2.1 Reminders on formal moduli problems

Reminders on formal $\mathbb{E}_{n}$ -moduli problems

Let $k$ be a field of arbitrary characteristic. Let $\text{Mod}_{k}$ denote the $\infty$ -category of $k$ -module spectra. Let $n\geqslant 0$ be an integer and let $\text{Alg}_{k}^{(n)}:=\text{Alg}_{\mathbb{E}_{n}}(\text{Mod}_{k})$ denote the $\infty$ -category of $\mathbb{E}_{n}$ -algebras over $k$ in the sense of [Reference LurieLur17, Definition 7.1.3.5]. Let $\text{Alg}_{k}^{(n),\text{aug}}=(\text{Alg}_{k}^{(n)})_{/k}$ denote the $\infty$ -category of augmented $\mathbb{E}_{n}$ -algebras. We call an $\mathbb{E}_{n}$ -algebra $A$ over $k$ artinian if the following conditions are satisfied.

• -- For $i<0$ and for $i\gg 0$ , $\unicode[STIX]{x1D70B}_{i}A=0$ .

• -- For every $i$ , the $k$ -vector space $\unicode[STIX]{x1D70B}_{i}A$ is finite-dimensional.

• -- Let $\mathfrak{r}$ denote the radical of the algebra $\unicode[STIX]{x1D70B}_{0}A$ . Then the natural map $(\unicode[STIX]{x1D70B}_{0}A)/\mathfrak{r}\rightarrow k$ is an isomorphism.

We denote by $\text{Alg}_{k}^{(n),\text{art}}\subseteq \text{Alg}_{k}^{(n)}$ the full subcategory spanned by artinian $\mathbb{E}_{n}$ -algebras. For every integer $m\geqslant 0$ , we denote by $k\,\oplus \,k[m]$ the shifted square zero extension viewed as $\mathbb{E}_{n}$ -algebras over $k$ , which are examples of artinian $\mathbb{E}_{n}$ -algebras.

A functor $X:\text{Alg}_{k}^{(n),\text{art}}\rightarrow {\mathcal{S}}$ is called a formal $\mathbb{E}_{n}$ -moduli problem if the space $X(k)$ is contractible and if for every pullback diagram

in $\text{Alg}_{k}^{(n),\text{art}}$ such that the maps $\unicode[STIX]{x1D70B}_{0}A_{0}\rightarrow \unicode[STIX]{x1D70B}_{0}A_{01}\leftarrow \unicode[STIX]{x1D70B}_{0}A_{1}$ are surjective, the induced square

is a pullback diagram in ${\mathcal{S}}$ . Let $\text{Fun}_{\ast }(\text{Alg}_{k}^{(n),\text{art}},{\mathcal{S}})\subseteq \text{Fun}(\text{Alg}_{k}^{(n),\text{art}},{\mathcal{S}})$ denote the full subcategory spanned by those functors $X:\text{Alg}_{k}^{(n),\text{art}}\rightarrow {\mathcal{S}}$ such that $X(k)$ is a contractible space. We denote by $\text{Moduli}_{k}^{\mathbb{E}_{n}}\subseteq \text{Fun}_{\ast }(\text{Alg}_{k}^{(n),\text{art}},{\mathcal{S}})$ the full subcategory spanned by formal $\mathbb{E}_{n}$ -moduli problems. The class of pullback diagrams of the form

for $m\geqslant 0$ are used to defined the tangent complex of a formal $\mathbb{E}_{n}$ -moduli problem. Indeed if $X$ is a formal $\mathbb{E}_{n}$ -moduli problem over $k$ , the induced map $X(k\,\oplus \,k[m])\rightarrow \unicode[STIX]{x1D6FA}X(k\,\oplus \,k[m+1])$ is a homotopy equivalence, and defines a spectrum $T_{X}$ called the tangent complex of $X$ (see [Reference LurieLur11b, Definition 1.2.5]). By [Reference LurieLur11b, Proposition 1.2.10], a map of formal $\mathbb{E}_{n}$ -moduli problems $X\rightarrow Y$ is an equivalence if and only if the induced map $T_{X}\rightarrow T_{Y}$ is an equivalence of spectra.

By [Reference LurieLur11b, Theorem 4.0.8] there exists a functor

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}:\text{Alg}_{k}^{(n),\text{aug}}\longrightarrow \text{Moduli}_{k}^{\mathbb{E}_{n}}\end{eqnarray}$$

which is an equivalence of $\infty$ -categories. For an augmented $\mathbb{E}_{n}$ -algebra $B$ and for an artinian $\mathbb{E}_{n}$ -algebra $A$ over $k$ , there is an equivalence $\unicode[STIX]{x1D6F9}_{B}(A)\simeq \text{Map}_{\text{Alg}_{k}^{(n),\text{aug}}}(\mathfrak{D}^{(n)}(A),B)$ where $\mathfrak{D}^{(n)}:(\text{Alg}_{k}^{(n),\text{aug}})^{\text{op}}\rightarrow \text{Alg}_{k}^{(n),\text{aug}}$ is the Koszul duality functor defined in [Reference LurieLur11b, Proposition 4.4.1]. Moreover there exists a commutative diagram of $\infty$ -categories

where $\mathfrak{m}$ is given by taking the augmentation ideal.

Reminders on axiomatic deformation theory

We recall some definitions and constructions from [Reference LurieLur11b, §1].

For $\mathbf{A}$ a presentable $\infty$ -category, we denote by $\text{Sp}(\mathbf{A})$ the $\infty$ -category of spectra objects of $\mathbf{A}$ in the sense of [Reference LurieLur17, Definition 1.4.2.8]. By [Reference LurieLur17, Remark 1.4.2.25] there exists an equivalence of $\infty$ -categories $\text{Sp}(\mathbf{A})\simeq \lim (\cdots \rightarrow \mathbf{A}_{\ast }\stackrel{\unicode[STIX]{x1D6FA}}{\rightarrow }\mathbf{A}_{\ast }\stackrel{\unicode[STIX]{x1D6FA}}{\rightarrow }\mathbf{A}_{\ast })$ . We denote by $\unicode[STIX]{x1D6FA}^{\infty }:\text{Sp}(\mathbf{A})\rightarrow \mathbf{A}$ the evaluation on the $0$ -sphere $S^{0}$ and for every integer $n\in \mathbf{Z}$ , we denote by $\unicode[STIX]{x1D6FA}^{\infty -n}:\text{Sp}(\mathbf{A})\rightarrow \mathbf{A}$ the composition of $\unicode[STIX]{x1D6FA}^{\infty }$ with the $n$ th shift functor on $\text{Sp}(\mathbf{A})$ .

A deformation context in the sense of [Reference LurieLur11b, Definition 1.1.3] is a pair $(\mathbf{A},(E_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in T})$ where $\mathbf{A}$ is a presentable $\infty$ -category and $(E_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in T}$ is a sequence of objects of the $\infty$ -category $\text{Sp}(\mathbf{A})$ of spectra objects in $\mathbf{A}$ . Given a deformation context $(\mathbf{A},(E_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in T})$ , a map $f:A\rightarrow A^{\prime }$ in $\mathbf{A}$ is called elementary if there exists an index $\unicode[STIX]{x1D6FC}\in T$ , an integer $n\geqslant 0$ and a pullback diagram

in $\mathbf{A}$ , where $\ast _{\mathbf{A}}$ is a final object of $\mathbf{A}$ . A map $f:A\rightarrow A^{\prime }$ in $\mathbf{A}$ is called small if it can be written as a finite composition of elementary maps $A_{0}=A\rightarrow A_{1}\rightarrow \cdots \rightarrow A_{m}=A^{\prime }$ in $\mathbf{A}$ . An object $A\in \mathbf{A}$ is called artinian if the map $A\rightarrow \ast _{\mathbf{A}}$ is small. We denote by $\mathbf{A}^{\text{art}}$ the full subcategory of $\mathbf{A}$ spanned by artinian objects.

Given a deformation context $(\mathbf{A},(E_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in T})$ , recall that a formal moduli problem in the sense of [Reference LurieLur11b, Definition 1.1.14] is a functor $X:\mathbf{A}^{\text{art}}\rightarrow {\mathcal{S}}$ which satisfies the following two conditions.

1. (i) The space $X(\ast _{\mathbf{A}})$ is contractible.

2. (ii) For every pullback diagram

   
   in $\mathbf{A}^{\text{art}}$ such that the map $A_{0}\rightarrow A_{01}$ is small, the induced square
   
   is a pullback diagram in ${\mathcal{S}}$ .

By [Reference LurieLur11b, Proposition 1.1.15] we are allowed to test condition (2) on the smaller class of pullback diagrams of the form

in $\mathbf{A}^{\text{art}}$ where $\unicode[STIX]{x1D6FC}\in T$ and $n\geqslant 0$ .

We denote by $\text{Fun}_{\ast }(\mathbf{A}^{\text{art}},{\mathcal{S}})\subseteq \text{Fun}(\mathbf{A}^{\text{art}},{\mathcal{S}})$ the subcategory spanned functors satisfying condition (1) above, or in other words sending $\ast _{\mathbf{A}}$ to a contractible space. We denote by $\text{Moduli}_{\mathbf{A}}\subseteq \text{Fun}_{\ast }(\mathbf{A}^{\text{art}},{\mathcal{S}})$ the subcategory spanned by the formal moduli problems.

Example 2.1. Let $k$ be a field. Let $n\geqslant 1$ and $\mathbf{A}=\text{Alg}_{k}^{(n),\text{aug}}$ be the presentable $\infty$ -category of augmented $\mathbb{E}_{n}$ -algebras over $k$ . Consider the deformation context $(\mathbf{A},\{k\})$ where $k$ is seen as a module over itself. The deloopings of $k$ are given by the shifted square zero extensions $k\,\oplus \,k[m]$ for $m\geqslant 0$ . By [Reference LurieLur11b, Proposition 4.5.1], an augmented $\mathbb{E}_{n}$ -algebra $A$ is artinian with respect to the deformation context $(\mathbf{A},\{k\})$ if and only if it is artinian in the sense of § 2.1. By [Reference LurieLur11b, Proposition 4.5.1], a morphism $A\rightarrow A^{\prime }$ in $\text{Alg}_{k}^{(n),\text{aug}}$ is small with respect to the deformation context $(\mathbf{A},\{k\})$ if and only if the induced map $\unicode[STIX]{x1D70B}_{0}A\rightarrow \unicode[STIX]{x1D70B}_{0}A^{\prime }$ is surjective. Then [Reference LurieLur11b, Corollary 4.5.4] implies that $X$ is a formal moduli problem with respect to the deformation context $(\mathbf{A},\{k\})$ if and only if it is a formal $\mathbb{E}_{n}$ -moduli problem in the sense of § 2.1.

