Topological Stacks Research Articles

Stratified homotopy theory of topological ∞-stacks: A toolbox

We exploit the theory of ∞-stacks to provide some basic definitions and calculational tools regarding stratified homotopy theory of stratified topological stacks.

Numerable open covers and representability of topological stacks

We prove that the class of numerable open covers of topological spaces is the smallest class that contains covers with pairwise disjoint elements and numerable covers with two elements, closed under composition and coarsening of covers. We apply this result to establish an analogue of the Brown–Gersten property for numerable open covers of topological spaces: a simplicial presheaf on the site of topological spaces satisfies the homotopy descent property for all numerable open covers if and only if it satisfies it for numerable covers with two elements and covers with pairwise disjoint elements. We also prove a strengthening of these results for manifolds, ensuring that covers with two elements can be taken to have a specific simple form. We apply these results to deduce a representability criterion for stacks on topological spaces similar to arXiv:1912.10544. We also use these results to establish new simple criteria for chain complexes of sheaves of abelian groups to satisfy the homotopy descent property. This article is available at arXiv:2203.03120v2.

English

A new approach to etale homotopy theory is presented which applies to a much broader class of objects than previously existing approaches, namely it applies not only to all schemes (without any local Noetherian hypothesis), but also to arbitrary higher stacks on the etale site of such schemes, and in particular to all algebraic stacks. This approach also produces a more refined invariant, namely a pro-object in the infinity category of spaces, rather than in the homotopy category. We prove a profinite comparison theorem at this level of generality, which states that if $\mathcal{X}$ is an arbitrary higher stack on the etale site of affine schemes of finite type over $\mathbb{C},$ then the etale homotopy type of $\mathcal{X}$ agrees with the homotopy type of the underlying stack $\mathcal{X}_{top}$ on the topological site, after profinite completion. In particular, if $\mathcal{X}$ is an Artin stack locally of finite type over $\mathbb{C}$, our definition of the etale homotopy type of $\mathcal{X}$ agrees up to profinite completion with the homotopy type of the underlying topological stack $\mathcal{X}_{top}$ of $\mathcal{X}$ in the sense of Noohi. In order to prove our comparison theorem, we provide a modern reformulation of the theory of local systems and their cohomology using the language of $\infty$-categories which we believe to be of independent interest.

On spin structures and orientations for gauge-theoretic moduli spaces

Let X be a compact manifold, G a Lie group, P→X a principal G-bundle, and BP the infinite-dimensional moduli space of connections on P modulo gauge, as a topological stack. For a real elliptic operator E• we previously studied orientations on the real determinant line bundle over BP, twisting E• by connections ∇Ad(P). These are used to construct orientations in the usual sense on smooth gauge theory moduli spaces, and have been extensively studied since the work of Donaldson [15–17].Here we consider complex elliptic operators F• and introduce the idea of spin structures, square roots of the complex determinant line bundle of F• twisted over BP. These may be used to construct spin structures in the usual sense on smooth complex gauge theory moduli spaces. We study the existence and classification of such spin structures.Our main result identifies spin structures on X with orientations on X×S1. Thus, if P→X and Q→X×S1 are principal G-bundles with Q|X×{1}≅P, we relate spin structures on (BP,F•) to orientations on (BQ,E•) for a certain class of operators F• on X and E• on X×S1.Combined with [25], we obtain canonical spin structures for positive Diracians on spin 6-manifolds and gauge groups G=U(m),SU(m). In a sequel [26] we will apply this to define canonical orientation data for all Calabi–Yau 3-folds X over C, as in Kontsevich and Soibelman [28, §5.2], solving a long-standing problem in Donaldson–Thomas theory.

