1 Introduction
Whatever it is that animates anima and breathes life into higher algebra, this something leaves its trace in the structure of a Dirac ring on the homotopy groups of a commutative algebra in spectra. Dirac geometry is built from Dirac rings in the same way as algebraic geometry is built from commutative rings. In the prequel to this paper, we developed the commutative algebra of Dirac rings and defined the category of Dirac schemes. Here, we first embed this category in the larger $\infty $ category of Dirac stacks, which also contains formal Dirac schemes. We next develop the coherent cohomology of Dirac stacks, which amounts to a functor that to a Dirac stack X assigns a presentably symmetric monoidal stable $\infty $ category ${\mathrm {QCoh}}(X)$ of quasicoherent $\mathcal {O}_X$ modules together with a symmetric monoidal subcategory ${\mathrm {QCoh}}(X)_{\geq 0} \subset {\mathrm {QCoh}}(X)$ , which is the connective part of a tstructure. As applications of the general theory to stable homotopy theory, we use Quillen’s theorem on complex cobordism and Milnor’s theorem on the dual Steenrod algebra to identify the Dirac stacks corresponding to $\operatorname {MU}$ and $\mathbb {F}_p$ in terms of their functors of points. Finally, in an appendix, we develop a rudimentary theory of accessible presheaves of anima on coaccessible $\infty $ categories.
To develop the theory of Dirac stacks and their coherent cohomology, we follow Lurie in [Reference Lurie23]. The category $\operatorname {\mathrm {Aff}}$ of affine Dirac schemes is the opposite of the category $\operatorname {\mathrm {CAlg}}(\operatorname {\mathrm {Ab}})$ of Dirac rings. The category of Dirac rings $\operatorname {\mathrm {CAlg}}(\operatorname {\mathrm {Ab}})$ and the $\infty $ category of anima ${\mathcal {S}}$ are both presentable, and we define the $\infty $ category of Dirac prestacks to be the $\infty $ category of accessible presheaves of anima on $\operatorname {\mathrm {Aff}}$ ,
We show in Appendix A that this $\infty $ category has excellent categorical properties and that it admits a number of equivalent descriptions. If $\kappa $ is a small regular cardinal, then we let ${\mathrm {Aff}}_{\kappa } \subset \operatorname {\mathrm {Aff}}$ be the full subcategory spanned by the affine Dirac schemes of cardinality $< \kappa $ . Since it is essentially small, the $\infty $ category ${\mathcal {P}}({\mathrm {Aff}}_{\kappa })$ of presheaves of anima on ${\mathrm {Aff}}_{\kappa }$ is presentable, and we show that
It follows that the $\infty $ category ${\mathcal {P}}(\operatorname {\mathrm {Aff}})$ admits all small colimits and limits, both of which are calculated pointwise. We also show that the Yoneda embedding restricts to a fully faithful functor
and that this functor exhibits the target as the cocompletion of the source.
The flat topology on $\operatorname {\mathrm {Aff}}$ is the Grothendieck topology, where a sieve $j \colon U \to h(S)$ is a covering sieve if there exists a finite family of maps $(T_i \to S)_{i \in I}$ such that the induced map $T \simeq \coprod _{i \in I} T_i \to S$ is faithfully flat and such that each $h(T_i) \to h(S)$ factors through j.
Definition 1.1. The $\infty $ category of Dirac stacks is the full subcategory
spanned by the sheaves for the flat topology.
It is not immediately clear that there exists a ‘sheafification’ functor, left adjoint to the canonical inclusion of sheaves in presheaves. That this is nevertheless the case is a consequence of a theorem of Waterhouse [Reference Waterhouse35, Theorem 5.1], which is perhaps not as well known as it deserves. It shows that the flat topology is wellbehaved. What is not wellbehaved is the $\infty $ category of all presheaves on $\operatorname {\mathrm {Aff}}$ ; once we limit ourselves to considering accessible presheaves only, the supposed problems with the flat topology disappear. In the Dirac context, Waterhouse’s theorem is the following statement, and the essential ingredient in its proof is the Dirac analogue of the equational criterion for faithful flatness, which we proved in [Reference Hesselholt and Pstrągowski12, Addendum 3.9].
Theorem 1.2 (Waterhouse’s theorem).
Let $\kappa < \kappa '$ be small regular cardinals. In this situation, the left Kan extension along the canonical inclusion
preserves sheaves with respect to the flat topology.
As we have already mentioned, an important consequence of Theorem 1.2 is the existence of a left exact ‘sheafification’ functor
left adjoint to the canonical inclusion functor. Indeed, if we replace $\operatorname {\mathrm {Aff}}$ by the essentially small ${\mathrm {Aff}}_{\kappa }$ , then Lurie has proved in [Reference Lurie20, Proposition 6.2.2.7] that a left exact sheafification functor exists, and Theorem 1.2 and [Reference Lurie17, Tag 02FV] show that these left adjoint functors assemble to give the stated left adjoint functor.
We can now deduce from [Reference Lurie20, Theorem 6.1.0.6] that the $\infty $ category of Dirac stacks satisfies the $\infty $ categorical Giraud axioms, except that it not presentable.
Theorem 1.3 (Giraud’s axioms for Dirac stacks).
The $\infty $ category $\,\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ of Dirac stacks has the following properties:

(i) It is complete and cocomplete, and it is generated under small colimits by the essential image of the Yoneda embedding $h \colon \operatorname {\mathrm {Aff}} \to \operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ , which consists of $\omega _1$ compact objects and is coaccessible.

(ii) Colimits in $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ are universal.

(iii) Coproducts in $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ are disjoint.