Notation 2.2. Let $(\mathbf{A},(E_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in T})$ be a deformation context. Consider the embedding $i_{\mathbf{A}}:\text{Moduli}_{\mathbf{A}}{\hookrightarrow}\text{Fun}_{\ast }(\mathbf{A}^{\text{art}},{\mathcal{S}})$ into the $\infty$ -category of functors $X:\mathbf{A}^{\text{art}}\rightarrow {\mathcal{S}}$ such that $X(\ast _{\mathbf{A}})$ is a contractible space. The $\infty$ -categories $\text{Moduli}_{\mathbf{A}}$ and $\text{Fun}_{\ast }(\mathbf{A}^{\text{art}},{\mathcal{S}})$ are presentable and by definition of formal moduli problems the functor $i_{\mathbf{A}}$ preserves all small limits. By the adjoint functor theorem [Reference LurieLur09, Corollary 5.5.2.9], this implies that $i_{\mathbf{A}}$ admits a left adjoint denoted by

For every functor $X:\mathbf{A}^{\text{art}}\rightarrow {\mathcal{S}}$ sending $\ast _{\mathbf{A}}$ to a contractible space we have the unit natural transformation $X\rightarrow i_{\mathbf{A}}(L_{\mathbf{A}}(X))$ of this adjunction. The formal moduli problem $L_{\mathbf{A}}(X)$ is therefore initial among formal moduli problems $Y$ having a natural transformation $X\rightarrow i_{\mathbf{A}}Y$ .

Let $(\mathbf{A},(E_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in T})$ be a deformation context. Recall from [Reference LurieLur11b, Definition 1.2.5] that every formal moduli problem $X:\mathbf{A}^{\text{art}}\rightarrow {\mathcal{S}}$ has a tangent spectrum $X(E_{\unicode[STIX]{x1D6FC}})\in \text{Sp}$ of $X$ at $\unicode[STIX]{x1D6FC}$ . Moreover a map $X\rightarrow Y$ of formal moduli problems is an equivalence if and only if for every $\unicode[STIX]{x1D6FC}$ the map $X(E_{\unicode[STIX]{x1D6FC}})\rightarrow Y(E_{\unicode[STIX]{x1D6FC}})$ is an equivalence of spectra.

In [Reference LurieLur11b, Definition 1.3.1 and 1.3.9], Lurie defines the notion of weak deformation theory and of deformation theory. In short a weak deformation theory for $(\mathbf{A},(E_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in T})$ consists of a functor $\mathfrak{D}:\mathbf{A}^{\text{op}}\rightarrow \mathbf{B}$ such that $\mathbf{B}$ is a presentable $\infty$ -category, $\mathfrak{D}$ has a left adjoint $\mathfrak{D}^{\prime }$ which behaves like an inverse equivalence to $\mathfrak{D}$ on a certain subcategory of $\mathbf{B}$ . In this situation, [Reference LurieLur11b, Corollary 1.3.6] asserts the existence of an $\infty$ -functor

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}:\mathbf{B}\rightarrow \text{Moduli}_{\mathbf{A}}\end{eqnarray}$$

which is given on objects by $\unicode[STIX]{x1D6F9}_{B}(A)=\text{Map}_{\mathbf{B}}(\mathfrak{D}(A),B)$ . By applying the tangent spectrum we obtain for every $\unicode[STIX]{x1D6FC}$ a functor $e_{\unicode[STIX]{x1D6FC}}:\mathbf{B}\rightarrow \text{Sp}$ given on objects by $e_{\unicode[STIX]{x1D6FC}}(B)=\unicode[STIX]{x1D6F9}_{B}(E_{\unicode[STIX]{x1D6FC}})$ . A deformation theory for $(\mathbf{A},(E_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in T})$ is a weak deformation theory such that each $e_{\unicode[STIX]{x1D6FC}}$ preserves small sifted colimits and such that a morphism $f$ in $\mathbf{B}$ is an equivalence if and only if each $e_{\unicode[STIX]{x1D6FC}}(f)$ is an equivalence of spectra. The main theorem [Reference LurieLur11b, Theorem 1.3.12] then asserts that under the existence of a deformation theory for $(\mathbf{A},(E_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in T})$ , the functor $\unicode[STIX]{x1D6F9}$ is an equivalence of $\infty$ -categories.

Formal spectrum and Koszul duality

We now set up some notations and intermediate results concerning the formal spectrum, pro-objects and formal Koszul duality.

Notation 2.3. Let $(\mathbf{A},(E_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in T})$ be a deformation context and let $A\in \mathbf{A}$ be any object. Consider the functor $\text{Spf}(A):\mathbf{A}^{\text{art}}\rightarrow {\mathcal{S}}$ which is co-represented by $A$ given by the formula $\text{Spf}(A)(R)=\text{Map}_{\mathbf{A}}(A,R)$ . Then $\text{Spf}(A)$ is a formal moduli problem which is called the formal spectrum of $A$ . Moreover the construction $A\mapsto \text{Spf}(A)$ determines a functor

$$\begin{eqnarray}\text{Spf}:\mathbf{A}^{\text{op}}\longrightarrow \text{Moduli}_{\mathbf{A}}\end{eqnarray}$$

which is fully faithful when restricted to the subcategory $(\mathbf{A}^{\text{art}})^{\text{op}}$ spanned by artinian objects.

Example 2.4. Let $n\geqslant 1$ and consider the deformation context $(\text{Alg}_{k}^{(n),\text{aug}},k)$ formed by augmented $\mathbb{E}_{n}$ -algebras over $k$ . Let $\unicode[STIX]{x1D716}:A\rightarrow k$ be an augmented $\mathbb{E}_{n}$ -algebra over $k$ . If we think about the $\mathbb{E}_{n}$ -algebra $A$ as determining a noncommutative scheme $\text{Spec}(A)$ , then the formal spectrum $\text{Spf}(A)$ can be thought of as the functor parametrizing deformations of the point of $\text{Spec}(A)$ determined by $\unicode[STIX]{x1D716}$ over artinian $\mathbb{E}_{n}$ -algebras.

The following lemma will be useful subsequently and follows from the fact that the Koszul duality functor $\mathfrak{D}^{(n)}$ is a deformation theory.

Lemma 2.5. Let $n\geqslant 1$ be an integer and denote by $\mathfrak{t}^{(n)}:\text{Moduli}_{k}^{\mathbb{E}_{n}}\rightarrow \text{Alg}_{k}^{(n),\text{aug}}$ the equivalence given by [Reference LurieLur11b, Theorem 4.0.8]. Let $B$ be an augmented $\mathbb{E}_{n}$ -algebra over $k$ . Then there exists a natural equivalence of augmented $\mathbb{E}_{n}$ -algebras $\mathfrak{t}_{\text{Spf}(B)}^{(n)}\simeq \mathfrak{D}^{(n)}(B)$ or equivalently an equivalence of formal moduli problems $\text{Spf}(B)\simeq \unicode[STIX]{x1D6F9}_{\mathfrak{D}^{(n)}(B)}$ .

Proof. Consider the natural transformation $\unicode[STIX]{x1D6FC}:\text{Spf}(B)\rightarrow \unicode[STIX]{x1D6F9}_{\mathfrak{D}^{(n)}(B)}$ induced by the natural map

$$\begin{eqnarray}\text{Map}_{\text{Alg}_{k}^{(n),\text{aug}}}(B,A)\stackrel{\unicode[STIX]{x1D6FC}_{A}}{\longrightarrow }\text{Map}_{\text{Alg}_{k}^{(n),\text{aug}}}(\mathfrak{D}^{(n)}(A),\mathfrak{D}^{(n)}(B))\end{eqnarray}$$

for each artinian $\mathbb{E}_{n}$ -algebra $A$ over $k$ . By adjunction the target space of $\unicode[STIX]{x1D6FC}_{A}$ is naturally equivalent to the space $\text{Map}_{\text{Alg}_{k}^{(n),\text{aug}}}(B,\mathfrak{D}^{(n)}\mathfrak{D}^{(n)}(A))$ and the map is induced by the unit map $A\rightarrow \mathfrak{D}^{(n)}\mathfrak{D}^{(n)}(A)$ . Let $m\geqslant 0$ be an integer and $A=k\,\oplus \,k[m]$ be the trivial square zero extension. We claim that the biduality map $A\rightarrow \mathfrak{D}^{(n)}\mathfrak{D}^{(n)}(A)$ is an equivalence. Indeed by [Reference LurieLur11b, Proposition 4.5.6] there exists an equivalence $\mathfrak{D}^{(n)}(\text{Free}_{\mathbb{E}_{n}}^{\text{aug}}(k[-m-n]))\simeq k\,\oplus \,k[m]$ of augmented $\mathbb{E}_{n}$ -algebras and by [Reference LurieLur11b, Proposition 4.1.13] the $\mathbb{E}_{n}$ -algebra $\text{Free}_{\mathbb{E}_{n}}^{\text{aug}}(k[-m-n])$ is $n$ -coconnective and locally finite whenever $m\geqslant 0$ , which by [Reference LurieLur11b, Theorem 4.4.5] implies that the biduality map is an equivalence for the $\mathbb{E}_{n}$ -algebra $\text{Free}_{\mathbb{E}_{n}}^{\text{aug}}(k[-m-n])$ , and thus also for $A=k\,\oplus \,k[m]$ . We deduce that the map $\unicode[STIX]{x1D6FC}_{A}$ is a homotopy equivalence and therefore that $\unicode[STIX]{x1D6FC}$ induces an equivalence on the tangent spectrum, and is therefore an equivalence, which proves the lemma.◻

2.2 Approximation of formal moduli and loop spaces

As explained above, the deformation functor of a linear $\infty$ -category over a field does not in general satisfy the derived Schlessinger conditions, but turns out to be rather close to satisfy them. Many examples of deformation functors of some algebro-geometric objects behave similarly. In [Reference LurieLur11b, §5.1], Lurie developed a formalism to study these functors under the name of $m$ -proximate formal moduli problems for $m\geqslant 0$ an integer. We will prove that the functor $L=L_{\mathbf{A}}:\text{Fun}_{\ast }(\mathbf{A}^{\text{art}},{\mathcal{S}})$ $\rightarrow \text{Moduli}_{\mathbf{A}}$ from Notation 2.2 admits a special description when restricted to the subcategory spanned by $m$ -proximate formal moduli problems in terms of $\mathbb{E}_{m}$ -maps between certain loop $\mathbb{E}_{m}$ -spaces (see Corollary 2.19). For this we will prove a result concerning the loop space functor on the $\infty$ -category of formal moduli problem (see Proposition 2.15).

