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In this paper, we consider coalgebra measurings between Hopf algebroids and show that they induce morphisms on cyclic homology and cyclic cohomology. We also consider comodule measurings between stable anti-Yetter Drinfeld (SAYD) modules over Hopf algebroids. These give an enrichment of the global category of SAYD modules over comodules. These measurings also induce morphisms on cyclic (co)homology of Hopf algebroids with SAYD coefficients, which are compatible with Hopf-Galois maps. Finally, we consider non-symmetric operads with multiplication and modules over them which have both cyclic and Gerstenhaber type structures, known as cyclic unital comp modules. We obtain an enrichment of cyclic unital comp modules over comodules, as well as morphisms on cyclic homology induced by comodule measurings of comp modules over operads with multiplication. [ABSTRACT FROM AUTHOR]

We relate the group structure of van der Kallen on orbit sets of unimodular rows ([26]) with values in a smooth algebra A over a field k with the motivic cohomotopy groups of X = Spec A with coefficients in A n ∖ 0 in the sense of [5]. In the last section, we compare the motivic cohomotopy theory studied in this paper and defined by A n + 1 ∖ 0 or, equivalently, by an A 1 -weakly equivalent quadric Q 2 n + 1 to that considered in [5] , defined by a quadric Q 2 n , by means of explicit morphisms Q 2 n + 1 → Q 2 n , Q 2 n × G m → Q 2 n + 1 of quadrics. [ABSTRACT FROM AUTHOR]

We study invariants, called shifting numbers, that measure the asymptotic amount by which an autoequivalence of a triangulated category translates inside the category. The invariants are analogous to Poincaré translation numbers that are widely used in dynamical systems. We additionally establish that in some examples the shifting numbers provide a quasimorphism on the group of autoequivalences. Additionally, the shifting numbers are related to the entropy function introduced by Dimitrov, Haiden, Katzarkov, and Kontsevich, as well as the phase functions of Bridgeland stability conditions. [ABSTRACT FROM AUTHOR]

Abstract Let R be the homogeneous coordinate ring of the Grassmannian G = Gr (2 , n) defined over an algebraically closed field of characteristic p > 0. In this paper we give a completely characteristic free description of the decomposition of R , considered as a graded R p -module, into indecomposables ("Frobenius summands"). As a corollary we obtain a similar decomposition for the Frobenius pushforward of the structure sheaf of G and we obtain in particular that this pushforward is almost never a tilting bundle. On the other hand we show that R provides a "noncommutative resolution" for R p when p ≥ n − 2 , generalizing a result known to be true for toric varieties. In both the invariant theory and the geometric setting we observe that if the characteristic is not too small the Frobenius summands do not depend on the characteristic in a suitable sense. In the geometric setting this is an explicit version of a general result by Bezrukavnikov and Mirković on Frobenius decompositions for partial flag varieties. We are hopeful that it is an instance of a more general " p -uniformity" principle. [ABSTRACT FROM AUTHOR]

Abstract The purpose of this article is to study the role of Artin fans in tropical and non-Archimedean geometry. Artin fans are logarithmic algebraic stacks that can be described completely in terms of combinatorial objects, so called Kato stacks, a stack-theoretic generalization of K. Kato's notion of a fan. Every logarithmic algebraic stack admits a tautological strict morphism ϕ X : X → A X to an associated Artin fan. The main result of this article is that, on the level of underlying topological spaces, the natural functorial tropicalization map of X is nothing but the non-Archimedean analytic map associated to ϕ X by applying Thuillier's generic fiber functor. Using this framework, we give a reinterpretation of the main result of Abramovich–Caporaso–Payne identifying the moduli space of tropical curves with the non-Archimedean skeleton of the corresponding algebraic moduli space. [ABSTRACT FROM AUTHOR]

Abstract We show that the set of Fano varieties (with arbitrary singularities) whose anticanonical divisors have large Seshadri constants satisfies certain weak and birational boundedness. We also classify singular Fano varieties of dimension n whose anticanonical divisors have Seshadri constants at least n , generalizing an earlier result of Liu and the author. [ABSTRACT FROM AUTHOR]

