hcju/stacksqa huggingface dataset:
Let $\mathcal{F}$ be a subsheaf of the final object $*$ of $\Sh(\mathcal{C})$ and let $E \subset \Sh(\mathcal{C})$ be the full subcategory consisting of those $\mathcal{G}$ such that $\mathcal{F} \times \mathcal{G} \to \mathcal{F}$ is an isomorphism. By Lemma \ref{lemma-closed-subtopos} the functor $i_*$ is isomorphic to the inclusion functor $\iota : E \to \Sh(\mathcal{C})$. \medskip\noindent Let $j_{\mathcal{F}} : \Sh(\mathcal{C})/\mathcal{F} \to \Sh(\mathcal{C})$ be the localization functor (Lemma \ref{lemma-localize-topos}). Note that $E$ can also be described as the collection of sheaves $\mathcal{G}$ such that $j_\mathcal{F}^{-1}\mathcal{G} = *$. \medskip\noindent Let $a, b : \mathcal{G}_1 \to \mathcal{G}_2$ be two morphism of $E$. To prove $\iota$ commutes with coequalizers it suffices to show that the coequalizer of $a$, $b$ in $\Sh(\mathcal{C})$ lies in $E$. This is clear because the coequalizer of two morphisms $* \to *$ is $*$ and because $j_\mathcal{F}^{-1}$ is exact. Similarly for pushouts. \medskip\noindent Thus $i_*$ satisfies properties (5), (6), and (7) of Lemma \ref{lemma-exactness-properties} and hence the morphism $i$ satisfies all properties mentioned in that lemma, in particular the ones mentioned in this lemma.
We have defined $f_* = u^p$ above. In the course of the proof of Proposition \ref{proposition-get-morphism} we saw that all the colimits used to define $u_p$ are filtered under the assumptions of the proposition. Hence we conclude from the definition of a type of algebraic structure that we may define $u_p$ by exactly the same colimits as a functor on presheaves of algebraic structures. Adjointness of $u_p$ and $u^p$ is proved in exactly the same way as the proof of Lemma \ref{lemma-adjoints-u}. The discussion of sheafification of presheaves of algebraic structures above then implies that we may define $f^{-1}(\mathcal{F}) = (u_p\mathcal{F})^\#$.
By our definition of subpresheaf, the collection of all subpresheaves of a presheaf $\mathcal{F}$ is a subset of $\prod_{U \in \Ob(\mathcal{C})} \mathcal{P}(\mathcal{F}(U))$. And this is a set. (Here $\mathcal{P}(A)$ denotes the powerset of $A$.) Hence the collection of sieves on $U$ is a set. \medskip\noindent The partial ordering is defined by: $S \leq S'$ if and only if $S(T) \subset S'(T)$ for all $T \to U$. Notation: $S \subset S'$. \medskip\noindent Given a collection of sieves $S_i$, $i \in I$ on $U$ we can define $\bigcup S_i$ as the sieve with values $(\bigcup S_i)(T) = \bigcup S_i(T)$ for all $T \in \Ob(\mathcal{C})$. We define the intersection $\bigcap S_i$ in the same way. \medskip\noindent Given $\{U_i \to U\}_{i\in I}$ as in the statement, consider the morphisms of presheaves $h_{U_i} \to h_U$. We simply define $S$ as the union of the images (Definition \ref{definition-image}) of these maps of presheaves. \medskip\noindent The last two statements of the lemma are obvious.
First we make a couple of general remarks. We will use that $S^\# = LLS$, and $h_U^\# = LLh_U$. In particular, by Lemma \ref{lemma-L-presheaf}, we see that $S^\# \to h_U^\#$ is injective. Note that $\text{id}_U \in h_U(U)$. Hence it gives rise to sections of $Lh_U$ and $h_U^\# = LLh_U$ over $U$ which we will also denote $\text{id}_U$. \medskip\noindent Suppose $S$ is a covering sieve. It clearly suffices to find a morphism $h_U \to S^\#$ such that the composition $h_U \to h_U^\#$ is the canonical map. To find such a map it suffices to find a section $s \in S^\#(U)$ wich restricts to $\text{id}_U$. But since $S$ is a covering sieve, the element $\text{id}_S \in \Mor_{\textit{PSh}(\mathcal{C})}(S, S)$ gives rise to a section of $LS$ over $U$ which restricts to $\text{id}_U$ in $Lh_U$. Hence we win. \medskip\noindent Suppose that $S^\# \to h_U^\#$ is an isomorphism. Let $1 \in S^\#(U)$ be the element corresponding to $\text{id}_U$ in $h_U^\#(U)$. Because $S^\# = LLS$ there exists a covering sieve $S'$ on $U$ such that $1$ comes from a $$ \varphi \in \Mor_{\textit{PSh}(\mathcal{C})}(S', LS). $$ This in turn means that for every $\alpha : V \to U$, $\alpha\in S'(V)$ there exists a covering sieve $S_{V, \alpha}$ on $V$ such that $\varphi(\text{id}_V)$ corresponds to a morphism of presheaves $S_{V, \alpha} \to S$. In other words $S_{V, \alpha}$ is contained in $S \times_U V$. By the second axiom of a topology we see that $S$ is a covering sieve.
