Proof. The assumptions of [Reference Lurie23, Proposition A.3.2.1] are satisfied, since coproducts in $\operatorname {\mathrm {Aff}}$ are universal and since flat maps in $\operatorname {\mathrm {Aff}}$ are stable under composition and under base-change along arbitrary maps.

Proof. This is [Reference Lurie23, Proposition A.3.3.1].

Proof. In the case of ordinary rings, this is proved in [Reference Waterhouse35, Lemma 3.1]. We recall the argument for the convenience of the reader and to verify that that it works without much change in the Dirac setting.

We first recall the necessary commutative algebra. By the equational criterion for flatness [Reference Hesselholt and Pstrągowski12, Theorem 3.8], a map of Dirac rings $f \colon A \to B$ is flat if and only if for every system of linear equations

(2.11) $$ \begin{align} \textstyle{\sum_{1 \leq k \leq n} y_kc_{ki} = 0 \hspace{12mm} (1 \leq i \leq m), } \end{align} $$

where the $c_{ki}$ are homogeneous elements of A, every solution $(y_k)_{1 \leq k \leq n}$ , consisting of homogeneous elements of B, is a linear combination

$$ \begin{align*}\textstyle{ y_k = \sum_{1 \leq j \leq p} b_jz_{jk} \hspace{12mm} (1 \leq k \leq n), }\end{align*} $$

where the $b_j$ are homogeneous elements of B, of solutions $(z_{jk})_{1 \leq k \leq n}$ to (2.11) consisting of homogeneous elements $z_{jk}$ of A.

By the equational criterion for faithful flatness, which we proved in [Reference Hesselholt and Pstrągowski12, Addendum 3.9], a flat map of Dirac rings $f \colon A \to B$ is faithfully flat if and only if it is a monomorphism and for every solution $(y_k)_{1 \leq k \leq n}$ consisting of homogeneous elements of B to a system of linear equations

(2.12) $$ \begin{align} \textstyle{\sum_{1 \leq k \leq n} y_kc_{ki} = d_i \hspace{12mm} (1 \leq i \leq m), } \end{align} $$

where the $c_{ki}$ and $d_i$ are homogeneous elements of A, there exists a solution $(x_k)_{1 \leq k \leq n}$ consisting of homogeneous elements of A.

Now, in order to prove the lemma, we let $f \colon A \to B$ and $g' \colon C' \to D'$ be as in the statement. We note that, as a consequence of [Reference Hesselholt and Pstrągowski12, Addendum 3.23], the map $f \colon A \to B$ necessarily is a monomorphism, and therefore, we may assume that it is the inclusion of a sub-Dirac ring.

We will define subextensions $g_n \colon C_n \to D_n$ of $f \colon A \to B$ recursively for $n \geq 0$ . We define $g_0 = g'$ , and assuming that $g_n$ has been defined for $n < r$ , we define $g_r$ as follows: We consider all systems of equations of the form (2.11) and (2.12) with $c_{ki}$ and $d_i$ homogeneous elements in $C_{r-1}$ . Since $f \colon A \to B$ is flat, every solution $(y_k)$ , consisting of homogeneous elements of B, to such a system of equations of the first kind can be written as a linear combination $y_k = \sum b_jz_{jk}$ with the $b_j$ and $z_{jk}$ homogeneous elements of B and A, respectively. Similarly, since $f \colon A \to B$ is faithfully flat, for every solution $(y_k)$ , consisting of homogeneous elements of B, to a system of equations of the second kind, there exists a solution $(x_k)$ to the same system of equations consisting of homogeneous elements of A. We now define $D_r \subset B$ to be the sub-Dirac ring obtained by adjoining to $D_{r-1}$ all the homogeneous elements $b_j$ , $z_{jk}$ , and $x_k$ of B obtained in this way and define $g_r \colon C_r \to D_r$ to be the inclusion of $C_r = A \cap D_r$ . This completes the definition of the subextensions $g_n \colon C_n \to D_n$ for $n \geq 0$ .

We claim that their union $g \colon C \to D$ is the desired subextension of $f \colon A \to B$ . Indeed, it is faithfully flat as both the above criteria hold for it by construction, and since $\kappa $ is uncountable, the Dirac rings C and D are $\kappa $ -small.

Proof. The first part is a restatement of Lemma 2.10. The second part is a consequence of the first part, since $\operatorname {\mathrm {SubExt}}_{\kappa }(f)$ is $\kappa $ -filtered as it is a poset which admits $\kappa $ -small upper bounds.

Proof. Using Joyal’s version of Quillen’s Theorem A, [Reference Clausen and Jansen8, Theorem 2.19], we must show that for any $k \colon R \to B^{\otimes _{A} [n]}$ with R a $\kappa $ -small Dirac ring, the slice category

$$ \begin{align*}\operatorname{\mathrm{SubExt}}_{\kappa}(f)_{k/}\end{align*} $$

is weakly contractible. Explicitly, the objects of this category are pairs

$$ \begin{align*}(g \colon C \to D, h \colon R \to D^{\otimes_{C} [n]})\end{align*} $$

of a $\kappa $ -small subextension of $f \colon A \to B$ and a map h such that the composite

$$ \begin{align*}R \to D^{\otimes_{C} [n]} \to B^{\otimes_{A} [n]}\end{align*} $$

coincides with k. Maps are inclusions of subextensions that commute with the given maps from R. In particular, the category in question is a poset. We will show that it is a $\kappa $ -filtered, which implies that it is weakly contractible.

Since we are considering a poset, it suffices to show that every subset

$$ \begin{align*}S = \{(C_i \xrightarrow{\;g_i\;} D_i, R \xrightarrow{\;h_i\;} D_i^{\otimes _{C_i}[n]}) \mid i \in I\}\end{align*} $$

indexed by a $\kappa $ -small index set I has an upper bound. By first choosing an upper bound $g \colon C \to D$ of the subset

$$ \begin{align*}S' = \{C_i \xrightarrow{\;g_i\;} D_i \mid i \in I\} \subset \operatorname{\mathrm{SubExt}}_{\lambda}(f),\end{align*} $$

we can assume that $C_i = C$ and $D_i = D$ for all $i \in I$ .

