast }:=f{|}_{\mathrm{Sh}(\mathcal{C})}$ is a right adjoint to the functor $g^{\ast }:=(\mathrm{Sh}(\mathcal{D})\to P(\mathcal{D})\to P(\mathcal{C})\stackrel{L_{\mathcal{C}}}{\to }\mathrm{Sh}(\mathcal{C}))$, which preserves finite limits. Therefore the adjoint pair $(g^{\ast },g_{\ast })$ is a geometric morphism from the $\infty$-topos $\mathrm{Sh}(\mathcal{C})$ to the $\infty$-topos $\mathrm{Sh}(\mathcal{D})$.

Definition 1.4 ([3] Definition 1.3.1.4). Let $\mathcal{X}$ be an $\infty$-topos, let $\mathcal{C}$ be an $\infty$-category. A $\mathcal{C}$-valued sheaf on $\mathcal{X}$ is a functor ${\mathcal{X}}^{\mathrm{op}}\to \mathcal{C}$ that preserves small limits. Let ${\mathrm{Sh}}_{\mathcal{C}}(\mathcal{X})$ denote the full subcategory of $\mathrm{Fun}({\mathcal{X}}^{\mathrm{op}},\mathcal{C})$ spanned by the $\mathcal{C}$-valued sheaves on $\mathcal{X}$.

Proposition 1.5 ([3] Remark 1.3.1.6). Let $\mathcal{C}$ be a presentable $\infty$-category and $\mathcal{X}$ an $\infty$-topos. Then ${\mathrm{Sh}}_{\mathcal{C}}(\mathcal{X})$ is presentable.

Proposition 1.6 ([3] Proposition 1.3.1.7). Let $\mathcal{T}$ be a small $\infty$-category equipped with a topology. Let $j:\mathcal{T}\to P(\mathcal{T})$ denote the Yoneda embedding, let $L:P(\mathcal{T})\to \mathrm{Sh}(\mathcal{T})$ be a left adjoint to the inclusion $\mathrm{Sh}(\mathcal{T})\to P(\mathcal{T})$. Let $\mathcal{C}$ be an arbitrary $\infty$-category which admits small limits. Then composition with $L\circ j$ induces an equivalence of $\infty$-categories ${\mathrm{Sh}}_{\mathcal{C}}(\mathrm{Sh}(\mathcal{T}))\to {\mathrm{Sh}}_{\mathcal{C}}(\mathcal{T})$.

2Sheaves of Spectra

Definition 2.1 ([3] Definition 1.3.2.1). Let $\mathcal{X}$ be an $\infty$-topos. A sheaf of spectra on $\mathcal{X}$ is a sheaf on $\mathcal{X}$ with values in the $\infty$-category $\mathrm{Sp}$ of spectra.

2.2 ([3] Remark 1.3.2.2). Let $\mathcal{X}$ be an $\infty$-topos. Recall that $\mathrm{Sp}$ is by definition the full subcategory of $\mathrm{Fun}({\mathcal{S}}_{\ast }^{\mathrm{fin}},\mathcal{S})$ spanned by the reduced and excisive functors. Therefore ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})$ is naturally isomorphic (as a simplicial set) to the full subcategory of $\mathrm{Fun}({\mathcal{S}}_{\ast }^{\mathrm{fin}},{\mathrm{Sh}}_{\mathcal{S}}(\mathcal{X}))$ spanned by the reduced and excisive functors. Since the Yoneda embedding induces an equivalence $\mathcal{X}\to {\mathrm{Sh}}_{\mathcal{S}}(\mathcal{X})$ by [2] Proposition 5.5.2.2, we obtain a natural equivalence $\mathrm{Sp}(\mathcal{X})\to {\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})$. In particular ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})$ is stable and presentable. Moreover we have a natural forgerful functor ${\Omega }^{\infty }:{\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})\to \mathcal{X}$ obtained by composition with the functor ${\Omega }^{\infty }:\mathrm{Sp}\to \mathcal{S}$ (together with the equivalence ${\mathrm{Sh}}_{\mathcal{S}}(\mathcal{X})\stackrel{\sim }{\to }\mathcal{X}$).