One direction is clear. For the other direction suppose that $\Sh(\mathcal{C}, J') \subset \Sh(\mathcal{C}, J)$. By formal nonsense this implies that if $\mathcal{F}$ is a presheaf of sets, and $\mathcal{F} \to \mathcal{F}^\#$, resp.\ $\mathcal{F} \to \mathcal{F}^{\#, \prime}$ is the sheafification wrt $J$, resp.\ $J'$ then there is a canonical map $\mathcal{F}^\# \to \mathcal{F}^{\#, \prime}$ such that $\mathcal{F} \to \mathcal{F}^\# \to \mathcal{F}^{\#, \prime}$ equals the canonical map $\mathcal{F} \to \mathcal{F}^{\#, \prime}$. Of course, $\mathcal{F}^\# \to \mathcal{F}^{\#, \prime}$ identifies the second sheaf as the sheafification of the first with respect to the topology $J'$. Apply this to the map $S \to h_U$ of Lemma \ref{lemma-sieve-sheafification}. We get a commutative diagram $$ \xymatrix{ S \ar[r] \ar[d] & S^\# \ar[r] \ar[d] & S^{\#, \prime} \ar[d] \\ h_U \ar[r] & h_U^\# \ar[r] & h_U^{\#, \prime} } $$ And clearly, if $S$ is a covering sieve for the topology $J$ then the middle vertical map is an isomorphism (by the lemma) and we conclude that the right vertical map is an isomorphism as it is the sheafification of the one in the middle wrt $J'$. By the lemma again we conclude that $S$ is a covering sieve for $J'$ as well.
We have already seen that composing the functors one way around is isomorphic to the identity functor. In the other direction, for any sheaf $\mathcal{H}$ there is a natural map of sheaves $$ can : \mathcal{H} \longrightarrow \mathcal{F}_{\mathcal{H}({}_GG)}. $$ Namely, for any object $U$ of $\mathcal{T}_G$ we let $can_U$ be the map $$ \begin{matrix} \mathcal{H}(U) & \longrightarrow & \mathcal{F}_{\mathcal{H}({}_GG)}(U) = \Mor_G(U, \mathcal{H}({}_GG)) \\ s & \longmapsto & (u \mapsto \alpha_u^*s). \end{matrix} $$ Here $\alpha_u : {}_GG \to U$ is the map $\alpha_u(g) = gu$ and $\alpha_u^* : \mathcal{H}(U) \to \mathcal{H}({}_GG)$ is the pullback map. A trivial but confusing verification shows that this is indeed a map of presheaves. We have to show that $can$ is an isomorphism. We do this by showing $can_U$ is an isomorphism for all $U \in \Ob(\mathcal{T}_G)$. We leave the (important but easy) case that $U = {}_GG$ to the reader. A general object $U$ of $\mathcal{T}_G$ is a disjoint union of $G$-orbits: $U = \coprod_{i\in I} O_i$. The family of maps $\{O_i \to U\}_{i \in I}$ is tautologically equivalent to a covering in $\mathcal{T}_G$ (by the properties of $\mathcal{T}_G$ listed at the beginning of this section). Hence by Lemma \ref{lemma-tautological-same-sheaf} the sheaf $\mathcal{H}$ satisfies the sheaf property with respect to $\{O_i \to U\}_{i \in I}$. The sheaf property for this covering implies $\mathcal{H}(U) = \prod_i \mathcal{H}(O_i)$. Hence it suffices to show that $can_U$ is an isomorphism when $U$ consists of a single $G$-orbit. Let $u \in U$ and let $H \subset G$ be its stabilizer. Clearly, $\Mor_G(U, \mathcal{H}({}_GG)) = \mathcal{H}({}_GG)^H$ equals the subset of $H$-invariant elements. On the other hand consider the covering $\{{}_GG \to U\}$ given by $g \mapsto gu$ (again it is just combinatorially equivalent to some covering of $\mathcal{T}_G$, and again this doesn't matter). Note that the fibre product $({}_GG)\times_U ({}_GG)$ is equal to $\{(g, gh), g\in G, h\in H\} \cong \coprod_{h\in H} {}_GG$. Hence the sheaf property for this covering reads as $$ \xymatrix{ \mathcal{H}(U) \ar[r] & \mathcal{H}({}_GG) \ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*} & \prod_{h \in H} \mathcal{H}({}_GG). } $$ The two maps $\text{pr}_i^*$ into the factor $\mathcal{H}({}_GG)$ differ by multiplication by $h$. Now the result follows from this and the fact that $can$ is an isomorphism for $U = {}_GG$.