With this accomplished, we consider the canonical comparison map

where the colimits are indexed by the partially ordered set $\operatorname {\mathrm {SubExt}}_{\kappa }(f)$ . Since this partially ordered set is $\kappa $ -filtered, and since I is $\kappa $ -small, the map in question is an equivalence by [Reference Lurie20, Proposition 5.3.3.3]. Similarly, the map

is an equivalence, because R is $\kappa $ -small, and therefore a $\kappa $ -compact object in the category of Dirac rings. Finally, the map

is an equivalence since $A \simeq \varinjlim C$ and $B \simeq \varinjlim D$ and since relative tensor products preserve filtered colimits in all three variables.

Considering the composite of these maps, we conclude that there exists a $\kappa $ -small subextension $g' \colon C' \to D'$ of $f \colon A \to B$ that contains $g \colon C \to D$ as a subextension and a map of Dirac rings $h \colon R \to D'{}^{\otimes _{C'}[n]}$ such that for all $i \in I$ , the map

is equal to the map h. Thus, the pair $(g' \colon C' \to D', h \colon R \to D'{}^{\otimes _{C'}[n]})$ is the desired upper bound of the subset S.

Proof. We extend Waterhouse’s argument in [Reference Waterhouse35, Theorem 5.1] to the case of sheaves of anima. So we wish to show that if $X \in {\mathcal {P}}({\mathrm {Aff}}_{\lambda })$ satisfies the hypotheses (i)–(ii) of Proposition 2.5, then so does $i_!(X) \in {\mathcal {P}}(\operatorname {\mathrm {Aff}})$ .

First, if X satisfies (i), then by [Reference Lurie20, Lemma 5.5.8.14], it can be written as a sifted colimit of representable presheaves. Moreover, since $i_!$ preserves colimits and takes representable presheaves to representable presheaves, we deduce that also $i_!(X)$ can be written as a sifted colimit of representable presheaves and hence preserves finite products by [Reference Lurie20, Proposition 5.5.8.10].

Second, suppose that X satisfies (ii) and let $f \colon A \to B$ be a faithfully flat map of Dirac rings. We must show that the induced map of anima

is an equivalence. Now, for any $[n] \in \Delta _+$ , the left Kan extension is given by

$$ \begin{align*}\textstyle{ i_!(X)(\operatorname{\mathrm{Spec}}(B^{\otimes_A[n]})) \simeq \varinjlim X(\operatorname{\mathrm{Spec}}(R)), }\end{align*} $$

where the colimit is indexed by $R \to B^{\otimes _A[n]}$ in $(({\mathrm {Aff}}_{\lambda })^{\operatorname {\mathrm {op}}})_{/B^{\otimes _A[n]}}$ , and a combination of Corollary 2.13 and Lemma 2.14 shows that the functor

that to the subextension $g \colon C \to D$ of $f \colon A \to B$ assigns $D^{\otimes _C[n]} \to B^{\otimes _A[n]}$ is a $\smash { \varinjlim }$ -equivalence. Hence, we can identify the map in question with the map

with the colimits indexed by $\operatorname {\mathrm {SubExt}}_{\lambda }^{{\operatorname {f}\!\operatorname {f}}}(f)$ . Since the latter is $\lambda $ -filtered, and since $\lambda $ is uncountable, we can commute the limit past the colimit, and therefore, it will suffice to show that the map

is an equivalence. However, since every $g \colon C \to D$ in $\operatorname {\mathrm {SubExt}}_{\lambda }^{{\operatorname {f}\!\operatorname {f}}}(f)$ is faithfully flat, and since X satisfies (ii), this map is a colimit of equivalences and hence is itself an equivalence. This shows that $i_!(X)$ also satisfies (ii) as desired.

Proof. Given $X \in {\mathcal {P}}(\operatorname {\mathrm {Aff}})$ , by Proposition A.2, we can write $X \simeq i_!(X_0)$ for some cardinal $\lambda $ and $X_0 \in {\mathcal {P}}({\mathrm {Aff}}_{\lambda })$ . Since ${\mathrm {Aff}}_{\lambda }$ is essentially small, the inclusion

$$\begin{align*}\iota_{\lambda} \colon \operatorname{\mathrm{Shv}}({\mathrm{Aff}}_{\lambda}) \to {\mathcal{P}}({\mathrm{Aff}}_{\lambda}) \end{align*}$$

admits a left adjoint sheafification functor $L_{\lambda } \colon {\mathcal {P}}({\mathrm {Aff}}_{\lambda }) \to \operatorname {\mathrm {Shv}}({\mathrm {Aff}}_{\lambda })$ because its domain and target are both presentable. Now, we claim that the map

is a sheafification. Indeed, by Theorem 2.15, the target is the underlying presheaf of a sheaf, and for every $Y \in \operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ , we have

$$ \begin{align*}\begin{aligned} {} & \operatorname{\mathrm{Map}}(i_!\iota_{\lambda}L_{\lambda}(X_0),\iota(Y)) \simeq \operatorname{\mathrm{Map}}(\iota_{\lambda}L_{\lambda}(X_0),i^*\iota(Y)) \cr {} & \quad \simeq \operatorname{\mathrm{Map}}(\iota_{\lambda}L_{\lambda}(X_0),\iota_{\lambda}i^*(Y)) \simeq \operatorname{\mathrm{Map}}(L_{\lambda}(X_0),i^*(Y)) \simeq \operatorname{\mathrm{Map}}(X_0,\iota_{\lambda}i^*(Y)) \cr {} & \quad \simeq \operatorname{\mathrm{Map}}(X_0,i^*\iota(Y)) \simeq \operatorname{\mathrm{Map}}(i_!(X_0),\iota(Y)) \simeq \operatorname{\mathrm{Map}}(X,\iota(Y)), \cr \end{aligned}\end{align*} $$

as desired. Hence, we conclude from [Reference Lurie17, Tag 02FV] that the desired left adjoint functor L exists. It preserves finite limits because each $L_{\lambda }$ does so.