Abstract Using skew-symmetric matrices of infinite rank, the unfolding approach is extended from skew-symmetrizable matrices to acyclic sign-skew-symmetric matrices. Applying this approach, a positive answer is provided to the question posed by Berenstein, Fomin, and Zelevinsky in [5] , namely whether an acyclic sign-skew-symmetric matrix is always totally sign-skew-symmetric. As applications, positivity for cluster algebras in acyclic sign-skew-symmetric case is given, and the F -polynomials of cluster algebras are proved to have constant term 1 in the acyclic sign-skew-symmetric case. [ABSTRACT FROM AUTHOR]

Abstract The relationship between the Giry monad and convex spaces is explored, by first showing the Giry monad can be factored, as an adjunction, through the category of convex spaces. This adjunction, in turn, is used to characterize the category of Giry algebras as being equivalent to the category of tame convex measurable spaces, where the objects are convex spaces endowed with a simple σ -algebra structure, and the morphisms are affine measurable functions. [ABSTRACT FROM AUTHOR]

Given a projective family of semi-stable curves over a complete discrete valuation ring of characteristic p > 0 with algebraically closed residue field, we construct a specialization functor between the category of continuous representations of the pro-étale fundamental group of the closed fibre and the category of stratified bundles on the geometric generic fibre. By Tannakian duality, this functor induces a morphism between the corresponding affine group schemes. We show that this morphism is a lifting of the specialization map, constructed by Grothendieck, between the étale fundamental groups. [ABSTRACT FROM AUTHOR]

The following well known open problem is answered in the negative: Given two compact spaces X and Y that admit minimal homeomorphisms, must the Cartesian product X × Y admit a minimal homeomorphism as well? Moreover, it is shown that such spaces can be realized as minimal sets of torus homeomorphisms homotopic to the identity. A key element of our construction is an inverse limit approach inspired by combination of a technique of Aarts & Oversteegen and the construction of Slovak spaces by Downarowicz & Snoha & Tywoniuk. This approach allows us also to prove the following result. Let ϕ : M × R → M be a continuous, aperiodic minimal flow on the compact, finite-dimensional metric space M . Then there is a generic choice of parameters c ∈ R , such that the homeomorphism h ( x ) = ϕ ( x , c ) admits a noninvertible minimal map f : M → M as an almost 1-1 extension. [ABSTRACT FROM AUTHOR]

Cluster categories were introduced in 2006 by Buan-Marsh-Reineke-Reiten-Todorov in order to categorify acyclic cluster algebras without coefficients. Their construction was generalized by Amiot (2009) [3] and Plamondon (2011) [50] to arbitrary cluster algebras associated with quivers. A higher dimensional generalization is due to Guo (2011) [17]. Cluster algebras with coefficients are important since they appear in nature as coordinate algebras of varieties like Grassmannians, double Bruhat cells, unipotent cells, etc. The work of Geiss-Leclerc-Schröer often yields Frobenius exact categories which allow us to categorify such cluster algebras. In this paper, we generalize the construction of (higher) cluster categories by Claire Amiot and by Lingyan Guo to the relative context. We prove the existence of an n -cluster tilting object in a Frobenius extriangulated category, namely the Higgs category (generalizing the Frobenius categories of Geiss-Leclerc-Schröer), which is stably n -Calabi–Yau and Hom-finite, arising from a left (n + 1) -Calabi–Yau morphism. Our results apply in particular to relative Ginzburg dg algebras coming from ice quivers with potential and higher Auslander algebras associated to n -representation-finite algebras. [ABSTRACT FROM AUTHOR]

We prove the Riemann–Roch theorem for homotopy invariant K -theory and projective local complete intersection morphisms between finite dimensional noetherian schemes, without smoothness assumptions. We also prove a new Riemann–Roch theorem for the relative cohomology of a morphism. In order to do so, we construct and characterize new Gysin morphisms for regular immersions between cohomologies represented by spectra (examples include homotopy invariant K -theory, motivic cohomology, their arithmetic counterparts, real absolute Hodge and Deligne–Beilinson cohomology, rigid syntomic cohomology, mixed Weil cohomologies) and we use this construction to prove a motivic version of the Riemann–Roch. [ABSTRACT FROM AUTHOR]