Let $(\mathbf{A},(E_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in T})$ be a deformation context and $m\geqslant 0$ an integer. Recall from [Reference LurieLur11b, Definition 5.1.5] that a functor $X:\mathbf{A}^{\text{art}}\rightarrow {\mathcal{S}}$ is called an $m$ -proximate formal moduli problem if the space $X(\ast _{\mathbf{A}})$ is contractible and if for every pullback square

in $\mathbf{A}^{\text{art}}$ such that the map $A_{0}\rightarrow A_{01}$ is small, the induced map

$$\begin{eqnarray}X(A)\rightarrow X(A_{0})\times _{X(A_{01})}X(A_{1})\end{eqnarray}$$

is $(m-2)$ -truncated (that is, all homotopy fibers of this map are $(m-2)$ -truncated). We observe that $0$ -proximate formal moduli problems are exactly the formal moduli problems.

Notation 2.6. We denote by $\text{Prox}(m)_{\mathbf{A}}\subseteq \text{Fun}_{\ast }(\mathbf{A}^{\text{art}},{\mathcal{S}})$ the full subcategory spanned by $m$ -proximate formal moduli problems. There is a tower of $\infty$ -categories

$$\begin{eqnarray}\text{Moduli}_{\mathbf{A}}=\text{Prox}(0)_{\mathbf{A}}\subseteq \text{Prox}(1)_{\mathbf{A}}\subseteq \text{Prox}(2)_{\mathbf{A}}\subseteq \cdots \subseteq \text{Fun}_{\ast }(\mathbf{A}^{\text{art}},{\mathcal{S}}).\end{eqnarray}$$

For example, we will see below that the deformation functor of an object in a linear $\infty$ -category is in general 1-proximate and that the deformation functor of a fixed linear $\infty$ -category itself is in general 2-proximate.

Notation 2.8. Let $(\mathbf{A},(E_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in T})$ be a deformation context and suppose that $\mathbf{A}$ is a pointed $\infty$ -category in the sense that any final object is also initial (e.g., the $\infty$ -category $\text{Alg}_{k}^{(n),\text{aug}}$ of augmented $\mathbb{E}_{n}$ -algebras over $k$ ). Then the $\infty$ -category $\text{Fun}_{\ast }(\mathbf{A}^{\text{art}},{\mathcal{S}})$ is also pointed. We denote by

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}:\text{Fun}_{\ast }(\mathbf{A}^{\text{art}},{\mathcal{S}})\longrightarrow \text{Fun}_{\ast }(\mathbf{A}^{\text{art}},{\mathcal{S}})\end{eqnarray}$$

the loop functor in the sense of [Reference LurieLur17, Remark 1.1.2.9]. The embedding $\text{Moduli}_{\mathbf{A}}\,{\hookrightarrow}\,$ $\text{Fun}_{\ast }(\mathbf{A}^{\text{art}},{\mathcal{S}})$ commutes with small limits and the loop functor restricts to a functor $\unicode[STIX]{x1D6FA}:\text{Moduli}_{\mathbf{A}}\rightarrow \text{Moduli}_{\mathbf{A}}$ . For $m\geqslant 0$ an integer, we denote by $\unicode[STIX]{x1D6FA}^{m}=\unicode[STIX]{x1D6FA}\circ \cdots \circ \unicode[STIX]{x1D6FA}$ ( $m$ times) the iterated loop functor.

Notation 2.12. Let ${\mathcal{X}}$ be an $\infty$ -category which admits finite products. Consider the cartesian symmetric monoidal structure ${\mathcal{X}}^{\times }$ with monoidal product given by the cartesian product (see [Reference LurieLur17, §2.4.1]). Let $0\leqslant m\leqslant \infty$ and consider the $\infty$ -category $\text{Mon}_{\mathbb{E}_{m}}({\mathcal{X}})=\text{Alg}_{\mathbb{E}_{m}}({\mathcal{X}}^{\times })$ of $\mathbb{E}_{m}$ -monoid objects in ${\mathcal{X}}$ . Recall that an $\mathbb{E}_{1}$ -monoid object $M$ in ${\mathcal{X}}$ with multiplication map $m:M\times M\rightarrow M$ is called grouplike if the two maps

$$\begin{eqnarray}(m,p_{1}):M\times M\rightarrow M\times M,\quad (m,p_{2}):M\times M\rightarrow M\times M\end{eqnarray}$$

are equivalences in ${\mathcal{X}}$ , where $p_{i}$ is the $i$ th projection map. Recall that an $\mathbb{E}_{m}$ -monoid object $M$ in ${\mathcal{X}}$ is called grouplike if the underlying $\mathbb{E}_{1}$ -monoid (for any embedding of $\infty$ -operads $\mathbb{E}_{1}^{\otimes }{\hookrightarrow}\mathbb{E}_{m}^{\otimes }$ ) is grouplike. Let $\text{Mon}_{\mathbb{E}_{m}}^{\text{gp}}({\mathcal{X}})\subseteq \text{Mon}_{\mathbb{ E}_{m}}({\mathcal{X}})$ denote the full subcategory spanned by grouplike $\mathbb{E}_{m}$ -monoid objects (see [Reference LurieLur17, Definition 5.2.6.6]). If ${\mathcal{X}}$ is a presentable $\infty$ -category, then an $\mathbb{E}_{m}$ -monoid object $M$ in ${\mathcal{X}}$ is grouplike if and only if the monoid $\unicode[STIX]{x1D70B}_{0}M$ is a group.

If ${\mathcal{X}}$ is a pointed presentable $\infty$ -category, by [Reference LurieLur17, Notation 5.2.6.11] the iterated loop functor $\unicode[STIX]{x1D6FA}^{m}:{\mathcal{X}}\rightarrow {\mathcal{X}}$ factorizes as a functor

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}^{m}:{\mathcal{X}}\rightarrow \text{Mon}_{\mathbb{ E}_{m}}^{\text{gp}}({\mathcal{X}}),\end{eqnarray}$$

where, by convention, if $m=\infty$ we set $\unicode[STIX]{x1D6FA}^{\infty }=\lim _{m}\unicode[STIX]{x1D6FA}^{m}$ via the equivalence $\lim _{m}\text{Mon}_{\mathbb{E}_{m}}({\mathcal{X}})\simeq \text{Mon}_{\mathbb{E}_{\infty }}({\mathcal{X}})$ .

Remark 2.14. Let ${\mathcal{X}}$ be an $\infty$ -category which admits finite products. Consider the full embedding $\text{Mon}_{\mathbb{E}_{m}}^{\text{gp}}({\mathcal{X}})\subseteq \text{Mon}_{\mathbb{ E}_{m}}({\mathcal{X}})$ of grouplike $\mathbb{E}_{m}$ -monoid objects into all $\mathbb{E}_{m}$ -monoid objects. This inclusion preserves finite product and therefore extends to a symmetric monoidal functor. We have an induced functor $\text{Mon}_{\mathbb{E}_{1}}(\text{Mon}_{\mathbb{E}_{m}}^{\text{gp}}({\mathcal{X}}))\rightarrow \text{Mon}_{\mathbb{ E}_{1}}(\text{Mon}_{\mathbb{E}_{m}}({\mathcal{X}}))$ on the $\infty$ -categories of $\mathbb{E}_{1}$ -monoid objects which is fully faithful. By the additivity theorem [Reference LurieLur17, Theorem 5.1.2.2] the target of this functor is equivalent to the $\infty$ -category $\text{Mon}_{\mathbb{E}_{m+1}}({\mathcal{X}})$ of $\mathbb{E}_{m+1}$ -monoid objects and by definition of grouplike objects the equivalence $\text{Mon}_{\mathbb{E}_{1}}(\text{Mon}_{\mathbb{E}_{m}}({\mathcal{X}}))\simeq \text{Mon}_{\mathbb{E}_{m+1}}({\mathcal{X}})$ restricts to an equivalence $\text{Mon}_{\mathbb{E}_{1}}^{\text{gp}}(\text{Mon}_{\mathbb{ E}_{m}}({\mathcal{X}}))\simeq \text{Mon}_{\mathbb{E}_{m+1}}^{\text{gp}}({\mathcal{X}})$ . Moreover the above embedding restricts to an embedding

$$\begin{eqnarray}\text{Mon}_{\mathbb{E}_{1}}^{\text{gp}}(\text{Mon}_{\mathbb{ E}_{m}}^{\text{gp}}({\mathcal{X}}))\rightarrow \text{Mon}_{\mathbb{ E}_{1}}^{\text{gp}}(\text{Mon}_{\mathbb{ E}_{m}}({\mathcal{X}}))\simeq \text{Mon}_{\mathbb{E}_{m+1}}^{\text{gp}}({\mathcal{X}})\end{eqnarray}$$

on the subcategories of grouplike $\mathbb{E}_{1}$ -monoid objects. This embedding is an equivalence of $\infty$ -categories. Indeed, it is also essentially surjective, for if $M$ is a grouplike $\mathbb{E}_{1}$ -object in $\text{Mon}_{\mathbb{E}_{m}}({\mathcal{X}})$ , it follows from Remark 2.13 that the underlying $\mathbb{E}_{m}$ -monoid object of $M$ is grouplike and lies in the essential image.

The following result provides conditions on the underlying deformation context under which the iterated loop functor is an equivalence for the $\infty$ -category $\text{Moduli}_{\mathbf{A}}$ of formal moduli problems.

Proposition 2.15 is a consequence of the following lemma.

Proof. If $m=0$ the assertion follows from the assumption that ${\mathcal{X}}$ is pointed, since there are equivalences $\text{Mon}_{\mathbb{E}_{0}}^{\text{gp}}({\mathcal{X}})\simeq \text{Mon}_{\mathbb{ E}_{0}}({\mathcal{X}})\simeq {\mathcal{X}}_{\ast }$ , where ${\mathcal{X}}_{\ast }$ is the $\infty$ -category of pointed objects. Otherwise for finite $m$ , by Remark 2.14 and ascending induction on $m$ it suffices to prove the result for $m=1$ . For this, consider the following diagram.