(iv) Every groupoid in $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ is effective.
The $\infty $ category of Dirac stacks is close enough to being an $\infty $ topos that it retains the important property of $\infty $ topoi that the contravariant functor
that to X assigns the slice $\infty $ category $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})_{/X}$ takes colimits of Dirac stacks to limits of $\infty $ categories. This statement is the source of all descent statements in the paper. In typical situations, we wish to show that flat descent holds for the full subcategory
spanned by the maps $f \colon Y \to X$ that have some property P. But this follows from the fundamental descent statement, once we prove that the property P satisfies descent for the flat topology in the sense that for every effective epimorphism $f \colon X' \to X$ , the diagram of $\infty $ categories
is cartesian. For example, we define a functor that to a Dirac stack X assigns the $\infty $ category $\operatorname {\mathrm {FGroup}}(X)$ of formal groups over X and use this strategy to show that it descends along effective epimorphisms.
It follows from Grothendieck’s faithfully flat descent for graded modules, which we proved in [Reference Hesselholt and Pstrągowski12, Theorem 1.5], that the flat topology on $\operatorname {\mathrm {Aff}}$ is subcanonical. Here, we prove that, more generally, the category of Dirac schemes, which we defined in [Reference Hesselholt and Pstrągowski12, Definition 2.29], embeds fully faithfully into the $\infty $ category of Dirac stacks.
Theorem 1.4 (Schemes are stacks).
Dirac schemes are Dirac stacks:

(1) If X is a Dirac scheme, then the functor $h(X) \colon \operatorname {\mathrm {Aff}}^{\operatorname {\mathrm {op}}} \to {\mathcal {S}}$ that to an affine scheme S assigns $\operatorname {\mathrm {Map}}(S,X)$ is accessible and a sheaf for the flat topology.

(2) The resulting functor $h \colon \operatorname {\mathrm {Sch}} \to \operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ is fully faithful.
This result is the starting point for the definition following Toën–Vezzosi [Reference Toën and Vezzosi33] and Lurie [Reference Lurie18] of the property of a map of Dirac stacks $f \colon Y \to X$ of being geometric. Informally, the geometric maps span the smallest full subcategory
that contains the maps $f \colon Y \to X$ that, locally on X, are equivalent to maps of Dirac schemes, and that is closed under the formation of the geometric realization of groupoids with flat face maps.Footnote ^{1} However, the formal definition, which we give in Section 2.3 below, is more complicated, because the notions of being geometric and being flat must be defined recursively together. The property of being geometric is wellbehaved: it is preserved under composition and basechange, it is local on the target, and maps in $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})_{/X}^{\operatorname {\mathrm {geom}}}$ are automatically geometric.
We now explain the theory of coherent cohomology of Dirac stacks, which we develop following Lurie [Reference Lurie23, Chapter 6]. It consists of the following data:

(Q1) A functor ${\mathrm {QCoh}} \colon \operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})^{\operatorname {\mathrm {op}}} \to \operatorname {\mathrm {CAlg}}({\operatorname {LPr}})$ ; it assigns to map of Dirac stacks $f \colon Y \to X$ a symmetric monoidal adjunction

