Coaffine Stacks in Algebraic Geometry

• Coaffine stacks are algebraic stacks defined by their affinity to affine schemes through hypergroupoid and tensor categorical presentations.
• They employ derived methods, abelian tensor categories, and 1-affineness to reconstruct global objects from local affine data.
• Their coherent completeness, descent stability, and Tannakian duality ensure well-behaved moduli spaces in algebraic and derived geometry.

Coaffine stacks are a class of algebraic and higher-categorical stacks characterized by their local or global affinity to affine schemes, tensorial structures, and descent-theoretic behaviors. The term encompasses contexts ranging from presentations via simplicial diagrams of affine schemes to stacks reconstructed from abelian tensor categories and those with affine stabilizers or coherent completeness properties. Coaffine stacks consistently appear where global objects are assembled from affine (or affine-like) local data, and where categorical or stack-theoretic properties are governed by algebraic and homological attributes closely paralleling those of affine schemes.

1. Presentations via Simplicial Diagrams of Affine Schemes

A prominent approach to coaffine stacks is through explicit simplicial (or cosimplicial) diagrams of affine schemes, known as “hypergroupoids” or “n-hypergroupoids” in the sense of Duskin–Glenn. In this framework, an Artin n-hypergroupoid is a simplicial affine scheme $X$ such that the so-called partial matching maps

$$X_m \to \operatorname{Hom}_{\mathrm{sSet}}(\Lambda^{m,k}, X)$$

are smooth surjections for all $m, k$ and are isomorphisms for all $m > n$. Such a diagram is “coskeletal,” meaning it is determined in degrees $\leq n+1$. The associated stack is recovered by hypersheafification, yielding a strongly quasi-compact $n$-geometric Artin stack. This explicit construction mimics the classical atlas of a manifold, but here every object and morphism is described entirely via diagrams of affine schemes.

In derived algebraic geometry, affine schemes are replaced by derived affine schemes (that is, Spec of simplicial commutative rings or connective dg-algebras) and the hypergroupoid conditions are appropriately derived: Reedy fibrancy and surjectivity is replaced by weak equivalence conditions on the matching maps. The derived stack is then constructed as the hypersheafification of this simplicial diagram. This method enables concrete paper of (higher, derived) stacks built as “homotopy colimits” of affine (or derived affine) pieces, with descent and cohomology encoded in finite diagrams, aligning precisely with the notion of coaffine stacks as those modeled entirely on affine objects (Pridham, 2011).

2. Categorical Reconstruction from Abelian Tensor Categories

An alternative and categorical perspective treats coaffine stacks as those arising from abelian tensor categories. Given an abelian tensor category $(A, \otimes)$, the associated stack $\mathcal{A}$ is constructed by considering, over a base $S$, the symmetric monoidal (tensor) functors $F: A \to \mathcal{B}_S$, where $\mathcal{B}_S$ is typically the category of quasi-coherent sheaves on $S$. This construction, formalized as

$$\mathcal{A}(S) := \operatorname{Hom}_\otimes(A, \mathcal{B}_S),$$

is highly sensitive to the tensor structure on $A$. For example, the stack defined from $(\mathrm{QCoh}(X), \otimes_X)$ recovers the scheme $X$, and the stack from $(G\text{-rep}, \otimes_{\mathcal{O}_G})$ is the classifying stack $BG$, while $(G\text{-rep}, \otimes_{z})$ yields disjoint unions of points indexed by irreducible representations—providing a “dual stack” to $BG$.

A coaffine stack, in this context, is characterized by the ability to patch the endomorphism rings $\mathcal{O}_{\mathcal{B}_S}(1_S)$ to form a sheaf of rings which governs the geometry in an affine-like manner. The construction’s functorial “Tannakian” flavor—where the stack is encoded by its tensor category—centralizes the concept of “affineness” or “coaffineness” to categorical descent data and monoidal structures (Liu et al., 2012).

Gluing techniques using 2-descent allow one to assemble global stacks from local affine data, which is crucial in constructing global G-gerbes and their duals and capturing the full extent of coaffine behavior through descent and cocycle conditions.

3. 1-Affineness and Sheaves of Categories

A classificatory and “categorified” notion arises in the paper of 1-affineness of (pre)stacks, which generalizes classical affineness to the context of sheaves of categories. For a prestack $Y$, 1-affineness is defined by the equivalence of categories

$$\operatorname{Loc}_Y : \mathrm{QCoh}(Y)\text{-mod} \stackrel{\sim}{\longleftrightarrow} \mathrm{ShvCat}(Y) : \Gamma^{\mathrm{enh}}_Y,$$

where $\mathrm{ShvCat}(Y)$ is the (higher) category of sheaves of DG categories on $Y$. This property captures when the module category over the monoidal DG category $\mathrm{QCoh}(Y)$ fully determines (and is determined by) the sheaves of categories on $Y$.

Key instances are classifying stacks $BG$ for affine algebraic groups $G$ of finite type, for which

$$\mathrm{ShvCat}(BG) \simeq \mathrm{Rep}(G)\text{-mod}$$

and 1-affineness robustly holds. Many quasi-compact, quasi-separated algebraic spaces and algebraic stacks fall into this class, although certain infinite-dimensional or non-finite-type examples do not. The flexibility of 1-affineness, or “coaffineness,” in this context extends the landscape of tractable stacks well beyond schemes with globally affine structure sheaves, and it is central to geometric representation theory and derived algebraic geometry (Gaitsgory, 2013).