Because $f$ preserves finite limits, there exists a natural equivalence $f^{\text{gp}}\unicode[STIX]{x1D6FA}_{{\mathcal{X}}}\simeq \unicode[STIX]{x1D6FA}_{{\mathcal{A}}}f$ making this diagram commutative up to equivalence. The $\infty$ -category ${\mathcal{A}}$ being stable, the functor $\unicode[STIX]{x1D6FA}_{{\mathcal{A}}}$ is an equivalence. The functors $\unicode[STIX]{x1D6FA}_{{\mathcal{X}}}$ and $\unicode[STIX]{x1D6FA}_{{\mathcal{A}}}$ admit as left adjoint the functors $B_{{\mathcal{X}}}$ and $B_{{\mathcal{A}}}$ respectively given by the bar construction. By virtue of the assumption that $f$ preserves geometric realizations of simplicial objects, and that the bar construction is given by the geometric realization of a simplicial object (see [Reference LurieLur17, Remark 5.2.2.8]), there exists a natural equivalence $fB_{{\mathcal{X}}}\simeq B_{{\mathcal{A}}}f^{\text{gp}}$ . Let $\unicode[STIX]{x1D716}:B_{{\mathcal{X}}}\unicode[STIX]{x1D6FA}_{{\mathcal{X}}}\rightarrow \text{id}_{{\mathcal{X}}}$ be the counit map. The composition of $\unicode[STIX]{x1D716}$ with the composite functor $\unicode[STIX]{x1D6FA}_{{\mathcal{A}}}f$ is equivalent to the natural transformation

$$\begin{eqnarray}f^{\text{gp}}\unicode[STIX]{x1D6FA}_{{\mathcal{X}}}\simeq \unicode[STIX]{x1D6FA}_{{\mathcal{A}}}B_{{\mathcal{A}}}f^{\text{gp}}\unicode[STIX]{x1D6FA}_{{\mathcal{X}}}\simeq \unicode[STIX]{x1D6FA}_{{\mathcal{A}}}fB_{{\mathcal{X}}}\unicode[STIX]{x1D6FA}_{{\mathcal{X}}}\rightarrow \unicode[STIX]{x1D6FA}_{{\mathcal{A}}}f,\end{eqnarray}$$

which is an equivalence because $f$ preserves finite limits. Because $\unicode[STIX]{x1D6FA}_{{\mathcal{A}}}$ is an equivalence and $f$ is conservative, this implies that $\unicode[STIX]{x1D716}$ is an equivalence. We can prove similarly that the unit map $\unicode[STIX]{x1D702}:\text{id}\rightarrow \unicode[STIX]{x1D6FA}_{{\mathcal{X}}}B_{{\mathcal{X}}}$ is an equivalence by composing with the functor $f^{\text{gp}}$ . The assertion for $m=\infty$ follows by passing to the limit over $m$ and form Remark 2.13.◻

Proof of Proposition 2.15.

Let $\mathfrak{D}:\mathbf{A}^{\text{op}}\rightarrow \mathbf{B}$ be a deformation theory for the deformation context $(\mathbf{A},(E_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in T})$ . By definition, we have at our disposal for every $\unicode[STIX]{x1D6FC}\in T$ , a functor $e_{\unicode[STIX]{x1D6FC}}:\mathbf{B}\rightarrow \text{Sp}$ which preserves small sifted colimits and such that the family $\{e_{\unicode[STIX]{x1D6FC}}\}_{\unicode[STIX]{x1D6FC}}$ is jointly conservative (see [Reference LurieLur11b, Corollary 1.3.8, Definition 1.3.9]). This implies that the functor

$$\begin{eqnarray}e=\mathop{\prod }_{\unicode[STIX]{x1D6FC}\in T}e_{\unicode[STIX]{x1D6FC}}:\mathbf{B}\rightarrow \mathop{\prod }_{\unicode[STIX]{x1D6FC}\in T}{\mathcal{A}}_{\unicode[STIX]{x1D6FC}}={\mathcal{A}},\end{eqnarray}$$

where ${\mathcal{A}}_{\unicode[STIX]{x1D6FC}}=\text{Sp}$ for every $\unicode[STIX]{x1D6FC}$ , is conservative and preserves small sifted colimits. By construction, the functor $e_{\unicode[STIX]{x1D6FC}}$ preserves finite limits, so that $e$ has the same property. Denote by $\mathfrak{t}:\text{Moduli}_{\mathbf{A}}\rightarrow \mathbf{B}$ the inverse of the equivalence $\unicode[STIX]{x1D6F9}:\mathbf{B}\rightarrow \text{Moduli}_{\mathbf{A}}$ (see [Reference LurieLur11b, Theorem 1.3.12]). Then the composite functor $e\circ \mathfrak{t}:\text{Moduli}_{\mathbf{A}}\rightarrow {\mathcal{A}}$ satisfies all the assumptions of Lemma 2.16, by which we deduce the claim.◻

Corollary 2.19. Let $(\mathbf{A},(E_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in T})$ be a deformation context admitting a deformation theory and such that $\mathbf{A}$ is a pointed $\infty$ -category. Let $m\geqslant 0$ and let $L_{m}:\text{Prox}(m)_{\mathbf{A}}\rightarrow \text{Moduli}_{\mathbf{A}}$ denote the restriction of the functor $L$ to the subcategory $\text{Prox}(m)_{\mathbf{A}}$ of $m$ -proximate formal moduli problems. Then there exists an equivalence of functors $L_{m}\simeq B^{m}\unicode[STIX]{x1D6FA}_{\text{prox}}^{m}:\text{Prox}(m)_{\mathbf{A}}\rightarrow \text{Moduli}_{\mathbf{A}}$ . In particular, the functor $L_{m}$ preserves all small limits. As a consequence, for every $m$ -proximate formal moduli problem $X$ , the functor $L_{m}X$ is given by the formula

$$\begin{eqnarray}L_{m}X(A)\simeq \text{Map}_{\text{Mon}_{\mathbb{ E}_{m}}^{\text{gp}}(\text{Moduli}_{\boldsymbol{ A}})}(\unicode[STIX]{x1D6FA}^{m}\text{Spf}(A),\unicode[STIX]{x1D6FA}^{m}X).\end{eqnarray}$$

Proof. By Lemma 2.11, there is an equivalence of functors $\unicode[STIX]{x1D6FA}_{\text{prox}}^{m}\simeq \unicode[STIX]{x1D6FA}^{m}L_{m}$ , which after composition with $B^{m}$ provides by Proposition 2.15 the expected natural equivalence $B^{m}\unicode[STIX]{x1D6FA}_{\text{prox}}^{m}\simeq L_{m}$ . Because $\unicode[STIX]{x1D6FA}_{\text{prox}}^{m}$ is given by a limit and that $B^{m}$ is an equivalence, we have that $L_{m}$ preserves all small limits. Let $X$ be an $m$ -proximate formal moduli problem. Then for $A\in \mathbf{A}^{\text{art}}$ we have natural homotopy equivalences

$$\begin{eqnarray}\displaystyle L_{m}X(A)\simeq B^{m}\unicode[STIX]{x1D6FA}_{\text{prox}}^{m}X(A) & \simeq & \displaystyle \text{Map}_{\text{Moduli}_{\boldsymbol{ A}}}(\text{Spf}(A),B^{m}\unicode[STIX]{x1D6FA}^{m}X)\nonumber\\ \displaystyle & \simeq & \displaystyle \text{Map}_{\text{Mon}_{\mathbb{ E}_{m}}^{\text{gp}}(\text{Moduli}_{\boldsymbol{ A}})}(\unicode[STIX]{x1D6FA}^{m}\text{Spf}(A),\unicode[STIX]{x1D6FA}^{m}X).\nonumber\end{eqnarray}$$

◻

3.1 Linear $\infty$ -categories

We denote by $\text{Pr}^{\text{L}}$ the $\infty$ -category of not necessarily small presentable $\infty$ -categories together with functors which are left adjoints (see [Reference LurieLur09, Definition 5.5.3.1] for a construction of $\text{Pr}^{\text{L}}$ ).

An $\infty$ -category ${\mathcal{C}}$ is said to be compactly generated if it is presentable and if in addition there exists a small $\infty$ -category ${\mathcal{C}}_{0}$ which admits finite colimits and an equivalence $\text{Ind}({\mathcal{C}}_{0})\simeq {\mathcal{C}}$ . In this situation, by [Reference LurieLur09, Theorem 5.4.2.2] the small $\infty$ -category ${\mathcal{C}}_{0}$ which generates ${\mathcal{C}}$ can be chosen to be the small idempotent complete full subcategory ${\mathcal{C}}^{c}\subseteq {\mathcal{C}}$ consisting of compact objects of ${\mathcal{C}}$ .

By [Reference LurieLur17, Proposition 4.8.1.14], the $\infty$ -category $\text{Pr}^{\text{L}}$ admits a symmetric monoidal structure in the sense of [Reference LurieLur17, Definition 2.1.2.13] and we denote the induced symmetric monoidal product by $\otimes$ . If ${\mathcal{C}},{\mathcal{C}}^{\prime }$ are presentable $\infty$ -categories of the form ${\mathcal{C}}={\mathcal{P}}({\mathcal{C}}_{0})$ and ${\mathcal{C}}^{\prime }={\mathcal{P}}({\mathcal{C}}_{0}^{\prime })$ with ${\mathcal{C}}_{0}$ , ${\mathcal{C}}_{0}$ small $\infty$ -categories, then we have ${\mathcal{C}}~\otimes ~{\mathcal{C}}={\mathcal{P}}({\mathcal{C}}_{0}\times {\mathcal{C}}_{0})$ where $\times$ is the cartesian symmetric monoidal structure on ${\mathcal{C}}\text{at}_{\infty }$ . The unit of $\text{Pr}^{\text{L},\otimes }$ is the $\infty$ -category ${\mathcal{S}}$ of small spaces. Moreover this symmetric monoidal structure is closed and for two presentable $\infty$ -categories ${\mathcal{C}}$ , ${\mathcal{C}}$ , the $\infty$ -category $\text{Fun}^{\text{L}}({\mathcal{C}},{\mathcal{C}})$ of colimit preserving $\infty$ -functors is presentable. By [Reference LurieLur17, Lemma 5.3.2.11], if ${\mathcal{C}}$ and ${\mathcal{C}}$ are compactly generated $k$ -linear $\infty$ -categories, the tensor product ${\mathcal{C}}~\otimes ~{\mathcal{C}}$ is again compactly generated and the subcategory $({\mathcal{C}}~\otimes ~{\mathcal{C}})^{c}$ is generated under finite colimits by ${\mathcal{C}}^{c}\times {\mathcal{C}}^{c}$ .