By Lemma \ref{lemma-plus-functorial} we get a commutative diagram $$ \xymatrix{ \mathcal{F} \ar[r] \ar[d] & \mathcal{F}^{+} \ar[r] \ar[d] & \mathcal{F}^{++} \ar[d] \\ \mathcal{G} \ar[r] & \mathcal{G}^{+} \ar[r] & \mathcal{G}^{++} } $$ and by Theorem \ref{theorem-plus} the lower horizontal maps are isomorphisms. The uniqueness follows from Lemma \ref{lemma-plus-surjective} which says that every section of $\mathcal{F}^\#$ locally comes from sections of $\mathcal{F}$.
Suppose first that $p$ is a point (Definition \ref{definition-point}) given by a functor $u$. Condition (2) is satisfied directly from the definition of a point. By Lemma \ref{lemma-points-recover} we have $(h_U)_p = u(U)$. By Lemma \ref{lemma-point-pushforward-sheaf} we have $(h_U^\#)_p = (h_U)_p$. Thus we see that $u$ is equal to the composition of functors $$ \mathcal{C} \xrightarrow{h} \textit{PSh}(\mathcal{C}) \xrightarrow{{}^\#} \Sh(\mathcal{C}) \xrightarrow{()_p} \textit{Sets} $$ Each of these functors is left exact, and hence we see $u$ satisfies (1). \medskip\noindent Conversely, suppose that $u$ satisfies (1) and (2). In this case we immediately see that $u$ satisfies the first two conditions of Definition \ref{definition-point}. And its stalk functor is exact, because it is a left adjoint by Lemma \ref{lemma-point-pushforward-sheaf} and it commutes with finite limits by Lemma \ref{lemma-neighbourhoods-directed}.
We have to show that given any sheaf $\mathcal{F}$ on $\mathcal{C}$, any $U \in \Ob(\mathcal{C})$, and any distinct sections $s, s' \in \mathcal{F}(U)$, there exists a point $p$ such that $s, s'$ have distinct image in $\mathcal{F}_p$. See Lemma \ref{lemma-enough}. Consider the system $(J, \geq, V_j, g_{jj'})$ with $J = \{1\}$, $V_1 = U$, $g_{11} = \text{id}_U$. Apply Lemma \ref{lemma-refine-all-at-once}. By the result of that lemma we get a system $(I, \geq, U_i, f_{ii'})$ refining our system such that $s_p \not = s'_p$ and such that moreover for every finite covering $\{W_k \to W\}$ of the site $\mathcal{C}$ the map $\coprod_k u(W_k) \to u(W)$ is surjective. Since every covering of $\mathcal{C}$ can be refined by a finite covering we conclude that $\coprod_k u(W_k) \to u(W)$ is surjective for {\it any} covering $\{W_k \to W\}$ of the site $\mathcal{C}$. This implies that $u = p$ is a point, see Proposition \ref{proposition-point-limits} (and the discussion at the beginning of this section which guarantees that $u$ commutes with finite limits).