Proof. Corollary 2.17 shows that $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ is a left exact localization of ${\mathcal {P}}(\operatorname {\mathrm {Aff}})$ , so completeness, cocompletness, the fact that $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ is generated under small colimits by the Yoneda image, and parts (ii)–(iv) follow from Theorem A.10. It remains to prove that if S is an affine Dirac scheme, then $h(S)$ is $\omega _1$ -compact. Now, for every Dirac stack X, we have

$$ \begin{align*}\operatorname{\mathrm{Map}}(h(S),X) \simeq X(S),\end{align*} $$

so the claim that $h(S)$ is $\omega _1$ -compact is equivalent to the statement that $\omega _1$ -filtered colimits in $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ are calculated pointwise. By Proposition A.9, all small colimits in ${\mathcal {P}}(\operatorname {\mathrm {Aff}})$ are calculated pointwise, so it suffices to show that $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}}) \subset {\mathcal {P}}(\operatorname {\mathrm {Aff}})$ is closed under $\omega _1$ -filtered colimits. But by Proposition 2.5, the sheaf condition is expressed in terms of $\omega _1$ -small limits, so the pointwise colimit of an $\omega _1$ -filtered diagram of sheaves is again a sheaf because $\omega _1$ -small limits and $\omega _1$ -filtered colimits of anima commute by [Reference Lurie20, Proposition 5.3.3.3], as $\omega _1$ is a regular cardinal.

Proof. Suppose that $X \simeq \varinjlim _{k \in K}X_k$ is a colimit of a small diagram in $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ . We wish to prove that the canonical map

is an equivalence. As in Remark 2.16, we may write

$$ \begin{align*}\textstyle{ \operatorname{\mathrm{Shv}}(\operatorname{\mathrm{Aff}}) \simeq \varinjlim_{\lambda \in L} \operatorname{\mathrm{Shv}}({\mathrm{Aff}}_{\lambda}) }\end{align*} $$

as the filtered colimit indexed by the partially ordered set L consisting of the small uncountable regular cardinals $\lambda $ for which the diagram in equation factors through $\operatorname {\mathrm {Shv}}({\mathrm {Aff}}_{\lambda }) \subset \operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ . Now, for every $\lambda \in L$ , $\operatorname {\mathrm {Shv}}({\mathrm {Aff}}_{\lambda })$ is an $\infty $ -topos, so

is an equivalence by [Reference Lurie20, Theorem 6.1.3.9]. Therefore, also the induced map of colimits

is an equivalence, so it remains to argue that the canonical map

is an equivalence. But there exists a regular cardinal $\kappa $ such that K is $\kappa $ -small and L is $\kappa $ -filtered, so this follows from [Reference Lurie20, Proposition 5.3.3.3] by the same argument as in the proof of [Reference Lurie23, Lemma 4.5.3.1].

Proof. The presheaf $h(X)$ is representable, and hence accessible, as the $\infty $ -category $\operatorname {\mathrm {Aff}}$ is coaccessible. The fact that $h(X)$ is a sheaf for flat topology is a restatement of faithfully flat descent, which we proved in [Reference Hesselholt and Pstrągowski12, Addendum 3.23].

Proof. We must verify that $h(X)$ satisfies conditions (i) and (ii) of Proposition 2.5. In the case of (i), there is nothing to prove, and in the case of (ii), we must show that if $h \colon T \to S$ is a faithfully flat map of affine Dirac schemes, then

is an equivalence of anima. Now, since the presheaf $h(X)$ takes values in the full subcategory ${\mathcal {S}}_{\leq 0} \subset {\mathcal {S}}$ of $0$ -truncated anima, the restriction

is an equivalence by [Reference Hesselholt and Pstrągowski12, Proposition A.1]. So it suffices to show that

is a limit diagram of sets.

We first verify that $h^*$ is injective. So we let $f_1, f_2 \colon S \to X$ be maps of Dirac schemes with $f_1h = f_2h \colon T \to X$ . Since h induces a surjective map of underlying topological spaces by [Reference Hesselholt and Pstrągowski12, Proposition 3.15], we deduce that $f_1$ and $f_2$ have the same underlying map $p \colon |S| \to |X|$ of topological spaces. So we have $f_i = (p,\phi _i)$ and wish to show that the maps $\phi _1, \phi _2 \colon p^*\mathcal {O}_{X} \to \mathcal {O}_S$ of sheaves of Dirac rings coincide. This can be checked stalkwise. So we write $h = (q,\psi )$ , let $s \in |X|$ with $x = p(x)$ and choose $t \in |T|$ such that $s = q(t)$ , where we use that $q \colon |T| \to |S|$ is surjective. Now, by the assumption that $f_1h = f_2h$ , the two composites

coincide. But the latter map is a flat local homomorphism of local Dirac rings, and therefore, it is injective by [Reference Hesselholt and Pstrągowski12, Proposition 3.16]. This shows that $\phi _{1,x} = \phi _{2,x}$ and hence that $f_1 = f_2$ , as desired.

Next, we show that $g \colon T \to X$ with $gd_0 = gd_1 \colon T \times _ST \to X$ factors through a map $f \colon S \to X$ . Such a factorization is necessarily unique, by what was proved above. We choose a covering $(W_i)_{i \in I}$ of X by affine open subschemes and consider the covering $(V_i)_{i \in I}$ of T by the open subschemes $V_i \simeq T \times _XW_i \to T$ . Since

$$ \begin{align*}|V_i \times_ST| = |T \times_SV_i| \subset |T \times_ST|,\end{align*} $$

we conclude from [Reference Hesselholt and Pstrągowski12, Proposition 3.24] that there exists a unique covering $(U_i)_{i \in I}$ of S by open subschemes such that $V_i \simeq T \times _SU_i \to T$ for all $i \in I$ . We observe that any $f \colon S \to X$ with the desired property necessarily will satisfy $f(U_i) \subset W_i$ . Thus, it suffices to show that for every $i \in I$ , there exists a map of Dirac schemes $f_i \colon U_i \to W_i$ with the property that the composite map

is equal to $g|_{V_i} \colon V_i \to X$ .