4. Tannakian Reconstruction and Hom-stacks

Coaffine stacks also emerge naturally in Tannaka-type duality theorems for algebraic stacks. For a stack $X$ with affine stabilizers, a central result is that

$$\omega_X(T) : \operatorname{Hom}(T, X) \to \operatorname{Hom}_{c\otimes, \simeq}(\mathrm{QCoh}(X), \mathrm{QCoh}(T))$$

is an equivalence for any locally excellent test stack $T$. Thus, $X$ is “tensorial,” i.e., determined by its symmetric monoidal quasi-coherent category. Critically, the global presence of affineness is not required; rather, the affine nature of local stabilizers suffices for full Tannakian reconstruction.

This advances the algebraicity and good behavior of Hom-stacks: if $X$ is quasi-compact and quasi-separated with affine stabilizers, then $\underline{\operatorname{Hom}}_S(Z, X)$ is algebraic under broad hypotheses—even when the diagonal of $X$ is not (quasi-)affine. Thus, “coaffine” in this setting refers to stacks with local affine features (e.g., affine inertia) guaranteeing that their categorical (and moduli) structures are well-behaved (Hall et al., 2014).

5. Cohomological Properties and Coherent Completeness

A theme in the paper of stacks in positive and mixed characteristic is the notion of coherent completeness, cohomological properness, and formal function theorems. For quotient stacks $[\,\mathrm{Spec}\,A/G\,]$ with $G$ reductive and $A^G$ complete local, coherent completeness along the residual gerbe means that every coherent sheaf on the formal completion arises from a unique (up to isomorphism) coherent sheaf on the stack itself. This ties into formal GAGA-type theorems and the effectiveness of deformation theory in stacks with affineness only in their stabilizers.

Coaffine stacks, in this context, are those where the map to their moduli space is cohomologically proper (structure sheaf has vanishing higher cohomology), and the formal completion functor is fully faithful—enabling algebraization results and strong local-to-global principles (Alper et al., 2023).

6. Descent Theory and Gluing Local Affine Data

Descent theory, particularly 2-descent for stacks, is foundational for coaffine stacks. If for a covering $S' \to S$ the pullback $X'$ of $X$ is coaffine (or has the needed affine-like properties), then, through effectivity of 2-descent, these properties descend to $X$ over $S$. For Galois covers, descent data is recast in terms of group actions with 1-isomorphisms and coherence 2-morphisms, and the construction of quotient stacks via group actions is interpreted as a descent mechanism for affineness.

Thus, the property of being coaffine is both local for the fppf or étale topology and stable under descent and gluing, which is essential for construction of moduli stacks, G-gerbes, and orbifold targets whose geometric or categorical structure is governed by local affine patches or presentations (Fortman, 18 Apr 2025).

7. Geometric and Dynamical Aspects: Coaffine Representations

In geometric representation theory, “coaffine” encompasses moduli of representations factoring through the affine group. In convex projective geometry, convex cocompact subgroups of $\mathrm{SL}(4,\mathbb R)$ with image in the coaffine subgroup yield moduli spaces (“coaffine stacks” in this sense) parameterized entirely by boundary bending data, expressed as affine measured laminations based on underlying convex projective structures with Hitchin holonomy. The projectivized space of bending data

$\mathbb{P}\mathcal{ML}^{\rho}(S) \simeq S^{6g-7}$

mirrors the Thurston-type topological structure but encodes subtler dynamical features of the affine factor, such as holonomy of the “slithering” connection (Bobb et al., 22 Apr 2024).

8. Summary Table: Coaffine Stack Constructions

Framework	Affine Datum	Key Notion of Coaffineness
Simplicial Diagram	Affine schemes	Gluing via hypergroupoids
Abelian Tensor Category	QCoh, Rep(G), etc.	Stack reconstructed, descent of O
1-Affineness	QCoh(Y), modules	Reconstruction of ShvCat
Tannakian Duality/Stacks	QCoh(X)	Affine stabilizers, categorical lift
Cohomological Completeness	Coh(X), cohomology	GAGA-type, formal completion
2-Descent, Group Quotients	Affine local pieces	Stability under descent
Convex Cocompact Representation	Hitchin linear part	Boundary data via affine lamination

Conclusion

Coaffine stacks unify presentations, descent theory, categorical reconstructions, and cohomological properties under the umbrella of affinity to affine data, whether at the level of local atlases, categorical tensor structures, stabilizer groups, or cohomological completeness. Their geometric and categorical tractability is preserved not only by explicit affine presentations but by sophisticated descent mechanisms and Tannakian duality principles, solidifying their foundational role in algebraic geometry, moduli theory, higher category theory, and geometric representation theory (Pridham, 2011, Liu et al., 2012, Gaitsgory, 2013, Hall et al., 2014, Alper et al., 2023, Bobb et al., 22 Apr 2024, Fortman, 18 Apr 2025).