Let $k$ be an $\mathbb{E}_{\infty }$ -ring. The $\infty$ -category $\text{Mod}_{k}$ of $k$ -module spectra admits a symmetric monoidal structure $\text{Mod}_{k}^{\otimes }$ given by the smash product over $k$ . Considering $\text{Mod}_{k}^{\otimes }$ as an $\mathbb{E}_{\infty }$ -algebra object in the symmetric monoidal $\infty$ -category $\text{Pr}^{\text{L},\otimes }$ , we define the $\infty$ -category of $k$ -linear $\infty$ -categories to be

$$\begin{eqnarray}\text{Pr}_{k}^{\text{L}}:=\text{Mod}_{\text{Mod}_{k}^{\otimes }}(\text{Pr}^{\text{L},\otimes }).\end{eqnarray}$$

Morphisms in $\text{Pr}_{k}^{\text{L}}$ are called $k$ -linear functors. Because $\text{Mod}_{k}$ is a stable $\infty$ -category in the sense of [Reference LurieLur17, Definition 1.1.1.9], the underlying $\infty$ -category of any $k$ -linear $\infty$ -category is stable. For example if $k=\mathbf{S}$ the sphere spectrum, then $\text{Pr}_{\mathbf{S}}^{\text{L}}$ is the $\infty$ -category of presentable stable $\infty$ -categories with exact functors between them. The symmetric monoidal structure on $\text{Pr}^{\text{L}}$ induces a symmetric monoidal structure on $\text{Pr}_{k}^{\text{L}}$ . We denote the corresponding tensor product by $\otimes _{k}$ . Moreover the symmetric monoidal structure on $\text{Pr}_{k}^{\text{L}}$ is closed. For $k$ -linear $\infty$ -categories ${\mathcal{C}}$ , ${\mathcal{C}}^{\prime }$ and ${\mathcal{C}}^{\prime \prime }$ there exists a $k$ -linear $\infty$ -category $\text{Fun}_{k}^{\text{L}}({\mathcal{C}},{\mathcal{C}})$ of $k$ -linear colimit preserving $\infty$ -functors, such that there exists an equivalence

$$\begin{eqnarray}\text{Map}_{\text{Pr}_{k}^{\text{L}}}({\mathcal{C}}\,\otimes _{k}\,{\mathcal{C}}^{\prime },{\mathcal{C}}^{\prime \prime })\simeq \text{Map}_{\text{Pr}_{k}^{\text{L}}}({\mathcal{C}},\text{Fun}_{k}^{L}({\mathcal{C}}^{\prime },{\mathcal{C}}^{\prime \prime })).\end{eqnarray}$$

In the case ${\mathcal{C}}={\mathcal{C}}^{\prime }$ , we denote by $\text{End}_{k}({\mathcal{C}})=\text{Fun}_{k}^{\text{L}}({\mathcal{C}},{\mathcal{C}})$ the $k$ -linear $\infty$ -category of $k$ -linear endofunctors of ${\mathcal{C}}$ . Any $k$ -linear $\infty$ -category ${\mathcal{C}}\in \text{Pr}_{k}^{\text{L}}$ is enriched over $\text{Mod}_{k}$ . Indeed if $E,F\in {\mathcal{C}}$ are objects, the $\infty$ -functor $\text{Mod}_{k}\rightarrow {\mathcal{S}}$ given on objects by $V\mapsto \text{Map}_{{\mathcal{C}}}(V\,\otimes \,E,F)$ commutes with small colimits and by presentability of ${\mathcal{C}}$ is therefore representable by an object $\text{}\underline{\text{Map}}_{{\mathcal{C}}}(E,F)$ $\in \text{Mod}_{k}$ satisfying the universal property

$$\begin{eqnarray}\text{Map}_{{\mathcal{C}}}(V\,\otimes \,E,F)\simeq \text{Map}_{\text{Mod}_{k}}(M,\text{}\underline{\text{Map}}_{{\mathcal{C}}}(E,F)).\end{eqnarray}$$

We set the notation $\text{Ext}_{{\mathcal{C}}}^{m}(E,F)=\unicode[STIX]{x1D70B}_{-m}\text{}\underline{\text{Map}}_{{\mathcal{C}}}(E,F)$ for the Ext groups in ${\mathcal{C}}$ .

We now recall the relation of the above with the homotopy theory of $k$ -linear differential graded categories. Let $k$ be a discrete commutative ring. We consider two $\infty$ -categories of $k$ -linear dg-categories (we use the terminology of [Reference RobaloRob14, §6.1.1]).

• -- $Dg(k)$ , the $\infty$ -category encoding the homotopy theory of presentable $k$ -linear dg-categories up to quasi-equivalence. Morphisms are dg-functors which commute with small sums. The dg-nerve construction provides an $\infty$ -functor

  $$\begin{eqnarray}\text{N}_{dg}^{\text{L}}:Dg(k)\longrightarrow \text{Pr}^{\text{L}}.\end{eqnarray}$$

• -- $Dg^{c}(k)$ , the $\infty$ -category encoding the homotopy theory of presentable compactly generated $k$ -linear dg-categories up to quasi-equivalence. Morphisms are dg-functors which commute with small sums and preserve compact objects. The dg-nerve restricts to an $\infty$ -functor

  $$\begin{eqnarray}\text{N}_{dg}^{c}:Dg^{c}(k)\longrightarrow \text{Pr}_{\unicode[STIX]{x1D714}}^{L}.\end{eqnarray}$$

By the work of Cohn [Reference CohnCoh13] there exists a commutative diagram of $\infty$ -categories

where $\text{Pr}_{\unicode[STIX]{x1D714},k}^{\text{L}}=\text{Mod}_{\text{Mod}_{k}^{\otimes }}(\text{Pr}_{\unicode[STIX]{x1D714}}^{L})$ is the $\infty$ -category of $k$ -linear compactly generated $\infty$ -categories together with $k$ -linear functors which preserve small colimits. Moreover the bottom horizontal arrow is an equivalence of $\infty$ -categories.

3.2 Deformations of objects

We recall from [Reference LurieLur11b, §5.2] the fundamental results about the deformation functor of an object in a linear $\infty$ -category over a field.

Let $k$ be a field, let ${\mathcal{C}}$ be a $k$ -linear $\infty$ -category and let $E\in {\mathcal{C}}$ be an object. Recall from [Reference LurieLur11b, Construction 5.2.1] that we can define a functor

$$\begin{eqnarray}\text{ObjDef}_{E}:\text{Alg}_{k}^{\text{art}}\rightarrow {\mathcal{S}}\end{eqnarray}$$

from artinian $\mathbb{E}_{1}$ -algebras over $k$ to spaces which satisfies the following properties.

• -- For every artinian $\mathbb{E}_{1}$ -algebra $A$ over $k$ , the space $\text{ObjDef}_{E}(A)$ classifies deformations of $E$ over $A$ in ${\mathcal{C}}$ , or in other words it is the classifying space of pairs $(E_{A},u)$ with $E_{A}\in \text{RMod}_{A}({\mathcal{C}})$ a right $A$ -module object in ${\mathcal{C}}$ and $u:E_{A}\,\otimes _{A}\,k\simeq E$ an equivalence in ${\mathcal{C}}$ .

• -- In general, the functor $\text{ObjDef}_{E}$ does not satisfy the derived Schlessinger conditions and is not a formal $\mathbb{E}_{1}$ -moduli problem. Let $\text{ObjDef}_{E}^{\wedge }=L(\text{ObjDef}_{E})$ denote the associated formal $\mathbb{E}_{1}$ -moduli problem.

• -- By [Reference LurieLur11b, Corollary 5.2.5], the functor $\text{ObjDef}_{E}$ is a $1$ -proximate formal $\mathbb{E}_{1}$ -moduli problem, or equivalently the unit map $\text{ObjDef}_{E}\rightarrow \text{ObjDef}_{E}^{\wedge }$ is $(-1)$ -truncated.

• -- Let $\text{}\underline{\text{End}}_{{\mathcal{C}}}(E)$ be the $\mathbb{E}_{1}$ -algebra of endomorphisms of $E$ in ${\mathcal{C}}$ . By [Reference LurieLur11b, Theorem 5.2.8] the augmented $\mathbb{E}_{1}$ -algebra which controls the formal $\mathbb{E}_{1}$ -moduli problem $\text{ObjDef}_{E}^{\wedge }$ is given by $k\,\oplus \,\text{}\underline{\text{End}}_{{\mathcal{C}}}(E)$ . In other words there exists for every $A\in \text{Alg}_{k}^{\text{art}}$ a natural equivalence

  $$\begin{eqnarray}\text{ObjDef}_{E}^{\wedge }(A)\simeq \text{Map}_{\text{Alg}_{k}}(\mathfrak{D}^{(1)}(A),\text{}\underline{\text{End}}_{{\mathcal{C}}}(E)).\end{eqnarray}$$
  In particular the tangent spectrum of $\text{ObjDef}_{E}^{\wedge }$ is given by $T_{\text{ObjDef}_{E}^{\wedge }}\simeq \text{}\underline{\text{End}}_{{\mathcal{C}}}(E)[1]$ .

• -- Koszul duality for modules implies that for every artinian $\mathbb{E}_{1}$ -algebra $A$ we have a natural homotopy equivalence

  $$\begin{eqnarray}\text{ObjDef}_{E}^{\wedge }(A)\simeq \text{LMod}_{A}^{!}({\mathcal{C}})^{\simeq }\times _{{\mathcal{C}}^{\simeq }}\{E\}\end{eqnarray}$$
  with the space of deformations of $E$ as an Ind-coherent right $A$ -modules in ${\mathcal{C}}$ , where $\text{LMod}_{A}^{!}({\mathcal{C}})=\text{LMod}_{A}^{!}\,\otimes _{k}\,{\mathcal{C}}$ is the $\infty$ -category of Ind-coherent right $A$ -modules in ${\mathcal{C}}$ (see [Reference LurieLur11b, Remark 5.2.16]).

• -- By [Reference LurieLur11b, Proposition 5.2.14], under the assumption that ${\mathcal{C}}$ admits a left complete $t$ -structure and that $E$ is connective, then $\text{ObjDef}_{E}$ is a formal $\mathbb{E}_{1}$ -moduli problem.

3.3 Deformations of categories

We first recall from [Reference LurieLur11b, §5.3] the fundamental results about the deformation functor of a linear $\infty$ -category over a field. Then we recall the construction of this functor.

Let $k$ be field and ${\mathcal{C}}$ a $k$ -linear $\infty$ -category. In [Reference LurieLur11b, §5.3] is constructed a functor

$$\begin{eqnarray}{\mathcal{C}}\text{atDef}_{{\mathcal{C}}}:\text{Alg}_{k}^{(2),\text{art}}\longrightarrow \widehat{{\mathcal{S}}}\end{eqnarray}$$

from artinian $\mathbb{E}_{2}$ -algebras over $k$ to not necessarily small spaces which satisfies the following properties.

• -- For any artinian $\mathbb{E}_{2}$ -algebra $A$ over $k$ , the space ${\mathcal{C}}\text{atDef}_{{\mathcal{C}}}(A)$ classifies deformations of ${\mathcal{C}}$ over $A$ , or in other words pairs $({\mathcal{C}}_{A},u)$ with ${\mathcal{C}}_{A}$ a right $A$ -linear $\infty$ -category and $u:{\mathcal{C}}_{A}\,\otimes _{A}\,k\simeq {\mathcal{C}}$ an equivalence in $\text{Pr}_{k}^{\text{L}}$ (see below for the definition of linear $\infty$ -category over an $\mathbb{E}_{2}$ -algebra). We observe that the space ${\mathcal{C}}\text{atDef}_{{\mathcal{C}}}(k)$ is contractible.