Let X [ n ] be the Fulton–MacPherson compactification of the configuration space of n ordered points on a smooth projective variety X . We prove that if either n ≠ 2 or dim ⁡ ( X ) ≥ 2 , then the connected component of the identity of Aut ( X [ n ] ) is isomorphic to the connected component of the identity of Aut ( X ) . When X = C is a curve of genus g ( C ) ≠ 1 we classify the dominant morphisms C [ n ] → C [ r ] , and thanks to this we manage to compute the whole automorphism group of C [ n ] , namely Aut ( C [ n ] ) ≅ S n × Aut ( C ) for any n ≠ 2 , while Aut ( C [ 2 ] ) ≅ S 2 ⋉ ( Aut ( C ) × Aut ( C ) ) . Furthermore, we extend these results on the automorphisms to the case where X = C 1 × . . . × C r is a product of curves of genus g ( C i ) ≥ 2 . Finally, using the techniques developed to deal with Fulton–MacPherson spaces, we study the automorphism groups of some Kontsevich moduli spaces M ‾ 0 , n ( P N , d ) . [ABSTRACT FROM AUTHOR]

We show that the metric structure of morphisms f : Y → X between quasi-smooth compact Berkovich curves over an algebraically closed field admits a finite combinatorial description. In particular, for a large enough skeleton Γ = ( Γ Y , Γ X ) of f , the sets N f , ≥ n of points of Y of multiplicity at least n in the fiber are radial around Γ Y with the radius changing piecewise monomially along Γ Y . In this case, for any interval l = [ z , y ] ⊂ Y connecting a point z of type 1 to the skeleton, the restriction f | l gives rise to a profile piecewise monomial function φ y : [ 0 , 1 ] → [ 0 , 1 ] that depends only on the type 2 point y ∈ Γ Y . In particular, the metric structure of f is determined by Γ and the family of the profile functions { φ y } with y ∈ Γ Y ( 2 ) . We prove that this family is piecewise monomial in y and naturally extends to the whole Y . In addition, we extend the classical theory of higher ramification groups to arbitrary real-valued fields and show that φ y coincides with the Herbrand function of H ( y ) / H ( f ( y ) ) . This gives a curious geometric interpretation of the Herbrand function, which also applies to non-normal and even inseparable extensions. [ABSTRACT FROM AUTHOR]

We prove a simultaneous generalization of the classical Riemann-Hurwitz and Plücker formulas, addressing the total number of inflection points of a morphism from a (smooth, projective) curve to an arbitrary (smooth, projective) higher-dimensional variety. Our definition of inflection point is relative to an algebraic family of divisors on the target variety, and our formula is obtained using the theory of localized top Chern classes. In assigning multiplicities to inflection points, we frequently have to consider excess degeneracy loci, but we are able to show nonetheless that the multiplicities are always nonnegative, and are positive under very mild hypotheses. [ABSTRACT FROM AUTHOR]

We show that an ∞-category M with a closed left action of a monoidal ∞-category V is completely determined by the V -valued graph of morphism objects resulting from closedness of the action equipped with the structure of a V -enrichment in the sense of Gepner-Haugseng. We prove a similar result when M is a V -enriched ∞-category in the sense of Lurie, an operadic generalization of the notion of ∞-category with closed action. Precisely, we prove that sending a V -enriched ∞-category in the sense of Lurie to the V -valued graph of morphism objects refines to an equivalence χ between the ∞-category of V -enriched ∞-categories in the sense of Lurie and of Gepner-Haugseng. Moreover if V is a presentably k + 1 -monoidal ∞-category for 1 ≤ k ≤ ∞ , we prove that χ restricts to a lax k-monoidal functor between the ∞-category of left V -modules in Pr L , the symmetric monoidal ∞-category of presentable ∞-categories, endowed with the relative tensor product, and the tensor product of V -enriched ∞-categories of Gepner-Haugseng. As an application of our theory we construct a lax symmetric monoidal embedding of the ∞-category of small stable ∞-categories into the ∞-category of small spectral ∞-categories. As a second application we produce an enriched Yoneda-embedding in the framework of Lurie's notion of enriched ∞-categories. [ABSTRACT FROM AUTHOR]