Now, for every $i \in I$ , we choose a covering $(U_{i,j})_{j \in J_i}$ of $U_i$ by affine open subschemes and note that also $(V_{i,j})_{j \in J_i}$ with $V_{i,j} \simeq T \times _SU_{i,j} \to T$ is a covering of $V_i$ by affine open subschemes. Moreover, the map $h|_{V_{i,j}} \colon V_{i,j} \to U_{i,j}$ is a faithfully flat map of affine Dirac schemes, and we have a map

with the property its restriction along the two projections

coincide. As everything is affine, we conclude from Lemma 2.23 that $g|_{V_{i,j}}$ admits a unique factorization through a map $f_{i,j} \colon U_{i,j} \to W_i$ . By uniqueness, the maps $f_{i,j} \colon U_{i,j} \to W_i$ glue to give the desired $f_i \colon U_i \to W_i$ . This completes the proof.

Proof. We know from Proposition 2.24 that $h(X) \in \operatorname {\mathrm {Fun}}(\operatorname {\mathrm {Aff}}^{\mathrm {op}},{\mathcal {S}})$ is a sheaf for the flat topology, and we wish to prove that it belongs to the full subcategory

$$ \begin{align*}\textstyle{ \varinjlim_{\kappa}\operatorname{\mathrm{Shv}}({\mathrm{Aff}}_{\kappa}) \simeq \operatorname{\mathrm{Shv}}(\operatorname{\mathrm{Aff}}) \subset \operatorname{\mathrm{Fun}}(\operatorname{\mathrm{Aff}}^{\mathrm{op}},{\mathcal{S}}). }\end{align*} $$

Since the inclusions $i_{\kappa } \colon {\mathrm {Aff}}_{\kappa } \to \operatorname {\mathrm {Aff}}$ exhibit $\operatorname {\mathrm {Aff}}$ as a colimit of the ${\mathrm {Aff}}_{\kappa }$ , it suffices to show that there exists a $\lambda $ such that for all $\kappa \geq \lambda $ , the diagram

exhibits $i_{\kappa }^*h(X)$ as a left Kan extension of $i_{\lambda }^*h(X)$ . Now, each $h(U)$ is accessible and $\mathfrak {B}$ is small, so we can find a $\lambda $ such that for all $\kappa \geq \lambda $ , the diagram

exhibits $i_{\kappa }^*h(U)$ as a left Kan extension of $i_{\lambda }^*h(U)$ . Since left Kan extension preserves sheaves by Theorem 2.15, we conclude that it will suffice to show that the map

in $\operatorname {\mathrm {Shv}}({\mathrm {Aff}}_{\lambda })$ is an equivalence.

To prove that this is so, let us write $h_{\lambda } \simeq i_{\lambda }^*h$ . Since, locally on S, every map $\eta \colon S \to X$ from an affine Dirac scheme factors through some $U \in \mathfrak {B}$ , the map

in $\operatorname {\mathrm {Shv}}({\mathrm {Aff}}_{\lambda })$ is an effective epimorphism. Therefore, it will suffice to show that the base-change

of the map in question along the map induced by the open immersion of any $U_0 \in \mathfrak {B}$ into X is an equivalence. Since colimits in $\operatorname {\mathrm {Shv}}({\mathrm {Aff}}_{\lambda })$ are universal, and since the Yoneda embedding preserves limits, the canonical map

in $\operatorname {\mathrm {Shv}}({\mathrm {Aff}}_{\lambda })$ is an equivalence. Hence, we wish to show that the map of colimits

induced by the functor $- \times _XU_0 \colon \mathfrak {B} \to \mathfrak {B}_{/U_0}$ , which exists by our assumption that the basis $\mathfrak {B}$ is closed under finite intersections, is an equivalence. In fact, we claim that this functor is a $\varinjlim $ -equivalence. Indeed, by [Reference Lurie20, Theorem 4.1.3.1], it suffices to show that

$$ \begin{align*}\mathfrak{B} \times_{\mathfrak{B}_{/U_0}}(\mathfrak{B}_{/U_0})_{V/}\end{align*} $$

is weakly contractible for every $V \in \mathfrak {B}_{/U_0}$ . But we can identify this category with the poset $\mathfrak {B}_{V/}$ , which has V as a minimal element, so its classifying anima is contractible, as required.

Proof. The statement holds for affine Dirac schemes by Lemma 2.23, and we now use Lemma 2.25 to bootstrap our way to general Dirac schemes.

First, if X admits an open immersion $j \colon X \to S$ into an affine Dirac scheme, then we take $\mathfrak {B}$ to be the poset consisting of the open subschemes $U \subset X$ with the property that $j(U) = S_f \subset S$ is a distinguished open subscheme. This $\mathfrak {B}$ is a basis for the topology on $|X|$ , it is stable under finite intersection, and $h(U)$ is accessible for every $U \in \mathfrak {B}$ , because $S_f$ is affine. So $h(X)$ is accessible by Lemma 2.25.

Finally, if X is any Dirac scheme, then we take $\mathfrak {B}$ to be the poset of open subschemes $U \subset X$ with the property that U admits an open immersion into some affine Dirac scheme. Again, this $\mathfrak {B}$ is a basis for the topology on $|X|$ , it is stable under finite intersection, and by the case considered above $h(U)$ is accessible for all $U \in \mathfrak {B}$ . So Lemma 2.25 shows that $h(X)$ is accessible.

Proof. It follows from Proposition 2.24 and Proposition 2.26 that the restricted Yoneda embedding from Definition 2.22 takes values in $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}}) \subset \operatorname {\mathrm {Fun}}(\operatorname {\mathrm {Aff}}^{\mathrm {op}},{\mathcal {S}})$ . So we get the desired functor and must prove that it is fully faithful. We let X and Y be Dirac schemes and wish to prove that the canonical map

is an equivalence of anima. (In fact, the domain and target of this map are both $0$ -truncated anima.) If Y is affine, then this follows from the Yoneda lemma.

In general, if $\mathfrak {B}$ is a basis for the topology on the underlying space $|Y|$ , which is stable under finite intersection, then the canonical map of Dirac schemes

is an equivalence by [Reference Hesselholt and Pstrągowski12, Theorem 2.32], and the canonical map of Dirac stacks

is an equivalence by the proof of Lemma 2.25. Thus, we can bootstrap our way from affine Dirac schemes to general Dirac schemes in the same way as we did in the proof of Proposition 2.26.