• -- In general, the functor ${\mathcal{C}}\text{atDef}_{{\mathcal{C}}}$ does not satisfy the derived Schlessinger conditions, and is not an $\mathbb{E}_{2}$ -formal moduli problem. Let ${\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{\wedge }:=L({\mathcal{C}}\text{atDef}_{{\mathcal{C}}})$ denote the associated $\mathbb{E}_{2}$ -formal moduli problem and

  $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{{\mathcal{C}}}:{\mathcal{C}}\text{atDef}_{{\mathcal{C}}}\rightarrow {\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{\wedge }\end{eqnarray}$$
  denote the unit map. In other words, the map $\unicode[STIX]{x1D70E}_{{\mathcal{C}}}$ is in general not an equivalence. We will see a simple example below.

• -- However, the map $\unicode[STIX]{x1D70E}_{{\mathcal{C}}}$ is rather close to being an equivalence as shown by Lurie. Namely, by [Reference LurieLur11b, Corollary 5.3.8] the functor ${\mathcal{C}}\text{atDef}_{{\mathcal{C}}}$ is a $2$ -proximate $\mathbb{E}_{2}$ -formal moduli problem in the sense of [Reference LurieLur11b, Definition 5.1.5] (with values in essentially small spaces). By [Reference LurieLur11b, Theorem 5.1.9], it means that the map $\unicode[STIX]{x1D70E}_{{\mathcal{C}}}$ has $0$ -truncated fibers, or in other words, that the map $\unicode[STIX]{x1D70E}_{{\mathcal{C}}}$ induces an isomorphism on $\unicode[STIX]{x1D70B}_{i}$ for $i\geqslant 2$ and is injective on $\unicode[STIX]{x1D70B}_{1}$ (all homotopy groups are based at the trivial deformation of ${\mathcal{C}}$ ).

• -- Let $\unicode[STIX]{x1D709}({\mathcal{C}})$ denote the $\mathbb{E}_{2}$ -algebra over $k$ given by the center of the $\infty$ -category ${\mathcal{C}}$ in the sense of [Reference LurieLur11b, Definition 5.3.10]. The underlying $k$ -module spectrum of $\unicode[STIX]{x1D709}({\mathcal{C}})$ is given by $\text{}\underline{\text{End}}_{\text{End}_{k}({\mathcal{C}})}(\text{id}_{{\mathcal{C}}})$ so that the homotopy groups of $\unicode[STIX]{x1D709}({\mathcal{C}})$ are isomorphic to the Hochschild cohomology groups of ${\mathcal{C}}$ . By [Reference LurieLur11b, Theorem 3.5.1], the augmented $\mathbb{E}_{2}$ -algebra which controls the $\mathbb{E}_{2}$ -formal moduli problem ${\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{\wedge }$ is given by $k~\oplus ~\unicode[STIX]{x1D709}({\mathcal{C}})$ . In other words, for any artinian $\mathbb{E}_{2}$ -algebra $A$ over $k$ there exists a natural equivalence

  $$\begin{eqnarray}{\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{\wedge }(A)\simeq \text{Map}_{\text{Alg}_{k}^{(2)}}(\mathfrak{D}^{(2)}(A),\unicode[STIX]{x1D709}({\mathcal{C}})),\end{eqnarray}$$
  where $\mathfrak{D}^{(2)}$ is the $\mathbb{E}_{2}$ -Kozsul duality functor. In particular the tangent spectrum of ${\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{\wedge }$ is given by $T_{{\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{\wedge }}\simeq \unicode[STIX]{x1D709}({\mathcal{C}})[2]$ .

We have stronger properties if we restrict to compactly generated deformations as observed by Lurie (see [Reference LurieLur11b, Variant 5.3.4]), a condition which is reasonable in most examples of interest. Let ${\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{c}$ denote the subfunctor of ${\mathcal{C}}\text{atDef}_{{\mathcal{C}}}$ spanned by pairs $({\mathcal{C}}_{A},u)$ where ${\mathcal{C}}_{A}$ is compactly generated. By [Reference LurieLur11b, Variant 5.3.6], the functor ${\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{c}$ has values in essentially small spaces, and can therefore be viewed as a functor

$$\begin{eqnarray}{\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{c}:\text{Alg}_{k}^{(2),\text{art}}\longrightarrow {\mathcal{S}}\end{eqnarray}$$

which satisfies the following properties.

• -- The natural embedding ${\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{c}\rightarrow {\mathcal{C}}\text{atDef}_{{\mathcal{C}}}$ has $0$ -truncated fibers, which implies that ${\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{c}$ is a $2$ -proximate $\mathbb{E}_{2}$ -formal moduli problem and that the induced map $L({\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{c})\rightarrow L({\mathcal{C}}\text{atDef}_{{\mathcal{C}}})={\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{\wedge }$ is an equivalence. We denote by

  $$\begin{eqnarray}\unicode[STIX]{x1D703}_{{\mathcal{C}}}:{\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{c}\rightarrow {\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{\wedge }\end{eqnarray}$$
  the unit map.

• -- Suppose that ${\mathcal{C}}$ is tamely compactly generated (see Definition 3.3). Then the functor ${\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{c}$ is a $1$ -proximate $\mathbb{E}_{2}$ -formal moduli problem by [Reference LurieLur11b, Proposition 5.3.21]. In other words, the map $\unicode[STIX]{x1D703}_{{\mathcal{C}}}$ induces an isomorphism on $\unicode[STIX]{x1D70B}_{i}$ for $i\geqslant 1$ and an injection on $\unicode[STIX]{x1D70B}_{0}$ . Our main result gives additional conditions under which the surjectivity on $\unicode[STIX]{x1D70B}_{0}$ holds for formal deformations.

We recall now the construction of the functor ${\mathcal{C}}\text{atDef}_{{\mathcal{C}}}$ from [Reference LurieLur11b, Construction 5.3.2]. By [Reference LurieLur17, Theorem 4.8.5.16] there exists a functor

$$\begin{eqnarray}\displaystyle \text{Alg}_{k} & \longrightarrow & \displaystyle \text{Pr}_{k}^{\text{L}}\nonumber\\ \displaystyle A & \longmapsto & \displaystyle \text{LMod}_{A}\nonumber\end{eqnarray}$$

which extends to a symmetric monoidal functor. Passing to algebra objects we obtain a functor

$$\begin{eqnarray}\displaystyle \text{Alg}_{k}^{(2)}\simeq \text{Alg}(\text{Alg}_{k}) & \longrightarrow & \displaystyle \text{Alg}(\text{Pr}_{k}^{\text{L}})\nonumber\\ \displaystyle A & \longmapsto & \displaystyle \text{LMod}_{A}^{\otimes }\nonumber\end{eqnarray}$$

which to an $\mathbb{E}_{2}$ -algebra $A$ associates the monoidal $\infty$ -category of left $A$ -modules.

Let $A$ be an $\mathbb{E}_{2}$ -algebra over $k$ . A right $A$ -linear $\infty$ -category (respectively left $A$ -linear $\infty$ -category) is by definition a right $\text{LMod}_{A}^{\otimes }$ -module in $\text{Pr}_{k}^{\text{L}}$ (respectively a left $\text{LMod}_{A}^{\otimes }$ -module in  $\text{Pr}_{k}^{\text{L}}$ ). If $A\rightarrow A^{\prime }$ is a morphism of $\mathbb{E}_{2}$ -algebras over $k$ , and ${\mathcal{C}}_{A}$ a right $A$ -linear $\infty$ -category, we denote by ${\mathcal{C}}_{A}\,\otimes _{A}\,A^{\prime }$ the extension of scalars ${\mathcal{C}}_{A}\,\widehat{\otimes }_{\text{LMod}_{A}}\text{LMod}_{A^{\prime }}$ .

Consider the cocartesian fibration $\text{RMod}(\text{Pr}_{k}^{\text{L}})\rightarrow \text{Alg}(\text{Pr}_{k}^{\text{L}})$ with fiber over a $k$ -linear monoidal $\infty$ -category ${\mathcal{D}}^{\otimes }$ given by the $\infty$ -category $\text{RMod}_{{\mathcal{D}}^{\otimes }}(\text{Pr}_{k}^{\text{L}})$ of $\infty$ -categories right tensored over ${\mathcal{D}}^{\otimes }$ . Set $\text{R}{\mathcal{C}}\text{at}_{k}=\text{Alg}_{k}^{(2)}\times _{\text{Alg}(\text{Pr}_{k}^{\text{L}})}\text{RMod}(\text{Pr}_{k}^{\text{L}})$ and consider the induced cocartesian fibration $p:\text{R}{\mathcal{C}}\text{at}_{k}\rightarrow \text{Alg}_{k}^{(2)}$ with fiber over an $\mathbb{E}_{2}$ -algebra $A$ given by the $\infty$ -category $\text{RMod}_{\text{LMod}_{A}^{\otimes }}(\text{Pr}_{k}^{\text{L}})$ of right $A$ -linear $\infty$ -categories. Consider the subcategory $\text{R}{\mathcal{C}}\text{at}_{k}^{\text{cocart}}$ consisting of $p$ -cocartesian morphisms so that we obtain a left fibration $\text{R}{\mathcal{C}}\text{at}_{k}^{\text{cocart}}\rightarrow \text{Alg}_{k}^{(2)}$ with fiber over $A$ the space of right $A$ -linear $\infty$ -categories.

Let ${\mathcal{C}}$ be a $k$ -linear $\infty$ -category and consider the pair $(k,{\mathcal{C}})$ as an object of $\text{R}{\mathcal{C}}\text{at}_{k}^{\text{cocart}}$ . The $\infty$ -category of deformations of ${\mathcal{C}}$ is the $\infty$ -category $\text{Deform}[{\mathcal{C}}]=(\text{R}{\mathcal{C}}\text{at}_{k}^{\text{cocart}})_{/(k,{\mathcal{C}})}$ . The induced left fibration $\text{Deform}[{\mathcal{C}}]\rightarrow (\text{Alg}_{k}^{(2)})_{/k}\simeq \text{Alg}_{k}^{(2),\text{aug}}$ is classified by a functor $\text{Alg}_{k}^{(2),\text{aug}}\rightarrow \widehat{{\mathcal{S}}}$ whose restriction to artinian $\mathbb{E}_{2}$ -algebras is denoted by ${\mathcal{C}}\text{atDef}_{{\mathcal{C}}}:\text{Alg}_{k}^{(2),\text{art}}\longrightarrow \widehat{{\mathcal{S}}}$ .

4.1 Simultaneous deformations

Let $k$ be a field. We now study the deformation functor of a pair $({\mathcal{C}},E)$ with ${\mathcal{C}}$ a $k$ -linear $\infty$ -category and $E\in {\mathcal{C}}$ an object.

We first give a construction of this functor which consists in mimicking the construction of ${\mathcal{C}}\text{atDef}_{{\mathcal{C}}}$ in [Reference LurieLur11b, Construction 5.3.2] but replacing the $\infty$ -category $\text{Pr}_{k}^{\text{L}}$ by the slice $\infty$ -category $(\text{Pr}_{k}^{\text{L}})_{\text{Mod}_{k}/}$ of pairs $({\mathcal{C}},E)$ with ${\mathcal{C}}$ a $k$ -linear $\infty$ -category and $E$ an object of ${\mathcal{C}}$ .