We relate the McMullen polynomial of a free-by-cyclic group to its Alexander polynomial. To do so, we introduce the notion of an orientable fully irreducible outer automorphism φ and use it to characterize when the homological stretch factor of φ is equal to its geometric stretch factor. [ABSTRACT FROM AUTHOR]

We prove the following results for projective klt pairs of dimension 3 over an algebraically closed field of characteristic p > 5 : the cone theorem, the base point free theorem, the contraction theorem, finiteness of minimal models, termination with scaling, existence of Mori fibre spaces, etc. [ABSTRACT FROM AUTHOR]

We construct groups G that are virtually torsion-free and have virtual cohomological dimension strictly less than the minimal dimension for any model for E _ G , the classifying space for proper actions of G . They are the first examples that have these properties and also admit cocompact models for E _ G . We exhibit groups G whose virtual cohomological dimension and Bredon cohomological dimension are two that do not admit any 2-dimensional contractible proper G -CW-complex. [ABSTRACT FROM AUTHOR]

Like quantum groups, quantum groupoids frequently appear in pairs of mutually dual objects. We develop a general Pontrjagin duality theory for quantum groupoids in the algebraic setting that extends Van Daele's duality theory for multiplier Hopf algebras and overcomes the finiteness restrictions of the approach of Kadison, Szlachányi, Böhm and Schauenburg. Our construction is based on the integration theory for multiplier Hopf algebroids and yields, as a corollary, a duality theory for weak multiplier Hopf algebras with integrals. We compute the duals in several examples and introduce morphisms of multiplier Hopf algebroids to succinctly describe their structure. Moreover, we show that such morphisms preserve the antipode. [ABSTRACT FROM AUTHOR]

In this paper local monomialization theorems are proven for morphisms of complex and real analytic spaces. This gives the generalization of the local monomialization theorem for morphisms of algebraic varieties over a field of characteristic zero proven in [16] and [18] to analytic spaces. [ABSTRACT FROM AUTHOR]

Motivated by the challenge of defining twisted quantum field theories in the context of higher categories, we develop a general framework for lax and oplax transformations and their higher analogs between strong ( ∞ , n ) -functors. We construct a double ( ∞ , n ) -category built out of the target ( ∞ , n ) -category governing the desired diagrammatics. We define (op)lax transformations as functors into parts thereof, and an (op)lax twisted field theory to be a symmetric monoidal (op)lax natural transformation between field theories. We verify that lax trivially-twisted relative field theories are the same as absolute field theories. As a second application, we extend the higher Morita category of E d -algebras in a symmetric monoidal ( ∞ , n ) -category C to an ( ∞ , n + d ) -category using the higher morphisms in C . [ABSTRACT FROM AUTHOR]

We show for a ring R of weak global dimension at most one that there is a bijection between the smashing subcategories of its derived category and the equivalence classes of homological epimorphisms starting in R . If, moreover, R is commutative, we prove that the compactly generated localizing subcategories correspond precisely to flat epimorphisms. We also classify smashing localizations of the derived category of any valuation domain, and provide an easy criterion for the Telescope Conjecture (TC) for any commutative ring of weak global dimension at most one. As a consequence, we show that the TC holds for any commutative von Neumann regular ring R , and it holds precisely for those Prüfer domains which are strongly discrete. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Given a generically étale morphism f : Y → X of quasi-smooth Berkovich curves, we define a different function δ f : Y → [ 0 , 1 ] that measures the wildness of the topological ramification locus of f . This provides a new invariant for studying f , which cannot be obtained by the usual reduction techniques. We prove that δ f is a piecewise monomial function satisfying a balancing condition at type 2 points analogous to the classical Riemann–Hurwitz formula, and show that δ f can be used to explicitly construct the simultaneous skeletons of X and Y . As another application, we use our results to completely describe the topological ramification locus of f when its degree equals to the residue characteristic p . [ABSTRACT FROM AUTHOR]