Addendum 2.28. The functor $h \colon \operatorname {\mathrm {Sch}} \to \operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ preserves coproducts.

Proof. The functor in question preserves finite coproducts, so it suffices to show that for a small family of Dirac schemes $(X_k)_{k \in K}$ , the canonical map

where the colimits range over the filtered category of finite subsets $I \subset K$ , is an equivalence. We claim that the left-hand colimit is calculated pointwise. Indeed, this claim is equivalent to the statement that the presheaf given by pointwise colimit satisfies the sheaf condition. But the flat topology on $\operatorname {\mathrm {Aff}}$ if finitary and the diagram in question takes values in ${\mathcal {S}}_{\leq 0}$ -valued presheaves, so by [Reference Hesselholt and Pstrągowski12, Proposition A.1], the sheaf condition amounts to the requirement that certain finite diagrams in ${\mathcal {S}}$ are limit diagrams. Hence, the claim follows from the fact that finite limits and filtered colimits of anima commute. Finally, given $T \in \operatorname {\mathrm {Aff}}$ , the canonical map

is an equivalence since T is quasi-compact.

Proof. We argue by induction on $n \geq 0$ . If f is $0$ -geometric, then, by definition, the base-change $f_S \colon Y_S \to S$ of f along any map $\eta \colon S \to X$ from a Dirac scheme is equivalent to a map of Dirac schemes. Since the identity map $\operatorname {\mathrm {id}} \colon Y_S \to Y_S$ is a $0$ -submersive map from a Dirac scheme, we conclude that f is $1$ -geometric.

So we let $n \geq 1$ and assume inductively that the statement has been proved for $k < n$ . If f is n-geometric, then for every base-change $f_S \colon Y_S \to S$ of f along a map $\eta \colon S \to X$ from a Dirac scheme S, there exits an $(n-1)$ -submersive map $p \colon T \to Y_S$ from a Dirac scheme. By the inductive hypothesis, the map p is also n-submersive, which shows that f is $(n+1)$ -geometric. The same argument shows that if f is n-submersive, then it is $(n+1)$ -submersive.

Proof. Since $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ is generated under small colimits by the image of the Yoneda embedding, and since the property of being an effective epimorphism is local on the base, we may assume that X is affine. We proceed by induction on $n \geq 0$ , the case $n = 0$ being trivial, because a $0$ -submersive map is an effective epimorphism by definition. So we let $n \geq 1$ and inductively assume the statement for $k < n$ . Since $f \colon Y \to X$ is n-submersive and X affine, there exists a factorization

with p an $(n-1)$ -submersive map from a Dirac scheme such that $fp$ is $0$ -submersive. Now, by the inductive hypothesis, both p and $fp$ are effective epimorphisms, so we conclude from [Reference Lurie20, Corollary 6.2.3.12] that also f is an effective epimorphism.

Proof. We argue by induction on $n \geq 0$ . The geometric part of case $n = 0$ follows from definitions, and the submersive part follows from the fact that both effective epimorphisms and flat maps of Dirac schemes are closed under composition. So we let $n \geq 1$ and assume inductively that the statements have been proved for $k < n$ . To prove the induction step, we consider the diagram

with S a Dirac scheme and the squares cartesian. Suppose first that f and g are n-geometric. We first choose an $(n-1)$ -submersive map $p \colon U \to Y_S$ from a Dirac scheme. Since g is n-geometric, so is $g_U$ , and hence, we may choose an $(n-1)$ -submersive map $q \colon V \to Z_U$ from a Dirac scheme. By the inductive hypothesis, the composition of the $(n-1)$ -submersive maps $p'$ and q is $(n-1)$ -submersive. This shows that $fg$ is n-geometric. If f and g are n-submersive, then we may choose the maps p and q such that $f_Sp$ and $g_Uq$ are $0$ -submersive. But then also

$$ \begin{align*}(fg)_S(p'q) \simeq f_Sg_Sp'q \simeq (f_Sp)(g_Uq)\end{align*} $$

is $0$ -submersive, so we conclude that $fg$ is n-submersive.

Proof. Let $(S_i)_{i \in I}$ be a family with $S_i \subset V$ affine open and $\bigcup _{i \in I}|S_i| = |V|$ . It follows from Remark 2.32 and Addendum 2.28 that the induced map

is $0$ -submersive. Applying Remark 2.31 to each $S_i$ separately, we find $0$ -submersive maps $q_i \colon T_i \to S_i$ with $T_i$ affine and factorizations of the composite maps

through $f \colon Y \to X$ . Thus, we obtain the desired diagram with $W \simeq \coprod _{i \in I}T_i$ .

Proof. For part (a), we consider the diagram

with S a Dirac scheme and the squares cartesian. Since $fg$ is $(n+1)$ -geometric, we can find an n-submersive map q from a Dirac scheme, and since g is n-submersive, so is its base-change $g_S$ and the composite map $g_Sq$ by Proposition 2.38. Hence, $g_Sq$ is the required n-submersive map to $Y_S$ .

For part (b), we consider the diagram

with S a Dirac scheme and with the squares cartesian. The maps p and q are an $(n+1)$ -submersive map from a Dirac scheme U and an n-submersive map from a Dirac scheme V, and they exist by the assumption that f and $fg$ be respectively $(n+2)$ -geometric and $(n+1)$ -geometric. Now, Proposition 2.37 shows that p is an effective epimorphism, so by Lemma 2.39, we can find a $0$ -submersive map of Dirac schemes $h \colon W \to V$ together with the indicated factorization of $g_S q h$ through p.

Since h is $0$ -submersive, it is n-submersive by Lemma 2.36, and since also q is n-submersive, it follows from Proposition 2.38 that $qh$ is n-submersive. Similarly, since k is $0$ -geometric, it is $(n+1)$ -geometric by Lemma 2.36, and since also p is $(n+1)$ -geometric, it follows from another application of Proposition 2.38 that $pk$ is $(n+1)$ -geometric. So part (a) shows that $g_S$ is $(n+1)$ -geometric, as desired.