Construction 4.1. Theorem 4.8.5.16 in Lurie’s book [Reference LurieLur17] provides a symmetric monoidal functor

$$\begin{eqnarray}\displaystyle \text{Alg}_{k} & \longrightarrow & \displaystyle (\text{Pr}_{k}^{\text{L}})_{\text{Mod}_{k}/}\nonumber\\ \displaystyle A & \longmapsto & \displaystyle (\text{LMod}_{A},A)\nonumber\end{eqnarray}$$

where the $\infty$ -category $(\text{Pr}_{k}^{\text{L}})_{\text{Mod}_{k}/}$ is endowed with the natural symmetric monoidal structure from [Reference LurieLur17, Remark 2.2.2.25] coming from the fact that $\text{Mod}_{k}$ is the unit in $\text{Pr}_{k}^{\text{L},\otimes }$ . Applying algebra objects we obtain a functor

$$\begin{eqnarray}\displaystyle \text{Alg}_{k}^{(2)}\simeq \text{Alg}(\text{Alg}_{k}) & \rightarrow & \displaystyle \text{Alg}((\text{Pr}_{k}^{\text{L}})_{\text{Mod}_{k}/})\simeq \text{Alg}(\text{Pr}_{k}^{\text{L}})\nonumber\\ \displaystyle A & \longmapsto & \displaystyle \text{LMod}_{A}^{\otimes }.\nonumber\end{eqnarray}$$

Consider the cocartesian fibration

$$\begin{eqnarray}\text{RMod}((\text{Pr}_{k}^{\text{L}})_{\text{Mod}_{k}/})\rightarrow \text{Alg}(\text{Pr}_{k}^{\text{L}})\end{eqnarray}$$

with fiber over a monoidal $\infty$ -category ${\mathcal{D}}^{\otimes }$ the $\infty$ -category $\text{RMod}_{{\mathcal{D}}^{\otimes }}((\text{Pr}_{k}^{\text{L}})_{\text{Mod}_{k}/})$ of pairs $({\mathcal{M}},M)$ where ${\mathcal{M}}$ is a $k$ -linear $\infty$ -category right tensored over ${\mathcal{D}}^{\otimes }$ and $M\in {\mathcal{M}}$ an object. Set $\text{R}{\mathcal{C}}\text{at}_{k}^{\ast }=\text{Alg}_{k}^{(2)}\times _{\text{Alg}(\text{Pr}_{k}^{\text{L}})}\text{RMod}((\text{Pr}_{k}^{\text{L}})_{\text{Mod}_{k}/})$ and consider the induced cocartesian fibration $q:\text{R}{\mathcal{C}}\text{at}_{k}^{\ast }\rightarrow \text{Alg}_{k}^{(2)}$ with fiber over an $\mathbb{E}_{2}$ -algebra $A$ the $\infty$ -category of right $A$ -linear $\infty$ -categories together with a distinguished object. Let $(\text{R}{\mathcal{C}}\text{at}_{k}^{\ast })^{\text{cocart}}$ be the subcategory of $q$ -cocartesian morphisms so that we have an induced left fibration $(\text{R}{\mathcal{C}}\text{at}_{k}^{\ast })^{\text{cocart}}\rightarrow \text{Alg}_{k}^{(2)}$ . Let ${\mathcal{C}}$ be a $k$ -linear $\infty$ -category and $E$ an object of ${\mathcal{C}}$ . The $\infty$ -category of deformations of $({\mathcal{C}},E)$ is the $\infty$ -category

$$\begin{eqnarray}\text{Deform}[{\mathcal{C}},E]:=(\text{R}{\mathcal{C}}\text{at}_{k}^{\ast })_{/(k,{\mathcal{C}},E)}^{\text{cocart}}.\end{eqnarray}$$

The induced left fibration $\text{Deform}[{\mathcal{C}},E]\rightarrow (\text{Alg}_{k}^{(2)})_{/k}\simeq \text{Alg}_{k}^{(2),\text{aug}}$ is classified by a functor $\text{Alg}_{k}^{(2),\text{aug}}\rightarrow \widehat{{\mathcal{S}}}$ whose restriction to artinian $\mathbb{E}_{2}$ -algebras is denoted by

$$\begin{eqnarray}{\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}:\text{Alg}_{k}^{(2),\text{art}}\longrightarrow \widehat{{\mathcal{S}}}.\end{eqnarray}$$

For any artinian $\mathbb{E}_{2}$ -algebra $A$ over $k$ , the space ${\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}(A)$ classifies $4$ -uples $({\mathcal{C}}_{A},E_{A},u,v)$ where ${\mathcal{C}}_{A}$ is a right $A$ -linear $\infty$ -category,  $u:{\mathcal{C}}_{A}\,\otimes _{A}\,k\simeq {\mathcal{C}}$ is an equivalence, $E_{A}\in {\mathcal{C}}_{A}$ is an object and $v:E_{A}\,\otimes _{A}\,k\simeq A$ is an equivalence in ${\mathcal{C}}$ . We have that ${\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}(k)$ is a contractible space.

The deformation functor ${\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}$ is closely related to the deformation functors ${\mathcal{C}}\text{atDef}_{{\mathcal{C}}}$ and $\text{ObjDef}_{E}$ . We construct the natural maps between them.

Construction 4.2. Let $k$ be a field, ${\mathcal{C}}$ a $k$ -linear $\infty$ -category and $E$ an object of ${\mathcal{C}}$ . The forgetful functor $(\text{Pr}_{k}^{\text{L}})_{\text{Mod}_{k}/}\rightarrow \text{Pr}_{k}^{\text{L}}$ induces a functor $\text{R}{\mathcal{C}}\text{at}_{k}^{\ast }\rightarrow \text{R}{\mathcal{C}}\text{at}_{k}$ which in turn induces a functor $\text{Deform}[{\mathcal{C}},E]\rightarrow \text{Deform}[{\mathcal{C}}]$ commuting with the projection to $\text{Alg}_{k}^{(2),\text{aug}}$ . We obtain natural transformations

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}_{({\mathcal{C}},E)}:{\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}\longrightarrow {\mathcal{C}}\text{atDef}_{{\mathcal{C}}} & & \displaystyle \nonumber\\ \displaystyle \unicode[STIX]{x1D70C}_{({\mathcal{C}},E)}:{\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}^{c}\longrightarrow {\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{c} & & \displaystyle \nonumber\end{eqnarray}$$

given on objects by forgetting the distinguished object deformation.

The homotopy fiber of the forgetful functor $(\text{Pr}_{k}^{\text{L}})_{\text{Mod}_{k}/}\rightarrow \text{Pr}_{k}^{\text{L}}$ at ${\mathcal{C}}$ is equivalent to the Kan complex $\text{Map}_{\text{Pr}_{k}^{\text{L}}}(\text{Mod}_{k},{\mathcal{C}})\simeq {\mathcal{C}}^{\simeq }$ . The induced functor ${\mathcal{C}}^{\simeq }\rightarrow (\text{Pr}_{k}^{\text{L}})_{\text{Mod}_{k}/}$ is given on object by sending an object $E$ of ${\mathcal{C}}^{\simeq }$ to the pair $({\mathcal{C}},E)$ . This functor induces a functor $\text{Deform}[E]\rightarrow \text{Deform}[{\mathcal{C}},E]$ commuting with the projection to $\text{Alg}_{k}^{(2),\text{aug}}$ . We therefore obtain a natural transformation

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{({\mathcal{C}},E)}:\text{ObjDef}_{E}^{\mathbb{E}_{2}}\longrightarrow {\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)},\end{eqnarray}$$

where $\text{ObjDef}^{\mathbb{E}_{2}}$ is the restriction of the functor $\text{ObjDef}_{E}$ to artinian $\mathbb{E}_{2}$ -algebras over $k$ along the forgetful functor $\text{Alg}_{k}^{(2),\text{art}}\rightarrow \text{Alg}_{k}^{\text{art}}$ . The functor $\unicode[STIX]{x1D711}_{({\mathcal{C}},E)}$ is given on object by sending a deformation $E_{A}$ of $E$ over an $\mathbb{E}_{2}$ -algebra $A$ to the pair $({\mathcal{C}}\,\otimes _{k}\,A,E_{A})$ . Moreover, whenever ${\mathcal{C}}$ is compactly generated, we have that the map $\unicode[STIX]{x1D711}_{({\mathcal{C}},E)}$ factors as a map $\text{ObjDef}_{E}^{\mathbb{E}_{2}}\rightarrow {\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}^{c}$ .

Proposition 4.3. Let $k$ be a field, ${\mathcal{C}}$ a $k$ -linear $\infty$ -category and $E\in {\mathcal{C}}$ an object. The sequence of natural transformations

$$\begin{eqnarray}\text{ObjDef}_{E}^{\mathbb{E}_{2}}\longrightarrow {\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}\longrightarrow {\mathcal{C}}\text{atDef}_{{\mathcal{C}}}\end{eqnarray}$$

from Construction 4.2 is a fiber sequence in $\text{Fun}_{\ast }(\text{Alg}_{k}^{(2),\text{art}},\widehat{{\mathcal{S}}})$ . Consequently, the functor ${\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}$ is a $2$ -proximate formal $\mathbb{E}_{2}$ -moduli problem (in a larger universe). Moreover it induces a homotopy commutative diagram

where the bottom line is a fiber sequence in $\text{Moduli}_{k}^{\mathbb{E}_{2}}$ .

Notation 4.5. Let $k$ be a field, ${\mathcal{C}}$ a $k$ -linear $\infty$ -category and $E\in {\mathcal{C}}$ an object. Consider the object $({\mathcal{C}},E)$ of the slice $\infty$ -category $(\text{Pr}_{k}^{\text{L}})_{\text{Mod}_{k}/}$ and its endomorphism monoidal $k$ -linear $\infty$ -category $\text{End}_{k}({\mathcal{C}},E)^{\otimes }:=\text{End}_{(\text{Pr}_{k}^{\text{L}})_{\text{Mod}_{k}/}}(({\mathcal{C}},E))^{\otimes }$ . Denote by $\unicode[STIX]{x1D709}({\mathcal{C}},E)=\text{}\underline{\text{End}}_{\text{End}_{k}({\mathcal{C}},E)}(\text{id}_{({\mathcal{C}},E)})$ the $\mathbb{E}_{2}$ -algebra over $k$ given by endomorphisms of the unit. By adjunction, this $\mathbb{E}_{2}$ -algebra satisfies the following universal property: for any $\mathbb{E}_{2}$ -algebra $A$ over $k$ , there exits a homotopy equivalence

$$\begin{eqnarray}\text{Map}_{\text{Alg}_{k}^{(2)}}(A,\unicode[STIX]{x1D709}({\mathcal{C}},E))\simeq \text{LMod}_{\text{LMod}_{A}^{\otimes }}((\text{Pr}_{k}^{\text{L}})_{\text{Mod}_{k}/})^{\simeq }\times _{((\text{Pr}_{k}^{\text{L}})_{\text{Mod}_{k}/})^{\simeq }}\{({\mathcal{C}},E)\},\end{eqnarray}$$

where the right-hand side is the space of left $A$ -linear structures on the pair $({\mathcal{C}},E)$ , or, in other words, the space of left $A$ -linear structures on ${\mathcal{C}}$ acting trivially on $E$ up to equivalence.