There are well-known constructions relating ring epimorphisms and tilting modules. The new notion of silting module provides a wider framework for studying this interplay. To every partial silting module we associate a ring epimorphism which we describe explicitly as an idempotent quotient of the endomorphism ring of the Bongartz completion. For hereditary rings, this assignment is used to parametrise homological ring epimorphisms by silting modules. We further show that homological ring epimorphisms of a hereditary ring form a lattice which completes the poset of noncrossing partitions in the case of finite dimensional algebras. [ABSTRACT FROM AUTHOR]

We construct a continuous-time non-commutative Markov operator on U ( gl N ) with corresponding morphisms U ( gl N ) → End ( L 2 ( U ( N ) ) ) ⊗ ∞ at any fixed times 0 ≤ t 1 < t 2 < … . This is an analog of a continuous-time non-commutative Markov operator on the group von Neumann algebra v N ( U ( N ) ) constructed in [16] , and is a variant of discrete-time non-commutative random walks on U ( gl N ) [2,9] . It is also shown that when restricting to the Gelfand–Tsetlin subalgebra of U ( gl N ) , the non-commutative Markov operator matches a ( 2 + 1 )-dimensional random surface model introduced in [7] . As an application, it is then proved that the moments converge to an explicit Gaussian field along time-like paths. Combining with [7] which showed convergence to the Gaussian free field along space-like paths, this computes the entire three-dimensional Gaussian field. In particular, it matches a Gaussian field from eigenvalues of random matrices [5] . [ABSTRACT FROM AUTHOR]

This paper introduces lax orthogonal algebraic weak factorisation systems on 2-categories and describes a method of constructing them. This method rests in the notion of simple 2-monad , that is a generalisation of the simple reflections studied by Cassidy, Hébert and Kelly. Each simple 2-monad on a finitely complete 2-category gives rise to a lax orthogonal algebraic weak factorisation system, and an example of a simple 2-monad is given by completion under a class of colimits. The notions of kz lifting operation , lax natural lifting operation and lax orthogonality between morphisms are studied. [ABSTRACT FROM AUTHOR]

The geometric Satake correspondence provides an equivalence of categories between the Satake category of spherical perverse sheaves on the affine Grassmannian and the category of representations of the dual group. In this note, we define a combinatorial version of the Satake category using irreducible components of fibres of the convolution morphism. We then prove an equivalence of coboundary categories between this combinatorial Satake category and the category of crystals of the dual group. [ABSTRACT FROM AUTHOR]

We prove inverse spectral results for differential operators on manifolds and orbifolds invariant under a torus action. These inverse spectral results involve the asymptotic equivariant spectrum, which is the spectrum itself together with “very large” weights of the torus action on eigenspaces. More precisely, we show that the asymptotic equivariant spectrum of the Laplace operator of any toric metric on a generic toric orbifold determines the equivariant biholomorphism class of the orbifold; we also show that the asymptotic equivariant spectrum of a T n -invariant Schrödinger operator on R n determines its potential in some suitably convex cases. In addition, we prove that the asymptotic equivariant spectrum of an S 1 -invariant metric on S 2 determines the metric itself in many cases. Finally, we obtain an asymptotic equivariant inverse spectral result for weighted projective spaces. As a crucial ingredient in these inverse results, we derive a surprisingly simple formula for the asymptotic equivariant trace of a family of semi-classical differential operators invariant under a torus action. [ABSTRACT FROM AUTHOR]

We show that recollements of module categories give rise to homomorphisms between the associated Hochschild cohomology algebras which preserve the strict Gerstenhaber structure, i.e., the cup product, the graded Lie bracket and the squaring map. We review various long exact sequences in Hochschild cohomology and apply our results in order to realise that the occurring maps preserve the strict Gerstenhaber structure as well. As a byproduct, we generalise a known long exact cohomology sequence of Koenig–Nagase to arbitrary surjective homological epimorphisms. We use our observations to motivate and formulate a variation of the finite generation conjecture by Snashall–Solberg. [ABSTRACT FROM AUTHOR]