(Q2) For every Dirac stack X, a symmetric monoidal adjunction
First, to produce the functor (Q1), we begin with the functor that to an affine Dirac scheme $S \simeq \operatorname {\mathrm {Spec}}(A)$ assigns the presentably symmetric monoidal category
of graded Amodules. It promotes via animation to a functor that to S assigns the presentably symmetric monoidal $\infty $ category
of animated graded Amodules, which, in turn, promotes via stabilization to a functor that to S assigns the presentably symmetric monoidal $\infty $ category
of spectra in animated graded Amodules. Following Grothendieck, we show that this functor is a sheaf for the flat topology on $\operatorname {\mathrm {Aff}}$ , a fact, which, in homotopy theory, was first observed and exploited by Hopkins [Reference Hopkins14]. This gives us (Q1):
Theorem 1.5 (Faithfully flat descent for quasicoherent modules).
The right Kan extension of the functor ${\mathrm {QCoh}} \colon \operatorname {\mathrm {Aff}}^{\operatorname {\mathrm {op}}} \to \operatorname {\mathrm {CAlg}}({\operatorname {LPr}})$ along $h \colon \operatorname {\mathrm {Aff}} \to {\mathcal {P}}(\operatorname {\mathrm {Aff}})$ admits a unique factorization
through the sheafification functor.
To spell out this definition of ${\mathrm {QCoh}}(X)$ , if we choose a regular cardinal $\kappa $ such that $X \in \operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ is the left Kan extension of $X_{\kappa } \in \operatorname {\mathrm {Shv}}({\mathrm {Aff}}_{\kappa })$ , then
where the limit is indexed by $(({\mathrm {Aff}}_{\kappa })_{/X})^{\operatorname {\mathrm {op}}}$ and is calculated in $\operatorname {\mathrm {CAlg}}({\operatorname {LPr}})$ .Footnote ^{2}
Second, to produce the symmetric monoidal adjunction (Q2), we show that for an affine Dirac scheme S, the $\infty $ category ${\mathrm {QCoh}}(S)_{\geq 0}$ is Grothendieck prestable in the sense of [Reference Lurie23, Definition C.1.4.2]. Thus, the symmetric monoidal adjunction
is the connective part of a tstructure, and we obtain (Q2) by right Kan extension. More concretely, if X is a Dirac stack, then $\mathcal {F} \in {\mathrm {QCoh}}(X)$ is connective if and only if $\eta ^*(\mathcal {F}) \in {\mathrm {QCoh}}(S)$ is connective for every map $\eta \colon S \to X$ with S affine. It is also true that if $\eta ^*(\mathcal {F}) \in {\mathrm {QCoh}}(S)$ is coconnective for every map $\eta \colon S \to X$ with S affine, then $\mathcal {F} \in {\mathrm {QCoh}}(X)$ is coconnective, but the converse is false. In particular, the tensor unit $\mathcal {O}_X$ belongs to the heart of the tstructure.
In general, the tstructure on ${\mathrm {QCoh}}(X)$ is not left complete or right complete, and the coconnective part is next to impossible to understand. However, for geometric Dirac stacks, including Dirac schemes, the situation improves:
Theorem 1.6 (Geometric stacks and tstructure).
If X is a geometric Dirac stack, then the tstructure on ${\mathrm {QCoh}}(X)$ is left and right complete and the coconnective part is closed under filtered colimits. Moreover, if $\eta \colon S \to X$ is submersive with S affine, then $\mathcal {F} \in {\mathrm {QCoh}}(X)$ is coconnective if and only if $\eta ^*(\mathcal {F}) \in {\mathrm {QCoh}}(S)$ is so.
We let $\pi _i^{\heartsuit }(\mathcal {F}) \in {\mathrm {QCoh}}(X)^{\heartsuit }$ be the ith homotopy $\mathcal {O}_X$ module of $\mathcal {F} \in {\mathrm {QCoh}}(X)$ with respect to the canonical tstructure. We refer to i as the ‘animated’ degree to separate it from the ‘spin’ degree intrinsic to Dirac geometry. If $f \colon Y \to X$ is a map of Dirac stacks, then the ‘correct’ relative coherent cohomology $\mathcal {O}_X$ modules of the $\mathcal {O}_Y$ module $\mathcal {G} \in {\mathrm {QCoh}}(Y)$ are given by
This functor can generally not be calculated as the derived functor of the leftexact functor between abelian categories $f_*^{\heartsuit }$ induced by the left texact functor $f_*$ .
Another special class of Dirac stacks that we consider are the formal Dirac schemes. If $\eta \colon S \to X$ is a closed immersion of Dirac schemes, whose defining quasicoherent ideal $\mathcal {I} \subset \mathcal {O}_X$ is of finite type, then we define the formal completion of X along S to be the colimit in Dirac stacks
of the infinitesimal thickenings $\eta ^{(m)} \colon S^{(m)} \to X$ . The map j is not geometric, but if the closed immersion $\eta \colon S \to X$ is regular in the sense that, locally on X, it is defined by a regular sequence, then it behaves as an open and affine immersion of a tubular neighborhood of S in X, as envisioned by Grothendieck.
Theorem 1.7 (Formal completion and recollement).
Let X be a Dirac scheme and let $\eta \colon S \to X$ be a regular closed immersion. Let $j \colon Y \to X$ be the formal completion of X along S and let $i \colon U \simeq X \smallsetminus S \to X$ be the inclusion of the open complement of S. In this situation, there is a stable recollement
and, in addition, the functor $j_*$ is texact.
The formal schemes that we will consider are the formal affine spaces, defined as follows. If S is a Dirac scheme and $\mathcal {E} \in {\mathrm {QCoh}}(S)^{\heartsuit }$ an $\mathcal {O}_S$ module locally free of finite rank, then the affine space associated with $\mathcal {E}$ is the affine map
and the formal affine space associated with $\mathcal {E}$ is its formal completion
along the zero section. We define a map of Dirac stacks $q \colon Y \to X$ to be a formal hyperplane if its base change along any map $\eta \colon S \to X$ from an affine Dirac scheme is equivalent to a formal affine space. This property is stable under base change, but it does not descend along effective epimorphisms in general, even if X is a Dirac scheme.Footnote ^{3} However, if we fix a section, then it does. We consider a formal affine space to be pointed by the zero section.
Definition 1.8. A pointed Dirac stack
is a pointed formal hyperplane if its base change along any map $\eta \colon S \to X$ from an affine Dirac scheme is equivalent to a pointed formal affine space.
By definition, the $\infty $ category of pointed formal hyperplanes over a Dirac stack is the full subcategory of the $\infty $ category of pointed Dirac stacks
spanned by the pointed formal hyperplanes. We show that it is equivalent to a $1$ category and that the property of being a pointed formal hyperplane descends along effective epimorphisms of Dirac stacks $f \colon X' \to X$ in the sense that
is a cartesian diagram of $\infty $ categories. So by Theorem 1.3, the $\infty $ category of pointed formal hyperplanes satisfies descent along effective epimorphisms:
Theorem 1.9 (Flat descent for pointed formal hyperplanes).
If $f \colon X' \to X$ is an effective epimorphism of Dirac stacks, then the canonical maps
are equivalences of $\infty $ categories.
We recall from [Reference Hesselholt and Pstrągowski12, Proposition A.1] that the righthand map in Theorem 1.9 is an equivalence, because the $\infty $ category of pointed formal hyperplanes is equivalent to a $1$ category, and moreover, the righthand term is equivalent to the $1$ category of pointed formal hyperplanes over $X'$ with descent data along $f \colon X' \to X$ .
Let $\mathcal {C}$ be an $\infty $ category which admits finite products. Following Lawvere, we let $\operatorname {\mathrm {Lat}}$ be the category of finitely generated free abelian groups and define the $\infty $ category $\operatorname {\mathrm {Ab}}(\mathcal {C})$ of abelian group objects in the $\infty $ category $\mathcal {C}$ to be the full subcategory
spanned by the functors that preserve finite products.
Definition 1.10. The $\infty $ category of formal groups over a Dirac stack X is the $\infty $ category
of abelian group objects in the $\infty $ category of formal hyperplanes over X.
The $\infty $ category $\operatorname {\mathrm {FGroup}}(X)$ is equivalent to a $1$ category. If $\mathcal {G}$ is a formal group over X, then the underlying hyperplane $\mathcal {G}(\mathbb {Z})$ is pointed by the zero section $\mathcal {G}(0) \to \mathcal {G}(\mathbb {Z})$ . Therefore, by Theorem 1.9, if $f \colon X' \to X$ is an effective epimorphism of Dirac stacks, then the canonical map
is an equivalence of $\infty $ categories, and the target $\infty $ category is equivalent to the $1$ category of formal groups over $X'$ with descent data along $f \colon X' \to X$ . Moreover, if $\mathcal {G}$ is a formal group over X, then we define its Lie algebra to be the vector bundle
associated with the conormal sheaf at the zero section. Since $\mathcal {G}$ is abelian, the Lie bracket on this vector bundle will be zero, so we do not take the trouble to define it.