Proof. Let $\eta \colon S \to X$ be a Dirac scheme and let $f_S \colon Y_S \to S$ be the base-change of f along $\eta $ . We wish to find an $(n-1)$ -submersive map $U \to Y_S$ from a Dirac scheme. Using Lemma 2.39, we can find the left-hand square

with q a $0$ -submersive map of Dirac schemes. Since $f'$ is n-geometric, we can find an $(n-1)$ -submersive map $r \colon U \to Y_T'$ from a Dirac scheme. But then

is $(n-1)$ -submersive. Indeed, the map $q_Y$ is the base-change of the $0$ -submersive map q and hence $0$ -submersive, so by Lemma 2.36, it is also $(n-1)$ -submersive, and finally, the composite map is $(n-1)$ -submersive by Proposition 2.38. This completes the proof of (a), and the proof of (b) is analogous.

Proof. We proceed by induction on $n \geq 0$ , the case $n = 0$ being trivial. So we let $n \geq 1$ and assume that the statement has been proved for $k < n$ . Since $f \colon T \to S$ is n-submersive, there exists, by definition, a scheme V and an $(n-1)$ -submersive map $p \colon V \to T$ such that the composite map

is $0$ -submersive. By the inductive hypothesis, the map p is $0$ -submersive, and since both $fp$ and p are flat, we deduce that f is flat as well. Similarly, since both $fp$ and p are effective epimorphisms, so is f. Hence, f is a $0$ -submersion.

Proof. Using Lemma 2.36 and downward induction, it suffices to show that if f is n-geometric and $(n+1)$ -submersive, then it is n-submersive. Since both properties are detected after base-change to a Dirac scheme, we may assume that X is a Dirac scheme. If $n = 0$ , then there is nothing to prove, so we assume that $n \geq 1$ . Since f is n-geometric, and since X is a Dirac scheme, there exists an $(n-1)$ -submersive map $p \colon U \to Y$ from a Dirac scheme. We claim that the composite map

is a $0$ -submersive map of Dirac schemes, so that f is also n-submersive.

To prove the claim, we consider the diagram

with q an n-submersive map from a Dirac scheme and the square cartesian. The map q exists, because f is $(n+1)$ -submersive. The base-change $q_U$ is also n-submersive, so we can find the $(n-1)$ -submersive map r from a Dirac scheme such that $q_Ur$ is $0$ -submersive. The composite map $(fp)(q_Ur)$ is an n-submersive map between Dirac schemes and hence $0$ -submersive by Lemma 2.43. Thus, $(fp)(q_Ur)$ and $q_Ur$ are both flat maps of Dirac schemes, so $fp$ is flat, and $(fp)(q_Ur)$ and $q_Ur$ are both effective epimorphisms, so $fp$ is an effective epimorphism. This proves the claim.

Proof. This is Proposition 2.38 and Proposition 2.40.

Proof. Only the claim concerning fiber products needs proof. If $n = 0$ , then X, $X'$ and Y are Dirac schemes, and hence, so is $Y'$ . If $n \geq 1$ , then by Proposition 2.47 (c), the map f is n-geometric, and therefore, so is $f'$ . Thus, Proposition 2.47 (a) implies that $Y'$ is n-geometric as stated.

Proof. We first show that $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})^{\operatorname {\mathrm {geom}}}$ satisfies (1) and (2). For (1), we recall from Addendum 2.28 that a small coproduct of affine Dirac schemes is a Dirac scheme – hence, a geometric Dirac stack. For (2), we let $S \colon \Delta ^{\mathrm {op}} \to \operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ be a groupoid with colimit X. By Theorem 2.19, S is effective in the sense that the square

is cartesian. In this situation, we wish to show that if $d_0$ is a submersive map between geometric Dirac stacks, then X is geometric. If we choose an $n \geq 0$ such that both $S_0$ and $S_1$ are n-geometric, then Proposition 2.47 (c) shows that $d_0$ is n-geometric, and thus Proposition 2.44 shows that $d_0$ is n-submersive. Since (the horizontal) f is an effective epimorphism, we conclude from Proposition 2.41 and from the fact that $d_0$ is n-submersive that (the vertical) f is n-submersive. But then Proposition 2.47 (b) shows that X is $(n+1)$ -geometric, as desired.

Finally, we let $\mathcal {C} \subset \operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})^{\operatorname {\mathrm {geom}}}$ be a full subcategory satisfying (1) and (2) and show, by induction on $n \geq 0$ , that $\mathcal {C}$ contains all n-geometric Dirac stacks. For $n = 0$ , we must show that $\mathcal {C}$ contains every Dirac scheme X. We first suppose that $X \simeq \coprod _{i \in I}X_i$ is a coproduct of Dirac schemes, each of which can be embedded as an open subscheme $X_i \subset S_i$ of an affine Dirac scheme. We can write each $X_i$ as the union of a family $(T_{i,j})_{j \in J_i}$ of distinguished open subschemes $T_{i,j} \subset S_i$ , and therefore, it follows from Addendum 2.28 that the canonical map

is $0$ -submersive. Now, since p is an effective epimorphism, its Čech nerve

is a colimit diagram, and since coproducts in $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ are disjoint, we have

$$ \begin{align*}\textstyle{ V_k \simeq V_0^{\times_X[k]} \simeq \coprod_{i \in I} \coprod_{j_0,\dots,j_k \in J_i} T_{i,j_0} \times_{S_i} \dots \times_{S_i} T_{i,j_k} }\end{align*} $$

for $k \geq 0$ . But each summand is an affine Dirac scheme, since distinguished open subschemes of an affine Dirac scheme are closed under finite intersections, so we conclude from properties (1) and (2) that X is contained in $\mathcal {C}$ .

Next, if X is a general Dirac scheme, then we choose a family $(S_i)_{i \in I}$ of affine open subschemes such that $\bigcup _{i \in I}|S_i| = |X|$ . In this case, the canonical map

is an effective epimorphism, and its Čech nerve $V \colon \Delta _+^{\mathrm {op}} \to \operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ has the property that for all $k \geq 0$ , the Dirac scheme $V_k$ is a coproduct of open subschemes of affine Dirac schemes. Hence, each $V_k$ is contained in $\mathcal {C}$ by what was proved in the previous paragraph, and therefore, so is X. This proves the case $n = 0$ .