Let R be the homogeneous coordinate ring of the Grassmannian G = Gr (2 , n) defined over an algebraically closed field k of characteristic p ≥ max ⁡ { n − 2 , 3 }. In this paper we give a description of the decomposition of R , considered as graded R p r -module, for r ≥ 2. This is a companion paper to [16] , where the case r = 1 was treated, and taken together, our results imply that R has finite F-representation type (FFRT). Though it is expected that all rings of invariants for reductive groups have FFRT, ours is the first non-trivial example of such a ring for a group which is not linearly reductive. As a corollary, we show that the ring of differential operators D k (R) is simple, that G has global finite F-representation type (GFFRT) and that R provides a noncommutative resolution for R p r . [ABSTRACT FROM AUTHOR]

Let X be a smooth projective variety with the action of ( C ⁎ ) n . The article describes the moduli space of ( C ⁎ ) n equivariant morphisms from stable toric varieties into X as the inverse limit of the GIT quotients of X and their flips when these spaces are enhanced with a naturally associated Deligne–Mumford stack structure. This description is used for constructing a class in the Chow group of the moduli space of dimension dim ( X ) − n which is invariant under equivariant deformations of X . [ABSTRACT FROM AUTHOR]

In this paper a further study is made of H -signatures of hermitian forms, introduced previously by the authors. It is shown that a tuple of reference forms H may be replaced by a single form and that the H -signature is invariant under Morita equivalence of algebras with involution. The “prime ideals” of the Witt group are studied, obtaining results that are analogues of the classification of prime ideals of the Witt ring by Harrison and Lorenz–Leicht. It follows that H -signatures canonically correspond to morphisms into the integers. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective curve of genus g ≥ 2 defined over an algebraically closed field k of characteristic p > 0 . For p > r ( r − 1 ) ( r − 2 ) ( g − 1 ) we construct an atlas for the locus of all Frobenius-destabilized bundles of rank r (i.e. we construct all Frobenius-destabilized bundles of rank r and degree zero up to isomorphism). This is done by exhibiting a surjective morphism from a certain Quot-scheme onto the locus of stable Frobenius-destabilized bundles. Further we show that there is a bijective correspondence between the set of stable vector bundles E over X such that the pull-back F ⁎ ( E ) under the Frobenius morphism of X has maximal Harder–Narasimhan polygon and the set of opers having zero p -curvature. We also show that, after fixing the determinant, these sets are finite, which enables us to derive the dimension of certain Quot-schemes and certain loci of stable Frobenius-destabilized vector bundles over X . The finiteness is proved by studying the properties of the Hitchin–Mochizuki morphism; an alternative approach to finiteness has been realized in [3] . In particular we prove a generalization of a result of Mochizuki to higher ranks. [ABSTRACT FROM AUTHOR]

We develop various properties of étale Borel–Moore homology and study its relationship with intersection theory. Using Gabber's localized cycle classes we define étale homological Gysin morphisms and show that they are compatible with the cycle class map and Gysin morphisms in intersection theory. We also study étale versions of bivariant operations, and establish their compatibility with Riemann–Roch transformations and Fulton–MacPherson bivariant operations. As an application of these techniques we show that in certain situations local terms for correspondences acting on étale cohomology are given by cycle classes. [ABSTRACT FROM AUTHOR]

Given a normal projective irreducible stack X over an algebraically closed field of characteristic zero we consider framed sheaves on X , i.e., pairs ( E , ϕ E ) , where E is a coherent sheaf on X and ϕ E is a morphism from E to a fixed coherent sheaf F . After introducing a suitable notion of (semi)stability, we construct a projective scheme, which is a moduli space for semistable framed sheaves with fixed Hilbert polynomial, and an open subset of it, which is a fine moduli space for stable framed sheaves. If X is a projective irreducible orbifold of dimension two and F a locally free sheaf on a smooth divisor D ⊂ X satisfying certain conditions, we consider ( D , F ) -framed sheaves , i.e., framed sheaves ( E , ϕ E ) with E a torsion-free sheaf which is locally free in a neighbourhood of D , and ϕ E | D an isomorphism. These pairs are μ -stable for a suitable choice of a parameter entering the (semi)stability condition, and of the polarization of X . This implies the existence of a fine moduli space parameterizing isomorphism classes of ( D , F ) -framed sheaves on X with fixed Hilbert polynomial, which is a quasi-projective scheme. In an appendix we develop the example of stacky Hirzebruch surfaces. This is the first paper of a project aimed to provide an algebro-geometric approach to the study of gauge theories on a wide class of 4-dimensional Riemannian manifolds by means of framed sheaves on “stacky” compactifications of them. In particular, in a subsequent paper [20] these results are used to study gauge theories on ALE spaces of type A k . [ABSTRACT FROM AUTHOR]