We end with a number of applications of Dirac geometry to algebraic topology, which we encode as a sixfunctor formalism on the $\infty $ category of anima following the general theory of Liu–Zheng [Reference Liu and Zheng16] and Mann [Reference Mann24, Appendix A.5]. The definition of this sixfunctor formalism is informed by the fact that anima are ‘discrete’ objects, as opposed to ‘continuous’ objects. So we declare that every map of anima $f \colon T \to S$ is a local isomorphism and that a map of anima $f \colon T \to S$ is proper if its fibers are equivalent to finite sets, including the empty set. By [Reference Mann24, Proposition A.5.10], this determines a sixfunctor formalism on $\mathcal {S}$ that to an anima S assigns the functor $\infty $ category
and that to a map of anima $f \colon T \to S$ assigns the adjoint functors
where $f^! \simeq f^*$ is the restriction along f, and $f_!$ and $f_*$ are the left and right Kan extensions along f.
Given a commutative algebra in spectra E and a group in anima G, we apply the general theory to construct a formal Dirac scheme
over $S \simeq \operatorname {\mathrm {Spec}}(R)$ with $R \simeq \pi _*(E)$ , and we will consider situations in which q is a formal hyperplane. By varying G, we will obtain formal groups, and by varying E, we will obtain formal groups over geometric Dirac stacks, which we characterize by exhibiting their functors of points.
To this end, we recall from [Reference Lurie19, Theorem 5.6.2.10] that every group in anima G is the loop group $\Omega BG$ of a pointed $1$ connective anima
and we apply the general theory to the unique map
The $\infty $ category $\operatorname {\mathrm {Sp}}^{BG}$ is the $\infty $ category of spectra with Gaction, and $p_!$ and $p_*$ are the functors that to a spectrum with Gaction Y assign its homotopy orbit spectrum $p_!(Y) \simeq Y_{hG}$ and its homotopy fixed point spectrum $p_*(Y) \simeq Y^{hG}$ , respectively, whereas $p^*$ is the functor that to a spectrum X assigns the spectrum with trivial Gaction $Y \simeq p^*(X)$ . We only consider spectra with trivial Gaction.
Given a commutative algebra in spectra E, the spectrum
promotes to a commutative algebra in spectra, and the unit maps
promote to maps of commutative algebras in spectra. Hence, if $R \simeq \pi _*(E)$ and $A \simeq \pi _*(p_*p^*(E))$ are the respective Dirac rings of homotopy groups, then these maps give A the structure of an augmented Ralgebra. We now define
to be the formal completion of $X \simeq \operatorname {\mathrm {Spec}}(A)$ along the closed immersion
defined by the augmentation.
First, we let G be the Pontryagin dual $\widehat {L} \simeq \operatorname {\mathrm {Hom}}(L,U(1))$ of $L \in \operatorname {\mathrm {Lat}}$ . If E is complex orientable, then the formal scheme
is a formal hyperplane, and as L varies, this defines the Quillen formal group
associated with E. Moreover, by considering the Postnikov filtration of E, we obtain a canonical isomorphism of line bundles
from the spin $1$ affine line over S to the Lie algebra of the Quillen formal group. We now let E vary through the tensor powers of the commutative algebra in spectra $\operatorname {MU}$ representing complex cobordism. In this way, we obtain a simplicial formal group
that to $[n]$ assigns the Quillen formal group associated with $\operatorname {MU}^{\otimes [n]}$ . Its geometric realization is a formal group $\mathcal {G}^Q$ over the geometric Dirac stack
equipped with a trivialization $\phi ^Q$ of its Lie algebra. We show that Quillen’s theorem on complex cobordism, [Reference Quillen31, Theorem 2], implies and is implied by the statement that $(\mathcal {G}^Q,\phi ^Q)$ is the universal $1$ dimensional spin $1$ formal group equipped with a trivialization of its Lie algebra.
Theorem 1.11 (Quillen’s theorem).
Let T be a Dirac stack. The functor that to $f \colon T \to X^Q$ assigns the pair $(f^*\mathcal {G}^Q,f^*\phi ^Q)$ is an equivalence of anima from $X^Q(T)$ to the groupoid of pairs $(\mathcal {G},\phi )$ of a formal group $\mathcal {G}$ over T and an isomorphism
of line bundles over T from the spin $1$ affine line.
We also show that if T is an affine Dirac scheme and $(\mathcal {G},\phi ) \in X^Q(T)$ , then the trivialization $\phi $ lifts to a spin $1$ global coordinate on the formal group $\mathcal {G}$ .
Second, we let p be an odd prime number and let G be the ptorsion subgroup
of the Pontryagin dual of $L \in \operatorname {\mathrm {Lat}}$ . If $k \simeq \mathbb {F}_p$ , then for every commutative kalgebra in spectra E,
is a formal hyperplane, and as L varies, this defines the Milnor formal group
associated with E. The canonical inclusion i induces a map of formal groups
to the Quillen formal group. It is surjective, and its kernel
is again a formal group over S. This filtration gives rise to a grading of the Lie algebra $\operatorname {\mathrm {Lie}}(\mathcal {G}_E^M)$ , which we refer to as the ‘charge’ grading.Footnote ^{4} Moreover, from the Postnikov filtration of E, we obtain a canonical isomorphism of graded vector bundles over S,
where e has spin and charge $1$ and where $\gamma $ has spin $1$ and charge $0$ .
Letting E vary through the tensor powers of k, we obtain a simplicial filtered formal group that to $[n] \in \Delta ^{\operatorname {\mathrm {op}}}$ assigns the Milnor formal group associated with $k^{\otimes [n]}$ with the filtration defined by the map to the Quillen formal group. Its geometric realization is a filtered formal group $\operatorname {\mathrm {Fil}}\mathcal {G}^M$ over the geometric Dirac stack
equipped with a trivialization $\phi ^M$ of its graded Lie algebra. We show that Milnor’s theorem [Reference Milnor29] on the structure of the dual Steenrod algebra implies and is implied by the following statement.
Theorem 1.12 (Milnor’s theorem).
Let $S \simeq \operatorname {\mathrm {Spec}}(\mathbb {F}_p)$ with p odd and let T be an SDirac stack. The functor that to a map of SDirac stacks $f \colon T \to X^M$ assigns the pair $(f^*\operatorname {\mathrm {Fil}}\mathcal {G}^M,f^*\phi ^M)$ is an equivalence from $X^M(T)$ to the groupoid of pairs $(\operatorname {\mathrm {Fil}}\mathcal {G},\phi )$ of a filtered formal group $\operatorname {\mathrm {Fil}}\mathcal {G}$ over T, locally isomorphic to a filtered additive formal group, and an isomorphism of graded vector bundles over T,
where e has spin and charge $1$ and where $\gamma $ has spin $1$ and charge $0$ .
Since $k \simeq \mathbb {F}_p$ is a prime field, the face maps $d_0,d_1 \colon S_{k^{\otimes [1]}} \to S_{k^{\otimes [0]}}$ are equal, so the groupoid Dirac scheme $S_{k^{\otimes []}}$ is, in fact, a group Dirac scheme $G^M$ and
is its classifying Dirac stack. The dual Steenrod algebra is the Hopf algebra in Dirac kvector spaces corresponding to $G^M$ .
Finally, we express the method of descent, which goes back to Adams [Reference Adams1], in Dirac geometric terms. Given a map of commutative algebras in spectra $\phi \colon k \to E$ , we may form the Dirac stack
This Dirac stack and its coherent cohomology is particularly useful if the face maps in the simplicial Dirac scheme are flat, in which case, we construct a functor
that to a kmodule in spectra V assigns a quasicoherent $\mathcal {O}_X$ module $\mathcal {F}(V)$ . This functor gives rise to an Adams filtration of V, and the associated spectral sequence is the Adams or descent spectral sequence
It starts from the coherent cohomology of X with coefficients in the halfinteger Serre twists of $\mathcal {F}(V)$ , and it converges conditionally to the homotopy groups of the completion of V with respect to the Adams filtration. A particularly understandable application of this method was given by Liu–Wang [Reference Liu and Wang15] in the situation of the basechange map in topological Hochschild homology
where the descent spectral sequence is concentrated on the lines $1 \leq i \leq 0$ and therefore degenerates for degreereasons.
Terminology. We write $\operatorname {\mathrm {Ab}}$ for the symmetric monoidal category of $\mathbb {Z}$ graded abelian groups with the Koszul sign in the symmetry isomorphism, and we define the category of Dirac rings to be the category $\operatorname {\mathrm {CAlg}}(\operatorname {\mathrm {Ab}})$ of commutative algebras therein. We use the terminology of Clausen–Jansen [Reference Clausen and Jansen8, Theorem 2.19] and say that a map $f \colon L \to K$ between small $\infty $ categories is a $\smash { \varinjlim }$ equivalence if for every diagram $X \colon K \to \mathcal {C}$ , the induced map
is an equivalence. These maps are referred to as cofinal in [Reference Lurie20, Section 4.1].
2 Dirac stacks
We wish to define a Dirac stack to be a presheaf $X \colon \operatorname {\mathrm {Aff}}^{\operatorname {\mathrm {op}}} \to {\mathcal {S}}$ of anima on the category of affine Dirac schemes, which is well behaved in the sense that