On orientations for gauge-theoretic moduli spaces

Let X be a compact manifold, D:Γ∞(E0)→Γ∞(E1) a real elliptic operator on X, G a Lie group, P→X a principal G-bundle, and BP the infinite-dimensional moduli space of all connections ∇P on P modulo gauge, as a topological stack. For each [∇P]∈BP, we can consider the twisted elliptic operator D∇Ad(P):Γ∞(Ad(P)⊗E0)→Γ∞(Ad(P)⊗E1) on X. This is a continuous family of elliptic operators over the base BP, and so has an orientation bundle OPE•→BP, a principal Z2-bundle parametrizing orientations of KerD∇Ad(P)⊕CokerD∇Ad(P) at each [∇P]. An orientation on (BP,E•) is a trivialization OPE•≅BP×Z2.In gauge theory one studies moduli spaces MPga of connections ∇P on P satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds (X,g). Under good conditions MPga is a smooth manifold, and orientations on (BP,E•) pull back to orientations on MPga in the usual sense of differential geometry under the inclusion MPga↪BP. This is important in areas such as Donaldson theory, where one needs an orientation on MPga to define enumerative invariants.We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on (BP,E•), after fixing some algebro-topological information on X. We use these to construct canonical orientations on gauge theory moduli spaces, including new results for moduli spaces of flat connections on 2- and 3-manifolds, instantons, the Kapustin–Witten equations, and the Vafa–Witten equations on 4-manifolds, and the Haydys–Witten equations on 5-manifolds.

Étale stacks as prolongations

We give a complete and categorical characterization of étale stacks (generalized orbifolds) in various geometric contexts, including differentiable stacks and topological stacks. This characterization states that étale stacks are precisely those stacks that arise as prolongations of stacks on a site of spaces and local homeomorphisms, and makes no mention of an atlas or groupoid presentation. Moreover, we show that the bicategory of étale differentiable stacks and local diffeomorphisms is equivalent to the bicategory of stacks on the site of smooth manifolds and local diffeomorphisms. An analogous statement holds for other flavors of manifolds (topological, Ck, complex, super...), and topological spaces locally homeomorphic to a given space X. A slight modification of this result also holds in an even more general context, including all étale topological stacks, and Zariski étale stacks, which are analogues of Deligne-Mumford stacks for the Zariski topology, and we also sketch a proof of an analogous characterization of classical Deligne-Mumford algebraic stacks (which we prove in full in [7] by different methods). We go on to characterize effective étale stacks (e.g. reduced orbifolds) as precisely those stacks arising as the prolongations of sheaves of sets (rather than stacks of groupoids). It follows that étale stacks (and in particular orbifolds) induce a small gerbe over their effective part, and all gerbes over effective étale stacks arise in this way. We show that well known Lie groupoids from foliation theory literature can be reproduced from this framework as natural presentations for moduli stacks of certain geometric structures. For example, there exists a classifying stack for Riemannian metrics, presented by Haefliger's groupoid RΓ [13] and submersions into this stack classify Riemannian foliations, and similarly for symplectic structures, with the role of RΓ replaced with ΓSp. We also prove some unexpected results, for example: the category of smooth n-manifolds and local diffeomorphisms has binary products.

Kato–Nakayama spaces, infinite root stacks and the profinite homotopy type of log schemes

For a log scheme locally of finite type over $\mathbb{C}$, a natural candidate for its profinite homotopy type is the profinite completion of its Kato-Nakayama space. Alternatively, one may consider the profinite homotopy type of the underlying topological stack of its infinite root stack. Finally, for a log scheme not necessarily over $\mathbb{C}$, another natural candidate is the profinite \'etale homotopy type of its infinite root stack. We prove that, for a fine saturated log scheme locally of finite type over $\mathbb{C}$, these three notions agree. In particular, we construct a comparison map from the Kato-Nakayama space to the underlying topological stack of the infinite root stack, and prove that it induces an equivalence on profinite completions. In light of these results, we define the profinite homotopy type of a general fine saturated log scheme as the profinite \'etale homotopy type of its infinite root stack.

Singular chains on topological stacks, I

We extend the functor Sing of singular chains to the category of topological stacks and establish its main properties. We prove that Sing respects weak equivalences and takes a morphism of topological stacks that is both a Serre and a Reedy fibration to a Kan fibration of simplicial sets. When restricted to the category of topological spaces Sing coincides with the usual singular functor.

Fibrations of topological stacks

In this note we define fibrations of topological stacks and establish their main properties. When restricted to topological spaces, our notion of fibration coincides with the classical one. We prove various standard results about fibrations (long exact sequence for homotopy groups, Leray–Serre and Eilenberg–Moore spectral sequences, etc.). We prove various criteria for a morphism of topological stacks to be a fibration, and use these to produce examples of fibrations. We prove that every morphism of topological stacks factors through a fibration and construct the homotopy fiber of a morphism of topological stacks. As an immediate consequence of the machinery we develop, we also prove van Kampenʼs theorem for fundamental groups of topological stacks.