Remark 4.6. Let $k$ be a field, ${\mathcal{C}}$ a $k$ -linear $\infty$ -category and $E\in {\mathcal{C}}$ an object. By definition of the slice $\infty$ -category $(\text{Pr}_{k}^{\text{L}})_{\text{Mod}_{k}/}$ , the $k$ -linear $\infty$ -category $\text{End}_{k}({\mathcal{C}},E)$ is the homotopy fiber of the $k$ -linear functor $\text{ev}_{E}:\text{End}({\mathcal{C}})\rightarrow {\mathcal{C}}$ given by evaluation at $E$ . We deduce that the induced sequence

$$\begin{eqnarray}\unicode[STIX]{x1D709}({\mathcal{C}},E)\rightarrow \unicode[STIX]{x1D709}({\mathcal{C}})\rightarrow \text{}\underline{\text{End}}_{{\mathcal{C}}}(E)\end{eqnarray}$$

is a fiber sequence of $k$ -module spectra. We observe that the first map in the sequence is a map of $\mathbb{E}_{2}$ -algebras over $k$ .

Proposition 4.7. Let $k$ be a field, ${\mathcal{C}}$ a $k$ -linear $\infty$ -category and $E\in {\mathcal{C}}$ an object. Consider the augmented $\mathbb{E}_{2}$ -algebra given by $k\,\oplus \,\unicode[STIX]{x1D709}({\mathcal{C}},E)$ and its associated $\mathbb{E}_{2}$ -formal moduli problem $\unicode[STIX]{x1D6F9}_{k\,\oplus \,\unicode[STIX]{x1D709}({\mathcal{C}},E)}$ . Then there exists a natural transformation ${\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}\rightarrow \unicode[STIX]{x1D6F9}_{k\,\oplus \,\unicode[STIX]{x1D709}({\mathcal{C}},E)}$ which is $0$ -truncated. Therefore the formal $\mathbb{E}_{2}$ -moduli problem ${\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}^{\wedge }$ is given by the formula

$$\begin{eqnarray}{\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}^{\wedge }(A)\simeq \text{Map}_{\text{Alg}_{k}^{(2)}}(\mathfrak{D}^{(2)}(A),\unicode[STIX]{x1D709}({\mathcal{C}},E)).\end{eqnarray}$$

Proof. We mimic the proof of [Reference LurieLur11b, Theorem 5.3.16], replacing the center $\unicode[STIX]{x1D709}({\mathcal{C}})$ by $\unicode[STIX]{x1D709}({\mathcal{C}},E)$ . Following the same lines as in [Reference LurieLur11b, Construction 5.3.18], we see that there exits a homotopy commutative diagram

where the upper functor sends a deformation $({\mathcal{C}}_{A},E_{A})$ over an augmented $\mathbb{E}_{2}$ -algebra $A$ to ${\mathcal{C}}$ seen as left tensored over $\mathfrak{D}^{(2)}(A)$ via the equivalence $(\mathfrak{D}^{(2)}(A)\,\otimes _{k}\,{\mathcal{C}}_{A})\,\otimes _{\mathfrak{D}^{(2)}(A)\,\otimes _{k}\,A}\,k\simeq {\mathcal{C}}$ together with the distinguished object $E$ . This functor induces a natural transformation

$$\begin{eqnarray}{\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}\rightarrow \unicode[STIX]{x1D6F9}_{k\,\oplus \,\unicode[STIX]{x1D709}({\mathcal{C}},E)}.\end{eqnarray}$$

The same proof as [Reference LurieLur11b, Proposition 5.3.17] but replacing the center $\unicode[STIX]{x1D709}({\mathcal{C}})$ by the $\mathbb{E}_{2}$ -algebra $\unicode[STIX]{x1D709}({\mathcal{C}},E)$ shows that this map is $0$ -truncated. Finally [Reference LurieLur11b, Theorem 5.1.9] implies that the induced map ${\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}^{\wedge }\rightarrow \unicode[STIX]{x1D6F9}_{k\,\oplus \,\unicode[STIX]{x1D709}({\mathcal{C}},E)}$ is an equivalence.◻

Remark 4.8. In the situation of Proposition 4.7, the universal property of the $\mathbb{E}_{2}$ -algebra $\unicode[STIX]{x1D709}({\mathcal{C}},E)$ (see Notation 4.5) implies that we have for each artinian $\mathbb{E}_{2}$ -algebra $A$ over $k$ a natural homotopy equivalence

$$\begin{eqnarray}{\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}^{\wedge }(A)\simeq \text{LMod}_{\text{LMod}_{\mathfrak{D}^{(2)}(A)}^{\otimes }}((\text{Pr}_{k}^{\text{L}})_{\text{Mod}_{k}/})^{\simeq }\times _{((\text{Pr}_{k}^{\text{L}})_{\text{Mod}_{k}/})^{\simeq }}\{({\mathcal{C}},E)\}\end{eqnarray}$$

with the space of left $\mathfrak{D}^{(2)}(A)$ -linear structures on the $\infty$ -category ${\mathcal{C}}$ such that $\mathfrak{D}^{(2)}(A)$ acts trivially on the object $E$ .

Remark 4.9. Let $k$ be a field, ${\mathcal{C}}$ a $k$ -linear $\infty$ -category and $E\in {\mathcal{C}}$ an object. Recall from Proposition 4.3 the existence of a fiber sequence of formal $\mathbb{E}_{2}$ -moduli problems

$$\begin{eqnarray}\text{ObjDef}_{E}^{\mathbb{E}_{2},\wedge }\longrightarrow {\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}^{\wedge }\longrightarrow {\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{\wedge }.\end{eqnarray}$$

It induces a fiber sequence

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}\,{\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}^{\wedge }\longrightarrow \unicode[STIX]{x1D6FA}\,{\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{\wedge }\longrightarrow \text{ObjDef}_{E}^{\mathbb{E}_{2},\wedge }\end{eqnarray}$$

which after taking the tangent spectrum gives a fiber sequence of $k$ -module spectra

$$\begin{eqnarray}\unicode[STIX]{x1D709}({\mathcal{C}},E)\longrightarrow \unicode[STIX]{x1D709}({\mathcal{C}})\longrightarrow \text{}\underline{\text{End}}_{{\mathcal{C}}}(E).\end{eqnarray}$$

It can be seen that for each artinian $\mathbb{E}_{2}$ -algebra $A$ over $k$ , the map $\unicode[STIX]{x1D6FA}\,{\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{\wedge }(A)\rightarrow \text{ObjDef}_{E}^{\mathbb{E}_{2},\wedge }(A)$ sends an $A$ -linear autoequivalence $f_{A}:{\mathcal{C}}\,\otimes _{k}\,A\simeq {\mathcal{C}}\,\otimes _{k}\,A$ of the trivial deformation to the image $f_{A}(E\,\otimes _{k}\,A)$ of the trivial deformation of $E$ in ${\mathcal{C}}\,\otimes _{k}\,A$ . From this we can deduce that the fiber sequence of tangent spectra above is equivalent to the fiber sequence given in Remark 4.6, although we will not use this fact. We denote by $\unicode[STIX]{x1D712}_{E}:\unicode[STIX]{x1D709}({\mathcal{C}})\rightarrow \text{}\underline{\text{End}}_{{\mathcal{C}}}(E)$ the map of $k$ -modules induced by evaluation at $E$ and call it the obstruction map of $E$ .

Remark 4.10. Let $k$ be a field, ${\mathcal{C}}$ a $k$ -linear $\infty$ -category and $E\in {\mathcal{C}}$ an object. Consider the homotopy commutative diagram

where the right square is induced by Proposition 4.3. The bottom left map is an equivalence because ${\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{\wedge }$ is a formal moduli problem. By [Reference LurieLur11b, Theorem 3.5.1], there exists an isomorphism $\unicode[STIX]{x1D70B}_{0}\,{\mathcal{C}}\text{atDef}_{{\mathcal{C}}}^{\wedge }(k\,\oplus \,k[m])\simeq \text{HH}^{m+2}({\mathcal{C}})$ . Moreover by [Reference LurieLur11b, Theorem 5.2.8] there is an isomorphism $\unicode[STIX]{x1D70B}_{0}\,\text{ObjDef}_{E}^{\wedge }(k\,\oplus \,k[m+1])\simeq \text{Ext}_{{\mathcal{C}}}^{m+2}(E,E)$ . From Proposition 4.3 and taking $\unicode[STIX]{x1D70B}_{0}$ , we have an induced commutative diagram

where the bottom line is exact in the middle. Let ${\mathcal{C}}_{1}$ be a deformation of ${\mathcal{C}}$ over $k\,\oplus \,k[m]$ and let $\unicode[STIX]{x1D719}=\unicode[STIX]{x1D70E}_{{\mathcal{C}}}({\mathcal{C}}_{1})$ . Suppose $E_{1}\in {\mathcal{C}}_{1}$ deforms $E$ or in other words that $E_{1}\,\otimes _{k\,\oplus \,k[m]}\,k\simeq E$ in ${\mathcal{C}}_{1}\,\otimes _{k\,\oplus \,k[m]}k$ $\simeq {\mathcal{C}}$ . In this case we have $\unicode[STIX]{x1D6FC}({\mathcal{C}}_{1},E_{1})={\mathcal{C}}_{1}$ and thus $\unicode[STIX]{x1D712}_{E}(\unicode[STIX]{x1D719})=0$ .

Notation 4.12. Let $k$ be a field, ${\mathcal{C}}$ a compactly generated $k$ -linear $\infty$ -category and $E\in {\mathcal{C}}$ an object. For each artinian $\mathbb{E}_{2}$ -algebra $A$ over $k$ we denote by ${\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}^{c}(A)$ the summand of ${\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}(A)$ consisting of $4$ -uplets $({\mathcal{C}}_{A},u,E_{A},v)$ with ${\mathcal{C}}_{A}$ being compactly generated. We obtain a functor

$$\begin{eqnarray}{\mathcal{S}}\text{imDef}_{({\mathcal{C}},E)}^{c}:\text{Alg}_{k}^{(2),\text{art}}\rightarrow \widehat{{\mathcal{S}}}.\end{eqnarray}$$