Notation 2.3 ([3] Notation 1.3.2.3). Let $\mathcal{X}$ be an $\infty$-topos and let ${\mathcal{X}}^{♡}:={\tau }_{\le 0}\mathcal{X}$ denote its underlying topos. We let ${\pi }_0$ denote the composite functor ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})\stackrel{{\Omega }^{\infty }}{\to }\mathcal{X}\stackrel{{\tau }_{\le 0}}{\to }{\mathcal{X}}^{♡}$. For $n\in \mathbb{Z}$ we let ${\pi }_n$ denote the composite functor ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})\stackrel{{\Omega }^n}{\to }{\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})\stackrel{{\pi }_0}{\to }{\mathcal{X}}^{♡}$. Notice that ${\pi }_n$ is equivalent to the composite functor ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})\stackrel{{\Omega }^{n-2}}{\to }{\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})\to {\mathrm{Sh}}_{{\mathcal{S}}_{\ast }}(\mathcal{X})\simeq {\mathcal{X}}_{\ast }\stackrel{{\pi }_2}{\to }\mathrm{Ab}({\mathcal{X}}^{♡})\to {\mathcal{X}}^{♡}$, where $\mathrm{Ab}({\mathcal{X}}^{♡})$ is the category of abelian group objects of ${\mathcal{X}}^{♡}$, in particular ${\pi }_n$ naturally factors through the functor $\mathrm{Ab}({\mathcal{X}}^{♡})\to {\mathcal{X}}^{♡}$.

Definition 2.4 ([3] Definition 1.3.2.5). Let $\mathcal{X}$ be an $\infty$-topos. For each $n\in \mathbb{Z}$ the functor ${\Omega }^{\infty -n}:\mathrm{Sp}\to \mathcal{S}$ induces a functor ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})\to {\mathrm{Sh}}_{\mathcal{S}}(\mathcal{X})\simeq \mathcal{X}$, which we also denote by ${\Omega }^{\infty -n}$. We say that an object $\mathcal{F}\in {\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})$ is $n$-truncated if ${\Omega }^{\infty +n}\mathcal{F}$ is a discrete object of $\mathcal{X}$. We will say that a sheaf of spectra $\mathcal{F}\in {\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})$ is $n$-connective if the homotopy groups ${\pi }_m\mathcal{F}$ vanish for $m<n$. We will say that $\mathcal{F}$ is connective if it is $0$-connective. We let ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X}{)}_{\ge n},{\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X}{)}_{\le n}\subseteq {\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})$ denote the full subcategories spanned by the $n$-connective, $n$-truncated objects, respectively. We denote ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X}{)}_{\ge 0}$ by ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X}{)}^{\mathrm{cn}}$.

Proposition 2.5 ([3] Remark 1.3.2.6). Let $\mathcal{X}$ be an $\infty$-topos, let $\mathcal{F}$ be a sheaf of spectra on $\mathcal{X}$. For $n\in \mathbb{Z}$, $\mathcal{F}$ is $n$-truncated if and only if for each object $U\in \mathcal{X}$, the spectrum $\mathcal{F}(U)$ is $n$-truncated.

Proposition 2.6 ([3] Proposition 1.3.2.7). Let $\mathcal{X}$ be an $\infty$-topos.

(i) The full subcategories $({\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X}{)}_{\ge 0},{\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X}{)}_{\le 0})$ determine a $t$-structure on ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})$.

(ii) The full subcategory ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X}{)}_{\le 0}\subseteq {\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})$ is stable under filtered colimits.