Homotopy types of topological stacks

We introduce the notion of classifying space of a topological stack X: it is a topological space X with a morphism φ:X→X that is a universal weak equivalence. We show that every topological stack X has a classifying space X that is well defined up to weak homotopy equivalence. Under a certain paracompactness condition on X, we show that X is indeed well defined up to homotopy equivalence. These results are formulated in terms of functors from the category of topological stacks to the (weak) homotopy category of topological spaces. We prove similar results for (small) diagrams of topological stacks.

Compactly generated stacks: A Cartesian closed theory of topological stacks

A convenient bicategory of topological stacks is constructed which is both complete and Cartesian closed. This bicategory, called the bicategory of compactly generated stacks, is the analogue of classical topological stacks, but for a different Grothendieck topology. In fact, there is an equivalence of bicategories between compactly generated stacks and those classical topological stacks which admit locally compact Hausdorff atlases. Compactly generated stacks are also equivalent to a bicategory of topological groupoids and principal bundles, just as in the classical case. If a classical topological stack and a compactly generated stack have a presentation by the same topological groupoid, then they restrict to the same stack over locally compact Hausdorff spaces and are homotopy equivalent.

Mapping stacks of topological stacks

We prove that the mapping stack Map(Y,X) of topological stacks X and Y is again a topological stack if Y admits a compact groupoid presentation. If Y admits a locally compact groupoid presentation, we show that Map(Y,X) is a paratopological stack. In particular, it has a classifying space (hence, a natural weak homotopy type). We prove an invariance theorem which shows that the weak homotopy type of the mapping stack Map(Y,X) does not change if we replace X by its classifying space, provided that Y is paracompact topological space. As an example, we describe the loop stack of the classifying stack BG of a topological group G in terms of twisted loop groups of G.

A Groupoid Approach to Noncommutative T-Duality

Topological T-duality is a transformation taking a gerbe on a principal torus bundle to a gerbe on a principal dual-torus bundle. We give a new geometric construction of T-dualization, which allows the duality to be extended in the following two directions. First, bundles of groups other than tori, even bundles of some nonabelian groups, can be dualized. Second, bundles whose duals are families of noncommutative groups (in the sense of noncommutative geometry) can be treated, though in this case the base space of the bundles is best viewed as a topological stack. Some methods developed for the construction may be of independent interest. These are a Pontryagin type duality that interchanges commutative principal bundles with gerbes, a nonabelian Takai type duality for groupoids, and the computation of certain equivariant Brauer groups.

Fundamental groups of topological stacks with the slice property

The main result of the paper is a formula for the fundamental group of the coarse moduli space of a topological stack. As an application, we find simple general formulas for the fundamental group of the coarse quotient of a group action on a topological space in terms of the fixed point data. The formulas seem, surprisingly, to be new. In particular, we recover, and vastly generalize, results of Armstrong, Bass, Higgins, Rhodes.

Inertia and delocalized twisted cohomology

We show that the inertia stack of a topological stack is again a topological stack. We further observe that the inertia stack of an orbispace is again an orbispace. We show how a U(1)-banded gerbe over an orbispace gives rise to a flat line bundle over its inertia stack. Via sheaf theory over topological stacks it gives rise to the twisted delocalized cohomology of the orbispace. With these results and constructions we generalize concepts, which are well-known in the smooth framework, to the topological case. In the smooth case we show, that our sheaf-theoretic definition of twisted delocalized cohomology of orbispaces coincides with former definitions using a twisted de Rham complex.

Sheaf theory for stacks in manifolds and twisted cohomology forS1–gerbes

This is the first of a series of papers on sheaf theory on smooth and topological stacks and its applications. The main result of the present paper is the characterization of the twisted (by a closed integral three-form) de Rham complex on a manifold. As an object in the derived category it will be related with the push-forward of the constant sheaf from a S^1-gerbe with Dixmier-Douady class represented by the three-form. In order to formulate and prove this result we develop in detail the foundations of sheaf theory for smooth stacks.