Let ( X , B ) be a complex projective klt pair, and let f : X → Z be a surjective morphism onto a normal projective variety with maximal albanese dimension such that K X + B is relatively big over Z . We show that such pairs have good log minimal models. [ABSTRACT FROM AUTHOR]

Let E ⊆ P 2 be a complex rational cuspidal curve contained in the projective plane and let ( X , D ) → ( P 2 , E ) be the minimal log resolution of singularities. Applying the log Minimal Model Program to ( X , 1 2 D ) we prove that if E has more than two singular points or if D , which is a tree of rational curves, has more than six maximal twigs or if P 2 ∖ E is not of log general type then E is Cremona equivalent to a line, i.e. the Coolidge–Nagata conjecture for E holds. We show also that if E is not Cremona equivalent to a line then the morphism onto the minimal model contracts at most one irreducible curve not contained in D . [ABSTRACT FROM AUTHOR]

Abstract: Let be a morphism of algebraic stacks that is locally of finite presentation with affine stabilizers. We prove that there is an algebraic S-stack—the Hilbert stack—parameterizing proper algebraic stacks mapping quasi-finitely to X. This was previously unknown, even for a morphism of schemes. [Copyright &y& Elsevier]

Abstract: The behaviour of limits of weak morphisms in 2-dimensional universal algebra is not 2-categorical in that, to fully express the behaviour that occurs, one needs to be able to quantify over strict morphisms amongst the weaker kinds. -categories were introduced to express this interplay between strict and weak morphisms. We express doctrinal adjunction as an -categorical lifting property and use this to give monadicity theorems, expressed using the language of -categories, that cover each weaker kind of morphism. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: We address the (pointed) homotopy theory of 2-crossed modules (of groups), which are known to faithfully represent Gray 3-groupoids, with a single object, and also connected homotopy 3-types. The homotopy relation between 2-crossed module maps will be defined in a similar way to Cransʼ 1-transfors between strict Gray functors, however being pointed, thus this corresponds to Bauesʼ homotopy relation between quadratic module maps. Despite the fact that this homotopy relation between 2-crossed module morphisms is not, in general, an equivalence relation, we prove that if and are 2-crossed modules, with the underlying group F of being free (in short is free up to order one), then homotopy between 2-crossed module maps yields, in this case, an equivalence relation. Furthermore, if a chosen basis B is specified for F, then we can define a 2-groupoid of 2-crossed module maps , homotopies connecting them, and 2-fold homotopies between homotopies, where the latter correspond to (pointed) Cransʼ 2-transfors between 1-transfors. We define a partial resolution , for a 2-crossed module , whose underlying group is free, with a canonical chosen basis, together with a projection map , defining isomorphisms at the level of 2-crossed module homotopy groups. This resolution (which is part of a comonad) leads to a weaker notion of homotopy (lax homotopy) between 2-crossed module maps, which we fully develop and describe. In particular, given 2-crossed modules and , there exists a 2-groupoid of (strict) 2-crossed module maps , and their lax homotopies and lax 2-fold homotopies, leading to the question of whether the category of 2-crossed modules and strict maps can be enriched over the monoidal category Gray. The associated notion of a (strict) 2-crossed module map to be a lax homotopy equivalence has the two-of-three property, and it is closed under retracts. This discussion leads to the issue of whether there exists a model category structure in the category of 2-crossed modules (and strict maps) where weak equivalences correspond to lax homotopy equivalences, and any free up to order one 2-crossed module is cofibrant. [Copyright &y& Elsevier]