(iii) The functor ${\pi }_0$ determines an equivalence from the heart of ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})$ to $\mathrm{Ab}({\mathcal{X}}^{♡})$. Moreover, for each $n\in \mathbb{Z}$, the functor ${\pi }_n:{\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})\to \mathrm{Ab}({\mathcal{X}}^{♡})$ as in Notation 2.3 is equivalent to the composite functor ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})\stackrel{{\pi }_n}{\to }{\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X}{)}^{♡}\stackrel{{\pi }_0}{\to }\mathrm{Ab}({\mathcal{X}}^{♡})$, where the first functor is the homotopy group functor induced by the $t$-structure on ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})$.

Remark 2.7. This $t$-structure is not necessarily nondegenerate.

2.8 ([3] Remark 1.3.2.8). Let $g^{\ast }:\mathcal{X}\to \mathcal{Y}$ be a geometric morphism of $\infty$-topoi (which is by definition a functor which preserves small colimits and finite limits). Then $g^{\ast }$ is left exact, and therefore induces a functor ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})\simeq \mathrm{Sp}(\mathcal{X})\to \mathrm{Sp}(\mathcal{Y})\simeq {\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{Y})$. We denote this functor also by $g^{\ast }$. It is a left adjoint to the functor $g_{\ast }:{\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{Y})\to {\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})$ given by composing with $g^{\ast }:\mathcal{X}\to \mathcal{Y}$.

The functor $g^{\ast }:{\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})\to {\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{Y})$ is $t$-exact.

Now let $\mathcal{C}$ and $\mathcal{D}$ be small $\infty$-categories equipped with Grothendieck topologies. Let $F:\mathcal{C}\to \mathcal{D}$ be a cocontinuous functor. Choose $\mathcal{X}=\mathrm{Sh}(\mathcal{D})$, $\mathcal{Y}=\mathrm{Sh}(\mathcal{C})$, and $g^{\ast }$ is as in Example 1.3. Under the identifications ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})\simeq {\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{D})$ and ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{Y})\simeq {\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{C})$, the functor $g^{\ast }:{\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})\to {\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{Y})$ corresponds to the composite functor ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{D})\to \mathrm{Fun}({\mathcal{D}}^{\mathrm{op}},\mathrm{Sp})\to \mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathrm{Sp})\to {\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{C})$, where the first map is the inclusion, the second map is given by composing with $F$, and the third map is given by sheafification (that is, left adjoint to the inclusion). The functor $g_{\ast }:{\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{Y})\to {\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})$ corresponds to the composite functor ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{C})\to \mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathrm{Sp})\to \mathrm{Fun}({\mathcal{D}}^{\mathrm{op}},\mathrm{Sp})$, where the first functor is the inclusion, and the second functor is right adjoint to the functor $\mathrm{Fun}({\mathcal{D}}^{\mathrm{op}},\mathrm{Sp})\to \mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathrm{Sp})$ (and in particular brings sheaves of spectra on $\mathcal{C}$ to sheaves of spectra on $\mathcal{D}$).

3Symmetric Monoidal Structure

Proposition 3.1 ([3] Lemma 1.3.4.3). Let $\mathcal{X}$ be an $\infty$-topos and $\mathcal{C}$ a presentable $\infty$-category. Then the inclusion ${\mathrm{Sh}}_{\mathcal{C}}(\mathcal{X})\to \mathrm{Fun}({\mathcal{X}}^{\mathrm{op}},\mathcal{C})$ admits a left adjoint.

3.2 ([3] Proposition 1.3.4.6). Let $K$ be a simplicial set, then the $\infty$-category $\mathrm{Fun}(K,\mathrm{Sp})$ admits a natural symmetric monoidal structure, by [1] Remark 2.1.3.4, which we will refer to as the pointwise smash product symmetric monoidal structure.

Let $\mathcal{X}$ be an $\infty$-topos and let $L:\mathrm{Fun}({\mathcal{X}}^{\mathrm{op}},\mathrm{Sp})\to {\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})$ be a left adjoint to the inclusion. Then this localization functor is compatible with the pointwise smash product symmetric monoidal structure in the sense of [1] Definition 2.2.1.6, and in particular [1] Proposition 2.2.1.9 gives ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})$ a natural symmetric monoidal structure with respect to which the functor $L$ is symmetric monoidal.