The final statement of the proposition holds simply since each of the listed categories, endowed with the obvious forgetful functor, is indeed a type of algebraic structure in the sense explained at the beginning of this section. See Sheaves, Lemma \ref{sheaves-lemma-list-algebraic-structures}. \medskip\noindent Proof of (2). We think of a sheaf of abelian groups as a quadruple $(\mathcal{F}, +, 0, i)$ as explained in the discussion preceding the proposition. If $(\mathcal{F}, +, 0, i)$ lives on $\mathcal{C}$, then its pullback is defined as $(f^{-1}\mathcal{F}, f^{-1}+, f^{-1}0, f^{-1}i)$. If $(\mathcal{G}, +, 0, i)$ lives on $\mathcal{D}$, then its pushforward is defined as $(f_*\mathcal{G}, f_*+, f_*0, f_*i)$. This works because $f_*(\mathcal{G} \times \mathcal{G}) = f_*\mathcal{G} \times f_*\mathcal{G}$. Adjointness follows from adjointness of the set based functors, since $$ \Mor_{\textit{Ab}(\mathcal{C})} ((\mathcal{F}_1, +, 0, i), (\mathcal{F}_2, +, 0, i)) = \left\{ \begin{matrix} \varphi \in \Mor_{\Sh(\mathcal{C})} (\mathcal{F}_1, \mathcal{F}_2) \\ \varphi \text{ is compatible with }+, 0, i \end{matrix} \right\} $$ Details left to the reader. \medskip\noindent This method also works for sheaves of rings by thinking of a sheaf of rings (with unit) as a sixtuple $(\mathcal{O}, + , 0, i, \cdot, 1)$ satisfying a list of axioms that you can find in any elementary algebra book. \medskip\noindent A sheaf of pointed sets is a pair $(\mathcal{F}, p)$, where $\mathcal{F}$ is a sheaf of sets, and $p : * \to \mathcal{F}$ is a map of sheaves of sets. \medskip\noindent A sheaf of groups is given by a quadruple $(\mathcal{F}, \cdot, 1, i)$ with suitable axioms. \medskip\noindent A sheaf of monoids is given by a pair $(\mathcal{F}, \cdot)$ with suitable axiom. \medskip\noindent Let $R$ be a ring. An sheaf of $R$-modules is given by a quintuple $(\mathcal{F}, +, 0, i, \{\lambda_r\}_{r \in R})$, where the quadruple $(\mathcal{F}, +, 0, i)$ is a sheaf of abelian groups as above, and $\lambda_r : \mathcal{F} \to \mathcal{F}$ is a family of morphisms of sheaves of sets such that $\lambda_r \circ 0 = 0$, $\lambda_r \circ + = + \circ (\lambda_r, \lambda_r)$, $\lambda_{r + r'} = + \circ \lambda_r \times \lambda_{r'} \circ (\text{id}, \text{id})$, $\lambda_{rr'} = \lambda_r \circ \lambda_{r'}$, $\lambda_1 = \text{id}$, $\lambda_0 = 0 \circ (\mathcal{F} \to *)$.
Let $A_1 \to A$ and $A_2 \to A$ be morphisms of $\mathcal{C}_\Lambda$. Assume $A_2 \to A$ is surjective. According to Formal Deformation Theory, Lemma \ref{formal-defos-lemma-RS-2-categorical} it suffices to show that the functor $\mathcal{F}(A_1 \times_A A_2) \to \mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)$ is an equivalence of categories. \medskip\noindent Thus we have to show that the category of finite projective modules over $A_1 \times_A A_2$ is equivalent to the fibre product of the categories of finite projective modules over $A_1$ and $A_2$ over the category of finite projective modules over $A$. This is a special case of More on Algebra, Lemma \ref{more-algebra-lemma-finitely-presented-module-over-fibre-product}. We recall that the inverse functor sends the triple $(M_1, M_2, \varphi)$ where $M_1$ is a finite projective $A_1$-module, $M_2$ is a finite projective $A_2$-module, and $\varphi : M_1 \otimes_{A_1} A \to M_2 \otimes_{A_2} A$ is an isomorphism of $A$-module, to the finite projective $A_1 \times_A A_2$-module $M_1 \times_\varphi M_2$.