(1) X to be determined by a small amount of data, and

(2) X to be determined by local data,
and both of these conditions require some care.
In the case of $(1)$ , the category $\operatorname {\mathrm {Aff}}$ of affine Dirac schemes is not small, so not every presheaf of anima on $\operatorname {\mathrm {Aff}}$ can be written as a small colimit of representable presheaves. Our solution, which is similar to the one used by Clausen–Scholze [Reference Clausen and Scholze9], is to only consider the presheaves which can be expressed as small colimits of representable presheaves. We show in Appendix A that this condition is equivalent to the functor X being accessible and that it leads to an excellent theory.
In the case of (2), this leads us to consider presheaves which satisfy the sheaf condition with respect to some Grothendieck topology. We would like to use the fpqctopology, which is very strong but still subcanonical. In general, the category of fpqccovering sieves of a fixed affine Dirac scheme does not have a small cofinal subcategory, and hence, it is not clear that a sheafication functor exists. We extend the work of Waterhouse [Reference Waterhouse35] to show our solution to (1) also solves (2) in that an accessible presheaf admits a sheafification which is again accessible.
Remark 2.1. We will show that there exists a fully faithful embedding
of the category of Dirac schemes, which we defined in [Reference Hesselholt and Pstrągowski12, Definition 2.29], into the $\infty $ category of Dirac stacks defined according to the above principles.
2.1 The flat topology
The category $\operatorname {\mathrm {Aff}}$ of affine Dirac schemes is not small, but it is coaccessible in the sense that its opposite category $\operatorname {\mathrm {Aff}}^{\operatorname {\mathrm {op}}} \simeq \operatorname {\mathrm {CAlg}}(\operatorname {\mathrm {Ab}})$ is accessible. We show in Proposition A.2 that a presheaf
is an accessible functor if and only if it is a small colimit of representable presheaves.
Definition 2.2. The $\infty $ category of Dirac prestacks is the full subcategory
spanned by the accessible presheaves.
We show in Theorem A.10 that ${\mathcal {P}}(\operatorname {\mathrm {Aff}})$ is complete and cocomplete and that it satisfies the Giraud axioms, with the exception that it is not presentable. We also show that the Yoneda embedding factors through a fully faithful functor
and that this functor exhibits $\mathcal {P}(\operatorname {\mathrm {Aff}})$ as the cocompletion of $\operatorname {\mathrm {Aff}}$ .
We proceed to define the flat topology on $\operatorname {\mathrm {Aff}}$ and to show that it gives rise to a wellbehaved subcategory $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}}) \subset {\mathcal {P}}(\operatorname {\mathrm {Aff}})$ of accessible flat sheaves. We recall the general notions of sieves and Grothendieck topologies in Appendix B.
Definition 2.3. A sieve $j \colon U \to h(S)$ on an affine Dirac scheme S is a covering sieve for the flat topology if there exists a finite family $(f_i \colon T_i \to S)_{i \in I}$ of maps of affine Dirac schemes such that the induced map
is faithfully flat and such that each $h(f_i) \colon h(T_i) \to h(S)$ factors through j.
Proposition 2.4. The collection of covering sieves for the flat topology forms a finitary Grothendieck topology on $\operatorname {\mathrm {Aff}}$ .
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 basechange along arbitrary maps.
Proposition 2.5. A presheaf $X \colon \operatorname {\mathrm {Aff}}^{\operatorname {\mathrm {op}}} \to {\mathcal {S}}$ is a sheaf for the flat topology if and only if it has the following properties:

(i) For every finite family $(S_i)_{i \in I}$ of objects in $\operatorname {\mathrm {Aff}}$ , the induced map

(ii) For every faithfully flat map $f \colon T \to S$ in $\operatorname {\mathrm {Aff}}$ , the induced map
Proof. This is [Reference Lurie23, Proposition A.3.3.1].
Definition 2.6. The $\infty $ category of Dirac stacks is the full subcategory
spanned by the sheaves for the flat topology.
Let us emphasize that for a presheaf $X \colon \operatorname {\mathrm {Aff}}^{\operatorname {\mathrm {op}}} \to {\mathcal {S}}$ to be a Dirac stack, it must be both accessible and a sheaf for the flat topology. The former is the case if and only if the equivalent conditions of Proposition A.2 are satisfied, and the latter is the case if and only if the conditions of Proposition 2.5 are satisfied.
Remark 2.7. We will only consider sheaves on $\operatorname {\mathrm {Aff}}$ for the flat topology. However, it is also possible to consider sheaves for the étale topology, whose covering sieves are defined by substituting étale for flat in Definition 2.3. Since étale maps in $\operatorname {\mathrm {Aff}}$ are flat by [Reference Hesselholt and Pstrągowski12, Theorem 1.9], the flat topology is finer than the étale topology.
A priori, it is not clear that the inclusion $\iota \colon {\mathcal {P}}(\operatorname {\mathrm {Aff}}) \to \operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ admits a left adjoint ‘sheafification’ functor. However, we now show that, by an argument of Waterhouse [Reference Waterhouse35, Theorem 5.1], such a functor does indeed exist. It will be convenient to work with Dirac rings instead of affine Dirac schemes, and we will use the following notion throughout the proof.
Definition 2.8. A subextension of a map of Dirac rings $f \colon A \to B$ is a diagram of rings of the form
with i and j inclusions of subDirac rings. The subextensions of $f \colon A \to B$ form a partially ordered set $\operatorname {\mathrm {SubExt}}(f)$ .
Remark 2.9. The partially ordered set $\operatorname {\mathrm {SubExt}}(f)$ of $f \colon A \to B$ is equivalent to the full subcategory of $\operatorname {\mathrm {Fun}}(\Delta ^1,\operatorname {\mathrm {CAlg}}(\operatorname {\mathrm {Ab}}))_{/f}$ spanned by the monomorphisms.
Lemma 2.10. Let $\kappa $ be an uncountable cardinal, let $f \colon A \to B$ be a faithfully flat map of Dirac rings, and let $g' \colon C' \to D'$ be a subextension of $f \colon A \to B$ such that $C'$ and $D'$ are $\kappa $ small. There exists a subextension $g \colon C \to D$ of $f \colon A \to B$ such that $g' \colon C' \to D'$ is a subextension of $g \colon C \to D$ , such that C and D are $\kappa $ small, and such that $g \colon C \to D$ is faithfully flat.
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
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
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
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 subDirac 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 $C_{r1}$ . 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 subDirac ring obtained by adjoining to $D_{r1}$ 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.
Corollary 2.13. Let $f \colon A \to B$ be a faithfully flat map of Dirac rings and $\kappa $ an uncountable regular cardinal. Let $\operatorname {\mathrm {SubExt}}_{\kappa }(f)$ be the poset of those subextensions $g \colon C \to D$ such that both C and D are $\kappa $ small, and let $\operatorname {\mathrm {SubExt}}^{{\operatorname {f}\!\operatorname {f}}}_{\kappa }(f)$ be the subposet of those $g \colon C \to D$ that, in addition, are faithfully flat. In this situation, the inclusion
is cofinal. In particular, the partially ordered set $\operatorname {\mathrm {SubExt}}^{{\operatorname {f}\!\operatorname {f}}}_{\kappa }(f)$ is $\kappa $ filtered.
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.
Lemma 2.14. Let $f \colon A \to B$ be a faithfully flat map of Dirac rings, let $\kappa $ be an uncountable regular cardinal, and let $[n] \in \Delta _{+}$ . The map
from the poset of $\kappa $ small subextensions of f to the category of $\kappa $ small Dirac rings over $B^{\otimes _{A} [n]}$ that to $g \colon C \to D$ assigns the induced map
is a $\varinjlim $ equivalence as a functor of $\infty $ categories.
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
is weakly contractible. Explicitly, the objects of this category are pairs
of a $\kappa $ small subextension of $f \colon A \to B$ and a map h such that the composite
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
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
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.
For every infinite cardinal $\kappa $ , the flat topology on $\operatorname {\mathrm {Aff}}$ induces a topology on the full subcategory ${\mathrm {Aff}}_{\kappa } \subset \operatorname {\mathrm {Aff}}$ . A presheaf $X \colon {\mathrm {Aff}}_{\kappa }^{\operatorname {\mathrm {op}}} \to {\mathcal {S}}$ is a sheaf for this topology if and only if the conditions (i)–(ii) of Proposition 2.5 hold for all finite families $(S_i)_{i \in I}$ of objects in ${\mathrm {Aff}}_{\kappa }$ and for all faithfully flat maps $f \colon T \to S$ in ${\mathrm {Aff}}_{\kappa }$ , respectively.
Theorem 2.15. For every uncountable regular cardinal $\lambda $ , the left Kan extension
preserves sheaves with respect to the flat topology.
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
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.
Remark 2.16. Let $\operatorname {\mathrm {Shv}}({\mathrm {Aff}}_{\kappa }) \subset {\mathcal {P}}({\mathrm {Aff}}_{\kappa })$ be the full subcategory spanned by the sheaves for the flat topology. Theorem 2.15 shows that the canonical map
from the colimit indexed by the large partially ordered set of uncountable regular cardinals $\lambda $ in the universe of discourse of the large $\infty $ categories $\operatorname {\mathrm {Shv}}({\mathrm {Aff}}_{\lambda })$ to the large $\infty $ category $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ of accessible stacks on $\operatorname {\mathrm {Aff}}$ is an equivalence.
Corollary 2.17. The inclusion of the full subcategory spanned by the sheaves for the flat topology admits a left adjoint sheafification functor,
and the sheafification functor L preserves finite limits.
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
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
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.
Remark 2.18. The sheafification functor $L \colon {\mathcal {P}}(\operatorname {\mathrm {Aff}}) \to \operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ is a localization with respect to the large set W consisting of the following colimitinterchange maps:

(i) For every finite family $(S_i)_{i \in I}$ of objects in $\operatorname {\mathrm {Aff}}$ , the map

(ii) For every faithfully flat map $f \colon T \to S$ in $\operatorname {\mathrm {Aff}}$ , the map
Indeed, this follows from Proposition 2.5.
We now show that the $\infty $ category of Dirac stacks satisfies the following variant on the $\infty $ categorical Giraud axioms.
Theorem 2.19. The $\infty $ category $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ has the following properties:

(i) It is complete and cocomplete, and it is generated under small colimits by the essential image of the Yoneda embedding $h \colon \operatorname {\mathrm {Aff}} \to \operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ , which consists of $\omega _1$ compact objects and is coaccessible.

(ii) Colimits in $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ are universal.

(iii) Coproducts in $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ are disjoint.

(iv) Every groupoid in $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ is effective.
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
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.
Remark 2.20. In the language of Question A.20, part (i) of Theorem 2.19 would be phrased as saying that $\operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ is an $\omega _1$ macropresentable $\infty $ category.
We next deduce from [Reference Lurie20, Theorem 6.1.3.9] the following fundamental descent theorem. We will use later to prove that $\infty $ categories of Dirac formal groups over Dirac stacks descend along effective epimorphisms.
Theorem 2.21. The slice $\infty $ category functor
preserves small limits.
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
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].
2.2 Dirac schemes as stacks
We show that that the category of Dirac schemes embeds fully faithfully into the $\infty $ category of Dirac stacks.
Definition 2.22. The restricted Yoneda embedding is the composition
of the Yoneda embedding and the restriction along the canonical inclusion.
If X is a Dirac scheme, then we write $h(X)$ for its image by the restricted Yoneda embedding. We proceed to show that $h(X)$ is a Dirac stack.
Lemma 2.23. If X is affine, then the presheaf $h(X)$ is a Dirac stack.
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].
Proposition 2.24. If X is a Dirac scheme, then the presheaf $h(X)$ is a sheaf for the flat topology.
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
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.
Lemma 2.25. Let X be a Dirac scheme and let $\mathfrak {B}$ be a basis for the topology on its underlying space $X$ such that if $U,V \in \mathfrak {B}$ , then $U \cap V \in \mathfrak {B}$ . If $h(U)$ is accessible for all $U \in \mathfrak {B}$ , then so is $h(X)$ .
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
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 basechange
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
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.
Proposition 2.26. If X is a Dirac scheme, then the presheaf $h(X)$ is accessible.
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.
Theorem 2.27. The restricted Yoneda embedding defines a fully faithful functor
from the category of Dirac schemes to the $\infty $ category of Dirac stacks.
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.
A small diagram $X \colon K \to \operatorname {\mathrm {Sch}}$ may not admit a colimit, and even if it does, then the colimit is generally not meaningful. Instead, it is the colimit of the composite diagram $h(X) \colon K \to \operatorname {\mathrm {Shv}}(\operatorname {\mathrm {Aff}})$ , which always exists, that is meaningful. We show, however, that if K is static, then the two colimits do agree.
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 lefthand 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 quasicompact.
2.3 Geometric maps
The $\infty $ category of all Dirac stacks is quite large, and in practice, one is often interested in the smaller subcategory of geometric Dirac stacks, which, informally, is the closure of the full subcategory of Dirac schemes under quotients by flat groupoids. The precise definition, however, is more delicate and follows Lurie [Reference Lurie18, Chapter 26] and Töen–Vezzosi [Reference Toën and Vezzosi33]. Geometric Dirac stacks are formally similar to Artin stacks, the main differences being the following:

1. We consider sheaves of anima, as opposed to sheaves of groupoids.

2. We consider sheaves on affine Dirac schemes, as opposed to affine schemes.

3. We allow quotients by flat groupoids, as opposed to only smooth grouoids.
More generally, we define the relative notion of a geometric map between Dirac stacks. We follow Lurie [Reference Lurie18, Chapter 26], who sets up the analogous theory in derived algebraic geometry. The definition is recursive.
Definition 2.29. Let $f \colon Y \to X$ be a map of Dirac stacks.