Definition 3.3 ([3] Remark 1.3.5.1). For any $\infty$-topos $\mathcal{X}$ we have a natural isomorphism of simplicial sets ${\mathrm{Sh}}_{\mathrm{CAlg}}(\mathcal{X})\simeq \mathrm{CAlg}({\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X}))$, where $\mathrm{CAlg}$ is the $\infty$-category of $E_{\infty }$-rings. We will refer to elements in ${\mathrm{Sh}}_{\mathrm{CAlg}}(\mathcal{X})$ as sheaves of $E_{\infty }$-rings on $\mathcal{X}$.

Example 3.4. Let $\mathcal{C}$ be a small $\infty$-category. Under the identification $\mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathrm{Sp})\simeq {\mathrm{Sh}}_{\mathrm{Sp}}(P(\mathcal{C}))$ (which is obtained by equipping $\mathcal{C}$ with the indiscrete Grothendieck topology), one can show that the symmetric monoidal structure on ${\mathrm{Sh}}_{\mathrm{Sp}}(P(\mathcal{C}))$ corresponds to the pointwise smash product symmetric monoidal structure on $\mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathrm{Sp})$.

Now if we equip $\mathcal{C}$ with a Grothendieck topology (not necessarily indiscrete), then the inclusion ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{C})\to \mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathrm{Sp})$ admits a left adjoint, which is compatible with the pointwise smash product symmetric monoidal structure in the sense of [1] Definition 2.2.1.6, and in particular equips ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{C})$ with a symmetric monoidal structure. One can show that this symmetric monoidal structure corresponds to the symmetric monoidal sturcture on ${\mathrm{Sh}}_{\mathrm{Sp}}(P(\mathcal{C}))$ constructed in 3.2 under the identification ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{C})\simeq {\mathrm{Sh}}_{\mathrm{Sp}}(P(\mathcal{C}))$.

Example 3.5. Let $g^{\ast }:\mathcal{X}\to \mathcal{Y}$ be a geometric morphism of $\infty$-topoi. Then we have a pair of adjunctions $(g^{\ast },g_{\ast })$ between ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{X})$ and ${\mathrm{Sh}}_{\mathrm{Sp}}(\mathcal{Y})$. One can show that $g^{\ast }$ is symmetric monoidal ($g^{\ast }$ can be extended to a morphism of underlying simplicial sets over ${\mathrm{Comm}}^{\otimes }$ of the $\infty$-operads, which is a left adjoint functor of isofibrations over ${\mathrm{Comm}}^{\otimes }$, thus is symmetric monoidal by [1] Proposition 7.3.2.6), and consequently $g_{\ast }$ is lax symmetric monoidal.

4Miscellany

Cech Nerves

Definition 4.1. Let ${\Delta }_{+}$ denote the category ${\Delta }^{◃}$, which we thought of as the category $\Delta$ added the empty linearly ordered set $[-1]$ as an additional object. Let $\mathcal{C}$ be an $\infty$-category, a simplicial object of $\mathcal{C}$ is a functor ${\Delta }^{\mathrm{op}}\to \mathcal{C}$, an augmented simplicial object of $\mathcal{C}$ is a functor ${\Delta }_{+}^{\mathrm{op}}\to \mathcal{C}$.

Definition 4.2. Let $\mathcal{C}$ be an $\infty$-category. An augmented simplicial object $U:{\Delta }_{+}^{\mathrm{op}}\to \mathcal{C}$ is called a Cech nerve if $U$ is a right Kan extension of $U{|}_{({\Delta }_{+}^{\le 0}{)}^{\mathrm{op}}}$, where ${\Delta }_{+}^{\le 0}$ is the full subcategory of ${\Delta }_{+}$ spanned by $[-1]$ and $[0]$.

See [2] Proposition 6.1.2.11 for equivalent characterizations.