With $\mathcal{F}$ as in Example \ref{example-finite-projective-modules} set $x_0 = (k, V) \in \Ob(\mathcal{F}(k))$. Recall that $T\Deformationcategory_V = T_{x_0}\mathcal{F}$ is the set of isomorphism classes of pairs $(x, \alpha)$ consisting of an object $x$ of $\mathcal{F} $ over the dual numbers $k[\epsilon]$ and a morphism $\alpha : x \to x_0$ of $\mathcal{F}$ lying over $k[\epsilon] \to k$. \medskip\noindent Up to isomorphism, there is a unique pair $(M, \alpha)$ consisting of a finite projective module $M$ over $k[\epsilon]$ and $k[\epsilon]$-linear map $\alpha : M \to V$ which induces an isomorphism $M \otimes_{k[\epsilon]} k \to V$. For example, if $V = k^{\oplus n}$, then we take $M = k[\epsilon]^{\oplus n}$ with the obvious map $\alpha$. \medskip\noindent Similarly, $\text{Inf}(\Deformationcategory_V) = \text{Inf}_{x_0}(\mathcal{F})$ is the set of automorphisms of the trivial deformation $x'_0$ of $x_0$ over $k[\epsilon]$. See Formal Deformation Theory, Definition \ref{formal-defos-definition-infinitesimal-auts} for details. \medskip\noindent Given $(M, \alpha)$ as in the second paragraph, we see that an element of $\text{Inf}_{x_0}(\mathcal{F})$ is an automorphism $\gamma : M \to M$ with $\gamma \bmod \epsilon = \text{id}$. Then we can write $\gamma = \text{id}_M + \epsilon \psi$ where $\psi : M/\epsilon M \to M/\epsilon M$ is $k$-linear. Using $\alpha$ we can think of $\psi$ as an element of $\text{End}_k(V)$ and this finishes the proof.
Let $A_1 \to A$ and $A_2 \to A$ be morphisms of $\mathcal{C}_\Lambda$. Assume $A_2 \to A$ is surjective. According to Formal Deformation Theory, Lemma \ref{formal-defos-lemma-RS-2-categorical} it suffices to show that the functor $\mathcal{F}(A_1 \times_A A_2) \to \mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)$ is an equivalence of categories. \medskip\noindent Consider an object $$ ((A_1, M_1, \rho_1), (A_2, M_2, \rho_2), (\text{id}_A, \varphi)) $$ of the category $\mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)$. Then, as seen in the proof of Lemma \ref{lemma-finite-projective-modules-RS}, we can consider the finite projective $A_1 \times_A A_2$-module $M_1 \times_\varphi M_2$. Since $\varphi$ is compatible with the given actions we obtain $$ \rho_1 \times \rho_2 : \Gamma \longrightarrow \text{GL}_{A_1 \times_A A_2}(M_1 \times_\varphi M_2) $$ Then $(M_1 \times_\varphi M_2, \rho_1 \times \rho_2)$ is an object of $\mathcal{F}(A_1 \times_A A_2)$. This construction determines a quasi-inverse to our functor.
We first deal with the infinitesimal automorphisms. Let $M = V \otimes_k k[\epsilon]$ with induced action $\rho_0' : \Gamma \to \text{GL}_n(M)$. Then an infinitesimal automorphism, i.e., an element of $\text{Inf}(\Deformationcategory_{V, \rho_0})$, is given by an automorphism $\gamma = \text{id} + \epsilon \psi : M \to M$ as in the proof of Lemma \ref{lemma-finite-projective-modules-TI}, where moreover $\psi$ has to commute with the action of $\Gamma$ (given by $\rho_0$). Thus we see that $$ \text{Inf}(\Deformationcategory_{V, \rho_0}) = H^0(\Gamma, \text{End}_k(V)) $$ as predicted in the lemma. \medskip\noindent Next, let $(k[\epsilon], M, \rho)$ be an object of $\mathcal{F}$ over $k[\epsilon]$ and let $\alpha : M \to V$ be a $\Gamma$-equivariant map inducing an isomorphism $M/\epsilon M \to V$. Since $M$ is free as a $k[\epsilon]$-module we obtain an extension of $\Gamma$-modules $$ 0 \to V \to M \xrightarrow{\alpha} V \to 0 $$ We omit the detailed construction of the map on the left. Conversely, if we have an extension of $\Gamma$-modules as above, then we can use this to make a $k[\epsilon]$-module structure on $M$ and get an object of $\mathcal{F}(k[\epsilon])$ together with a map $\alpha$ as above. It follows that $$ T\Deformationcategory_{V, \rho_0} = \Ext^1_{k[\Gamma]}(V, V) $$ as predicted in the lemma. This is equal to $H^1(\Gamma, \text{End}_k(V))$ by \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-ext-modules-hom}. \medskip\noindent The statement on dimensions follows from \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-finite-dim-group-cohomology}.