Finally, we let $n \geq 1$ and assume, inductively, that $\mathcal {C}$ contains all $(n-1)$ -geometric Dirac stacks. Given an n-geometric Dirac stack X, we choose an $(n-1)$ -submersive map $p \colon V_0 \to X$ from a Dirac scheme and consider its Čech nerve

Since p is $(n-1)$ -submersive and $V|_{\Delta ^{\mathrm {op}}}$ an effective groupoid, we conclude that all face maps in $V|_{\Delta ^{\mathrm {op}}}$ are $(n-1)$ -submersive – hence, by Proposition 2.38, so is every iterated face map $V_k \to V_0$ . But $V_0$ is $(n-1)$ -geometric, being a Dirac scheme, so Proposition 2.47 shows that $V_k$ is $(n-1)$ -geometric for all $k \geq 0$ . By the inductive hypothesis, $V|_{\Delta ^{\mathrm {op}}}$ is a groupoid in $\mathcal {C}$ with submersive face maps, so we conclude from property (2) that X is contained in $\mathcal {C}$ , as desired.

Proof. The compact $1$ -projective objects in the abelian category $\mathcal {C} \simeq \operatorname {\mathrm {Mod}}_R(\operatorname {\mathrm {Ab}})$ are the graded R-modules which are finitely generated and projective, so $\mathcal {C}^{\operatorname {\mathrm {cp}}} \subset \mathcal {C}$ generates $\mathcal {C}$ under sifted colimits. Therefore, the animation $i \colon \mathcal {C}^{\operatorname {\mathrm {cp}}} \to \mathcal {C}$ exists and agrees, by [Reference Lurie20, Proposition 5.5.8.15], with the Yoneda embedding $h \colon \mathcal {C}^{\operatorname {\mathrm {cp}}} \to {\mathcal {P}}_{\Sigma }(\mathcal {C}^{\operatorname {\mathrm {cp}}})$ into the full subcategory ${\mathcal {P}}_{\Sigma }(\mathcal {C}^{\operatorname {\mathrm {cp}}}) \subset {\mathcal {P}}(\mathcal {C}^{\operatorname {\mathrm {cp}}})$ spanned by the functors $(\mathcal {C}^{\operatorname {\mathrm {cp}}})^{\mathrm {op}} \to {\mathcal {S}}$ that preserve finite products. Moreover, since $\mathcal {C}^{\operatorname {\mathrm {cp}}}$ is additive, ${\mathcal {P}}_{\Sigma }(\mathcal {C}^{\operatorname {\mathrm {cp}}})$ is complete Grothendieck prestable by [Reference Lurie23, Proposition C.1.5.7 and Remark C.1.5.9].

Proof. The $\infty $ -categories (1)–(3) are all presentable and stable and equipped with right complete t-structures. So it suffices to compare connective parts, and by the universal property of animation, it suffices to verify that (2) and (3) are generated under sifted colimits by the subcategories spanned by compact projective objects, which coincide with $\operatorname {\mathrm {Mod}}_R(\operatorname {\mathrm {Ab}})^{\operatorname {\mathrm {cp}}}$ . In the case of (2), this is [Reference Lurie19, Proposition 1.3.3.14], and in the case of (3), the heart of the Beilinson t-structure is given by

$$ \begin{align*}\operatorname{\mathrm{Mod}}_R(\operatorname{\mathrm{Fun}}(\mathbb{Z},\operatorname{\mathrm{Sp}}))^{\heartsuit} \simeq \operatorname{\mathrm{Mod}}_R(\operatorname{\mathrm{Fun}}(\mathbb{Z},\operatorname{\mathrm{Sp}})^{\heartsuit}) \simeq \operatorname{\mathrm{Mod}}_R(\operatorname{\mathrm{Ab}}),\end{align*} $$

and the connective part $\operatorname {\mathrm {Mod}}_R(\operatorname {\mathrm {Fun}}(\mathbb {Z},\operatorname {\mathrm {Sp}})_{\geq 0}$ is generated under sifted colimits by subcategory spanned by the compact projective objects in the heart, which precisely is $\operatorname {\mathrm {Mod}}_R(\operatorname {\mathrm {Ab}})^{\operatorname {\mathrm {cp}}}$ .

Proof. This is [Reference Lurie19, Propositions 4.8.1.10 and 2.2.1.9].

Proof. To prove (1), we use Proposition A.2 to write X as a colimit of a small diagram $S \colon K \to {\mathcal {P}}(\operatorname {\mathrm {Aff}})$ of affine Dirac stacks. Hence,

$$ \begin{align*}\textstyle{ {\mathrm{QCoh}}(X) \simeq \varprojlim_K {\mathrm{QCoh}}(S), }\end{align*} $$

and we wish to show that

$$ \begin{align*}\textstyle{ {\mathrm{QCoh}}(X)_{\geq 0} \simeq \varprojlim_K {\mathrm{QCoh}}(S)_{\geq 0} }\end{align*} $$

is the connective part of a t-structure. But ${\mathrm {QCoh}}(X)_{\geq 0}$ and ${\mathrm {QCoh}}(X)$ are both presentable, and therefore, it suffices by [Reference Lurie19, Proposition 1.4.4.11] to show that

$$ \begin{align*}{\mathrm{QCoh}}(X)_{\geq 0} \subset {\mathrm{QCoh}}(X)\end{align*} $$

is closed under small colimits and extensions, which is clear. This proves (1).