4.3. Let $\mathcal{C}$ be an $\infty$-category with finite limits. In this case a Cech nerve $U:{\Delta }_{+}^{\mathrm{op}}\to \mathcal{C}$ is determined by the morphism $U([0])\to U([-1])$, and every morphism $X\to Y$ in $\mathcal{C}$ can be (uniquely) extended to a Cech nerve.

Definition 4.4. Let $\mathcal{X}$ be an $\infty$-topos. A morphism $f:U\to X$ in $\mathcal{X}$ is called an effective epimorphism if the cech nerve of $f$ is a colimit diagram.

See [2] Corollary 6.2.3.5 for equivalent characterizations.

Example 4.5 (Weakly Final Objects and Cech-Alexander Resolution). Let $\mathcal{C}$ be a small $\infty$-category. Let $F:{\mathcal{C}}^{\mathrm{op}}\to \mathcal{S}$ be a presheaf on $\mathcal{C}$, let $G:{\mathcal{C}}^{\mathrm{op}}\to \ast \to \mathcal{S}$ be the final presheaf on $\mathcal{C}$. Then $F\to G$ is an effective epimorphism in $P(\mathcal{C})$ if and only if for every object $C\in \mathcal{C}$, the space $F(C)$ is nonempty, by [2] Corollary 6.2.3.5.

Let $X\in \mathcal{C}$ be an object. It is said to be weakly final if $h_X\to G$ is an effective epimorphism in $P(\mathcal{C})$, where $h_X$ is the presheaf represented by $X$. Then $X$ is weakly final if and only if for every object $Y\in \mathcal{C}$, ${\mathrm{Hom}}_{\text{h}\mathcal{C}}(Y,X)$ is nonempty.

Let $X\in \mathcal{C}$ be a weakly final object, let $U_{+}:{\Delta }_{+}^{\mathrm{op}}\to P(\mathcal{C})$ be the Cech nerve of $h_X\to G$, then $U_{+}$ is a colimit diagram in $P(\mathcal{C})$. Let $U:U_{+}{|}_{{\Delta }^{\mathrm{op}}}$ denote the underlying simplicial object of $U_{+}$. If $\mathcal{C}$ admits finite nonempty products, then $U$ factors through the Yoneda embedding $\mathcal{C}\to P(\mathcal{C})$.

Let $\mathcal{F}:P(\mathcal{C}{)}^{\mathrm{op}}\to \mathrm{CAlg}$ be a sheaf of $E_{\infty }$-rings (for example. Similar constructions apply with $\mathrm{CAlg}$ replaced with, for example, $\mathrm{Sp}$.) on $P(\mathcal{C})$, where $\mathrm{CAlg}$ is the $\infty$-category of $E_{\infty }$-rings. Then the composite functor ${\Delta }_{+}\stackrel{U^{\mathrm{op}}}{\to }P(\mathcal{C}{)}^{\mathrm{op}}\stackrel{\mathcal{F}}{\to }\mathrm{CAlg}$ is a limit diagram in $\mathrm{CAlg}$. If $\mathcal{C}$ admits nonempty finite products, then this functor exhibits $\mathcal{F}(G)$ as a limit of objects in the essential image of the composite functor ${\mathcal{C}}^{\mathrm{op}}\to P(\mathcal{C}{)}^{\mathrm{op}}\stackrel{\mathcal{F}}{\to }\mathrm{CAlg}$, where the first map is the Yoneda embedding.

References

[1]	Jacob Lurie, “Higher Algebra”, Avalible for download at https://www.math.ias.edu/ lurie/papers/HA.pdf
[2]	Jacob Lurie, “Higher Topos Theory”, Avalible for download at https://www.math.ias.edu/ lurie/papers/HTT.pdf
[3]	Jacob Lurie, “Spectral Algebraic Geometry”, February 3, 2018 version, Avalible for download at https://www.math.ias.edu/ lurie/papers/SAG-rootfile.pdf