The existence of a hull follows from Lemmas \ref{lemma-representations-RS} and \ref{lemma-representations-TI} and Formal Deformation Theory, Lemma \ref{formal-defos-lemma-RS-implies-S1-S2} and Remark \ref{formal-defos-remark-compose-minimal-into-iso-classes}. \medskip\noindent Assume $H^0(\Gamma, \text{End}_k(V)) = k$. To see that $F$ is prorepresentable it suffices to show that $F$ is a deformation functor, see Formal Deformation Theory, Theorem \ref{formal-defos-theorem-Schlessinger-prorepresentability}. In other words, we have to show $F$ satisfies (RS). For this we can use the criterion of Formal Deformation Theory, Lemma \ref{formal-defos-lemma-RS-associated-functor}. The required surjectivity of automorphism groups will follow if we show that $$ A \cdot \text{id}_M = \text{End}_{A[\Gamma]}(M) $$ for any object $(A, M, \rho)$ of $\mathcal{F}$ such that $M \otimes_A k$ is isomorphic to $V$ as a representation of $\Gamma$. Since the left hand side is contained in the right hand side, it suffices to show $\text{length}_A \text{End}_{A[\Gamma]}(M) \leq \text{length}_A A$. Choose pairwise distinct ideals $(0) = I_n \subset \ldots \subset I_1 \subset A$ with $n = \text{length}(A)$. By correspondingly filtering $M$, we see that it suffices to prove $\Hom_{A[\Gamma]}(M, I_tM/I_{t + 1}M)$ has length $1$. Since $I_tM/I_{t + 1}M \cong M \otimes_A k$ and since any $A[\Gamma]$-module map $M \to M \otimes_A k$ factors uniquely through the quotient map $M \to M \otimes_A k$ to give an element of $$ \text{End}_{A[\Gamma]}(M \otimes_A k) = \text{End}_{k[\Gamma]}(V) = k $$ we conclude.
Let $A_1 \to A$ and $A_2 \to A$ be morphisms of $\mathcal{C}_\Lambda$. Assume $A_2 \to A$ is surjective. According to Formal Deformation Theory, Lemma \ref{formal-defos-lemma-RS-2-categorical} it suffices to show that the functor $\mathcal{F}(A_1 \times_A A_2) \to \mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)$ is an equivalence of categories. \medskip\noindent Consider an object $$ ((A_1, P_1), (A_2, P_2), (\text{id}_A, \varphi)) $$ of the category $\mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)$. Then we consider $P_1 \times_\varphi P_2$. Since $\varphi : P_1 \otimes_{A_1} A \to P_2 \otimes_{A_2} A$ is an isomorphism of graded algebras, we see that the graded pieces of $P_1 \times_\varphi P_2$ are finite projective $A_1 \times_A A_2$-modules, see proof of Lemma \ref{lemma-finite-projective-modules-RS}. Thus $P_1 \times_\varphi P_2$ is an object of $\mathcal{F}(A_1 \times_A A_2)$. This construction determines a quasi-inverse to our functor and the proof is complete.
Let $A_1 \to A$ and $A_2 \to A$ be morphisms of $\mathcal{C}_\Lambda$. Assume $A_2 \to A$ is surjective. According to Formal Deformation Theory, Lemma \ref{formal-defos-lemma-RS-2-categorical} it suffices to show that the functor $\mathcal{F}(A_1 \times_A A_2) \to \mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)$ is an equivalence of categories. This is a special case of More on Algebra, Lemma \ref{more-algebra-lemma-properties-algebras-over-fibre-product}.
Recall that $\text{Inf}(\Deformationcategory_P)$ is the set of automorphisms of the trivial deformation $P[\epsilon] = P \otimes_k k[\epsilon]$ of $P$ to $k[\epsilon]$ equal to the identity modulo $\epsilon$. By Deformation Theory, Lemma \ref{defos-lemma-huge-diagram} this is equal to $\Hom_P(\Omega_{P/k}, P)$ which in turn is equal to $\text{Der}_k(P, P)$ by Algebra, Lemma \ref{algebra-lemma-universal-omega}. \medskip\noindent Recall that $T\Deformationcategory_P$ is the set of isomorphism classes of flat deformations $Q$ of $P$ to $k[\epsilon]$, more precisely, the set of isomorphism classes of $\Deformationcategory_P(k[\epsilon])$. Recall that a $k[\epsilon]$-algebra $Q$ with $Q/\epsilon Q = P$ is flat over $k[\epsilon]$ if and only if $$ 0 \to P \xrightarrow{\epsilon} Q \to P \to 0 $$ is exact. This is proven in More on Morphisms, Lemma \ref{more-morphisms-lemma-deform} and more generally in Deformation Theory, Lemma \ref{defos-lemma-deform-module}. Thus we may apply Deformation Theory, Lemma \ref{defos-lemma-choices} to see that the set of isomorphism classes of such deformations is equal to $\text{Ext}^1_P(\NL_{P/k}, P)$.