Abstract: Let be a morphism between normal complex varieties, where is Kawamata log terminal. Given any differential form , defined on the smooth locus of , we construct a “pull-back form” on . The pull-back map obtained by this construction is -linear, uniquely determined by natural universal properties and exists even in cases where the image of is entirely contained in the singular locus of . One relevant setting covered by the construction is that where is the inclusion (or normalisation) of the singular locus . As an immediate corollary, we show that differential forms defined on the smooth locus of induce forms on every stratum of the singularity stratification. The same result also holds for many Whitney stratifications. [Copyright &y& Elsevier]

Abstract: Let be an exact category and a subfunctor. A morphism in is an -phantom if the pullback of an -conflation along is a conflation in . If the exact category has enough injective objects and projective morphisms, it is proved that an ideal of is special precovering if and only if there is a subfunctor with enough injective morphisms such that is the ideal of -phantom morphisms. A crucial step in the proof is a generalization of Salce’s Lemma for ideal cotorsion pairs: if is a special precovering ideal, then the ideal cotorsion pair generated by in is complete. This theorem is used to verify: (1) that the ideal cotorsion pair cogenerated by the pure-injective modules of is complete; (2) that the ideal cotorsion pair cogenerated by the contractible complexes in the category of complexes is complete; and, using Auslander and Reiten’s theory of almost split sequences, (3) that the ideal cotorsion pair cogenerated by the Jacobson radical of the category of finitely generated representations of an Artin algebra is complete. [Copyright &y& Elsevier]

Abstract: For a toric Deligne–Mumford (DM) stack , we can consider a certain generalization of the Frobenius endomorphism. For such an endomorphism on a 2-dimensional toric DM stack , we show that the push-forward of the structure sheaf generates the bounded derived category of coherent sheaves on . We also choose a full strong exceptional collection from the set of direct summands of in several examples of two dimensional toric DM orbifolds . [Copyright &y& Elsevier]

Abstract: We study the self-dual Hopf algebra of special posets introduced by Malvenuto and Reutenauer and the Hopf algebra morphism from to the Hopf algebra of free quasi-symmetric functions given by linear extensions. In particular, we construct two Hopf subalgebras both isomorphic to ; the first one is based on plane posets, the second one on heap-ordered forests. An explicit isomorphism between these two Hopf subalgebras is also defined, with the help of two combinatorial transformations on special posets. The restriction of the Hopf pairing of to these Hopf subalgebras and others is also studied, as well as certain isometries between them. These problems are solved using duplicial and dendriform structures. [Copyright &y& Elsevier]

Abstract: It is known that monoidal functors between internal groupoids in the category of groups constitute the bicategory of fractions of the 2-category of internal groupoids, internal functors and internal natural transformations in , with respect to weak equivalences (that is, internal functors which are internally fully faithful and essentially surjective on objects). Monoidal functors can be equivalently described by a kind of weak morphisms introduced by B. Noohi under the name of butterflies. In order to internalize monoidal functors in a wide context, we introduce the notion of internal butterflies between internal crossed modules in a semi-abelian category , and we show that they are morphisms of a bicategory . Our main result states that, when in the notions of Huq commutator and Smith commutator coincide, then the bicategory of internal butterflies is the bicategory of fractions of with respect to weak equivalences. [Copyright &y& Elsevier]

Abstract: In this paper, we classify the epimorphisms of irreducible spherical Moufang buildings (of rank ) defined over a field. As an application, we characterize indecomposable epimorphisms of these buildings as those epimorphisms arising from -buildings. [Copyright &y& Elsevier]

Abstract: In this sequel to Bierstone and Milman [4], we find the smallest class of singularities in four variables with which we necessarily end up if we resolve singularities except for normal crossings. The main new feature is a characterization of singularities in four variables which occur as limits of triple normal crossings singularities, and which cannot be eliminated by a birational morphism that avoids blowing up normal crossings singularities. This result develops the philosophy of [4], that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities. [Copyright &y& Elsevier]