(1) The map $f \colon Y \to X$ is $0$ geometric if for every map $\eta \colon S \to X$ from a Dirac scheme, the basechange $f_S \colon Y_S \to S$ of f along $\eta $ is equivalent to a map of Dirac schemes.

(2) The map $f \colon Y \to X$ is $0$ submersive if for every map $\eta \colon S \to X$ from a Dirac scheme, the basechange $f_S \colon Y_S \to S$ of f along $\eta $ is an effective epimorphism and equivalent to a flat map of Dirac schemes.
Remark 2.30 (Affine source suffices).
In Definition 2.29, it suffices to consider maps $\eta \colon S \to X$ with S an affine Dirac scheme. However, the source of the basechange $f_S \colon Y_S \to S$ will typically not be an affine Dirac scheme, even if S is affine.
Remark 2.31 (Effective epimorphisms).
It follows from Proposition B.6 that a map of Dirac stacks $p \colon Y \to X$ is an effective epimorphism if and only if for every map $\eta \colon S \to X$ from an affine Dirac scheme, there exists a diagram
where q is a faithfully flat map of affine Dirac schemes. So $p \colon Y \to X$ is an effective epimorphism if and only if, locally for the flat topology on the source, every map $\eta \colon S \to X$ from an affine Dirac scheme admits a factorization through f.
Remark 2.32 (Submersive maps of Dirac schemes).
Let $f \colon Y \to X$ be a map of Dirac schemes. The map f is automatically $0$ geometric because the embedding of Dirac schemes in Dirac stacks preserves fiber products, and Remark 2.31 shows that f is $0$ submersive if and only if it is an $\operatorname {\mathrm {fpqc}}$ covering in the sense that f is flat and for every affine open $U \subset X$ , there exists a quasicompact open $V \subset f^{1}(U)$ such that $f_V \colon V \to U$ is surjective. This is the analogue for Dirac schemes of the definition of an $\operatorname {\mathrm {fpqc}}$ covering given in [32, Tag 022B].
Definition 2.33. Let $f \colon Y \to X$ be a map of Dirac stacks and $n \geq 1$ an integer.

(1) The map $f \colon Y \to X$ is ngeometric if for every basechange $f_S \colon Y_S \to S$ along a map $\eta \colon S \to X$ from a Dirac scheme, there exists an $(n1)$ submersive map $p \colon T \to Y_S$ from a Dirac scheme.

(2) The map $f \colon Y \to X$ is nsubmersive if for every basechange $f_S \colon Y_S \to S$ along a map $\eta \colon S \to X$ from a Dirac scheme, there exists an $(n1)$ submersive map $p \colon T \to Y_S$ from a Dirac scheme such that the composite map
The map $f \colon Y \to X$ is geometric if it is ngeometric for some $n \geq 0$ , and it is submersive if it is nsubmersive for some $n \geq 0$ .
Remark 2.34 (Stability under basechange).
In Definition 2.33, it suffices to consider maps $\eta \colon S \to X$ with S an affine Dirac scheme. The properties of being ngeometric and being nsubmersive are preserved under basechange along any map.
Warning 2.35 (The value of n is irrelevant).
Contrary to what Definition 2.33 might suggest, it is the properties of a map of Dirac stacks of being geometric or submersive that are important, whereas the particular n for which the map in question is ngeometric or nsubmersive is of little relevance. However, the recursive nature of the definitions is useful for making inductive proofs.
Lemma 2.36. Let $f \colon Y \to X$ be a map of Dirac stacks and let $n \geq 0$ be an integer. If f is ngeometric, then f is also $(n+1)$ geometric. If f is nsubmersive, then f is also $(n+1)$ submersive.
Proof. We argue by induction on $n \geq 0$ . If f is $0$ geometric, then, by definition, the basechange $f_S \colon Y_S \to S$ 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 ngeometric, then for every basechange $f_S \colon Y_S \to S$ of f along a map $\eta \colon S \to X$ from a Dirac scheme S, there exits an $(n1)$ submersive map $p \colon T \to Y_S$ from a Dirac scheme. By the inductive hypothesis, the map p is also nsubmersive, which shows that f is $(n+1)$ geometric. The same argument shows that if f is nsubmersive, then it is $(n+1)$ submersive.
Proposition 2.37. If a map of Dirac stacks $f \colon Y \to X$ is nsubmersive for some $n \geq 0$ , then it is an effective epimorphism.
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 nsubmersive and X affine, there exists a factorization
with p an $(n1)$ 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.
Proposition 2.38. Let $g \colon Z \to Y$ and $f \colon Y \to X$ be maps of Dirac stacks and let $n \geq 0$ . If f and g are both ngeometric or both nsubmersive, then so is $fg$ .
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 ngeometric. We first choose an $(n1)$ submersive map $p \colon U \to Y_S$ from a Dirac scheme. Since g is ngeometric, so is $g_U$ , and hence, we may choose an $(n1)$ submersive map $q \colon V \to Z_U$ from a Dirac scheme. By the inductive hypothesis, the composition of the $(n1)$ submersive maps $p'$ and q is $(n1)$ submersive. This shows that $fg$ is ngeometric. If f and g are nsubmersive, then we may choose the maps p and q such that $f_Sp$ and $g_Uq$ are $0$ submersive. But then also
is $0$ submersive, so we conclude that $fg$ is nsubmersive.
Lemma 2.39. If $f \colon Y \to X$ is an effective epimorphism of Dirac stacks, then for every map $\eta \colon V \to X$ from a Dirac scheme, there exists a diagram
with $p \colon W \to V$ a $0$ submersion from a Dirac scheme W.
Proof. Let $(S_i)_{i \in I}$ be a family with $S_i \subset V$ affine open and