To prove (2), given $\mathcal {G} \in {\mathrm {QCoh}}(X)$ be as in the statement, we must show that for every $\mathcal {F} \in {\mathrm {QCoh}}(X)_{\geq 0}$ , the mapping anima $\operatorname {\mathrm {Map}}(\mathcal {F},\mathcal {G})$ is $0$ -truncated. We again write $X \simeq \varinjlim _KS$ as a colimit of a small diagram $S \colon K \to \operatorname {\mathrm {Aff}}$ of affine Dirac stacks, and for $k \in K$ , we write $\eta _k \colon S_k \to X$ be the canonical map. This exhibits the mapping anima in question as a limit

$$ \begin{align*}\textstyle{ \operatorname{\mathrm{Map}}(\mathcal{F},\mathcal{G}) \simeq \varprojlim_{k \in K} \operatorname{\mathrm{Map}}(\eta_k^*(\mathcal{F}),\eta_k^*(\mathcal{G})) }\end{align*} $$

of anima, each of which is $0$ -truncated, by assumption. But then also the limit is $0$ -truncated since the inclusion ${\mathcal {S}}_{\leq 0} \to {\mathcal {S}}$ preserves limits. This proves (2).

Finally, we observe that (3) holds because $\eta ^*(\mathcal {O}_X(s)) \simeq \mathcal {O}_S(s) \in {\mathrm {QCoh}}(S)^{\heartsuit }$ for every map $\eta \colon S \to X$ with S affine.

Addendum 3.15. If $f \colon Y \to X$ is a map of Dirac stacks, then the extension of scalars functor and the restriction of scalars functor

are right t-exact and left t-exact, respectively.

Proof. Since $f^*$ and $f_*$ are adjoint, the statement that the former is right t-exact is equivalent to the statement that the latter is left t-exact. To show that $f^*$ is right t-exact, we let $\mathcal {F} \in {\mathrm {QCoh}}(X)_{\geq 0}$ and wish to show that $f^*(\mathcal {F}) \in {\mathrm {QCoh}}(Y)_{\geq 0}$ . But if $\eta \colon S \to Y$ is map from an affine Dirac scheme, then so is $f \circ \eta \colon S \to X$ , so

$$ \begin{align*}\eta^*(f^*(\mathcal{F})) \simeq (f \circ \eta)^*(\mathcal{F}) \in {\mathrm{QCoh}}(S)_{\geq 0},\end{align*} $$

since $\mathcal {F} \in {\mathrm {QCoh}}(X)_{\geq 0}$ .

Proof. We write $S \simeq \operatorname {\mathrm {Spec}}(A)$ , $S' \simeq \operatorname {\mathrm {Spec}}(A')$ , $T \simeq \operatorname {\mathrm {Spec}}(B)$ and $T' \simeq \operatorname {\mathrm {Spec}}(B')$ . The right adjoints $g_*$ and $g_*'$ exist, and since we consider affine Dirac schemes only, they are given by restriction of scalars, and hence, they preserve small colimits. Therefore, it suffices to show that for the generator $A'$ of the presentable stable $\infty $ -category ${\mathrm {QCoh}}(S')$ , the map $f^*g_*(A') \to g_*'f'{}^*(A')$ is an equivalence. But this map is given by the canonical projection

which is an equivalence by the assumption that either f or g or both are flat.

Proof. We first prove (1). Since $f \colon Y \to X$ is affine, we have an adjunction

with u given by composition with f and v given by base-change along f. A counit p of the adjunction induces a natural transformation

of functors from $({\mathrm {Aff}}_{/X})^{\mathrm {op}}$ to $\operatorname {\mathrm {Cat}}_{\infty }$ given by extension of scalars along the canonical projection $p \colon S \times _X Y \to S$ , and we have a commutative diagram

with the top horizontal map and the right-hand slanted map the maps of limits induced by $u^{\mathrm {op}}$ and $v^{\mathrm {op}}$ , respectively. Moreover, the latter map is an equivalence. Indeed, this follows from [Reference Lurie20, Example 2.21] since v admits a left adjoint and hence is a $\smash { \varinjlim }$ -equivalence. Hence, to prove that $f_*$ admits a right adjoint, we may instead prove that $p^*$ admits a right adjoint $p_*$ which itself admits a further right adjoint.

We may view $p^*$ as a map in ${\operatorname {LPr}}$ , so it is clear that it has a right adjoint $p_*$ , and to understand $p_*$ , we employ [Reference Lurie19, Proposition 4.7.4.19]. The assumption (*) in loc. cit. that for every map $h \colon T \to S $ in ${\mathrm {Aff}}_{/X}$ , the square

be right adjointable holds by Lemma 3.18 since $f \colon Y \to X$ is flat affine. So we conclude that $p_*$ is the map of limits induced by a natural transformation

of functors from $({\mathrm {Aff}}_{/X})^{\mathrm {op}}$ to ${\widehat {\mathrm {Cat}}}_{\infty }$ .

Now, for every $(S,S \to X)$ , we may identify $(p_{S})_{*}$ with the restriction of scalars along a map of Dirac rings, and hence, it is right t-exact. This implies that the induced map of limits $p_*$ is right t-exact, which shows that $f_*$ is right t-exact. But by Addendum 3.15, $f_*$ is also left t-exact, so we conclude that $f_*$ is t-exact.

Similarly, for every $(S,S \to X)$ , the functor $(p_S)_*$ admits a right adjoint given by coextension of scalars. Therefore, we may view $(p_S)_*$ as a natural transformation of functors from $({\mathrm {Aff}}_{/X})^{\mathrm {op}}$ to ${\operatorname {LPr}}$ , and since the forgetful functor

preserves limits, we conclude that $p_*$ admits a right adjoint.Footnote ⁵ So $f_*$ admits a right adjoint, which completes the proof of (1).

It remains to prove (2). First, if $X'$ is affine, then with notation as in the proof of (1) above, we may equivalently prove that the diagram

is right adjointable. But this follows from [Reference Lurie19, Proposition 4.7.4.19], since for every $\eta \colon S \to X$ and $h \colon T \to S$ with S and T affine, the diagram

is right adjointable by Lemma 3.18.

In general, we use that a map in ${\mathrm {QCoh}}(X')$ is an equivalence if and only if its image by the inverse image $h^* \colon {\mathrm {QCoh}}(X') \to {\mathrm {QCoh}}(X" )$ is an equivalence for every map $h \colon X" \to X'$ with $X" $ affine. So we consider the diagram

with $f'$ the base-change of f along g and $f" $ the base-change of $f'$ along h. Since $X" $ is affine, both the outer square and the bottom square are right adjointable by what was already proved. Thus, in the diagram of canonical maps