Model category structures on truncated multicomplexes for complex geometry

Model category structures on multicomplexes

Three models for the homotopy theory of homotopy theories

LetRbe a commutative ring with unit. We endow the categories of filtered complexes and of bicomplexes ofR-modules, with cofibrantly generated model structures, where the class of weak equivalences is given by those morphisms inducing a quasi-isomorphism at a certain fixed stage of the associated spectral sequence. For filtered complexes, we relate the different model structures obtained, when we vary the stage of the spectral sequence, using the functors shift and décalage.

Homotopic Descent over Monoidal Model Categories

A common technique for producing a new model category structure is to lift the fibrations and weak equivalences of an existing model structure along a right adjoint. Formally dual but technically much harder is to lift the cofibrations and weak equivalences along a left adjoint. For either technique to define a valid model category, there is a well-known necessary "acyclicity" condition. We show that for a broad class of "accessible model structures" - a generalization introduced here of the well-known combinatorial model structures - this necessary condition is also sufficient in both the right-induced and left-induced contexts, and the resulting model category is again accessible. We develop new and old techniques for proving the acyclity condition and apply these observations to construct several new model structures, in particular on categories of differential graded bialgebras, of differential graded comodule algebras, and of comodules over corings in both the differential graded and the spectral setting. We observe moreover that (generalized) Reedy model category structures can also be understood as model categories of "bialgebras" in the sense considered here.

Duflo isomorphism first appeared in Lie theory and representation theory. It is an isomorphism between invariant polynomials of a Lie algebra and the center of its universal enveloping algebra, generalizing the pioneering work of Harish-Chandra on semi-simple Lie algebras. Later on, Duflo’s result was refound by Kontsevich in the framework of deformation quantization, who also observed that there is a similar isomorphism between Dolbeault cohomology of holomorphic polyvector fields on a complex manifold and its Hochschild cohomology. The present book, which arose from a series of lectures by the first author at ETH, derives these two isomorphisms from a Duflo-type result for Q-manifolds. All notions mentioned above are introduced and explained in the book, the only prerequisites being basic linear algebra and differential geometry. In addition to standard notions such as Lie (super)algebras, complex manifolds, Hochschild and Chevalley–Eilenberg cohomologies, spectral sequences, Atiyah and Todd classes, the graphical calculus introduced by Kontsevich in his seminal work on deformation quantization is addressed in details. The book is well-suited for graduate students in mathematics and mathematical physics as well as for researchers working in Lie theory, algebraic geometry and deformation theory.

Classically, there are two model category structures on coalgebras in the category of chain complexes over a field. In one, the weak equivalences are maps which induce an isomorphism on homology. In the other, the weak equivalences are maps which induce a weak equivalence of algebras under the cobar functor. We unify these two approaches, realizing them as the two extremes of a partially ordered set of model category structures on coalgebras over a cooperad satisfying mild conditions.

We study some spectral sequences associated with a locally free {{mathscr {O}}}_X-module {{mathscr {A}}} which has a Lie algebroid structure. Here X is either a complex manifold or a regular scheme over an algebraically closed field k. One spectral sequence can be associated with {{mathscr {A}}} by choosing a global section V of {{mathscr {A}}}, and considering a Koszul complex with a differential given by inner product by V. This spectral sequence is shown to degenerate at the second page by using Deligne’s degeneracy criterion. Another spectral sequence we study arises when considering the Atiyah algebroid {{{mathscr {D}}}_{{{mathscr {E}}}}} of a holomolorphic vector bundle {{mathscr {E}}} on a complex manifold. If V is a differential operator on {{mathscr {E}}} with scalar symbol, i.e, a global section of {{{mathscr {D}}}_{{{mathscr {E}}}}}, we associate with the pair ({{mathscr {E}}},V) a twisted Koszul complex. The first spectral sequence associated with this complex is known to degenerate at the first page in the untwisted ({{mathscr {E}}}=0) case.

DG coalgebras as formal stacks

Let 𝕂 be a comonad on a model category M. We provide conditions under which the associated category M𝕂 of 𝕂-coalgebras admits a model category structure such that the forgetful functor M𝕂→M creates both cofibrations and weak equivalences. We provide concrete examples that satisfy our conditions and are relevant in descent theory and in the theory of Hopf–Galois extensions. These examples are specific instances of the following categories of comodules over a coring (co-ring). For any semihereditary commutative ring R, let A be a dg R-algebra that is homologically simply connected. Let V be an A-coring that is semifree as a left A-module on a degreewise R-free, homologically simply connected graded module of finite type. We show that there is a model category structure on the category MA of right A-modules satisfying the conditions of our existence theorem with respect to the comonad−⊗AV and conclude that the category MAV of V-comodules in MA admits a model category structure of the desired type. Finally, under extra conditions on R, A and V, we describe fibrant replacements in MAV in terms of a generalized cobar construction.

We show that the category of diagrams of topological spaces (or simplicial sets) admits many interesting model category structures in the sense of Quillen [8]. The strongest one renders any diagram of simplicial complexes and simplicial maps between them both fibrant and cofibrant. Namely, homotopy invertible maps between such are the weak equivalences and they are detectable by the "spaces of fixed points." We use a generalization of the method for defining model category structure of simplicial category given in [5].

Model category structures in bifibred categories

Using an integrable, homogeneous complex structure on the compact group SO(9), we show that the Hodge-de Rham spectral sequence for this non-Kahler compact complex manifold does not degenerate at Ei, contrary to a well-known conjecture. On any (compact) complex manifold M, the algebra of global complexvalued C°°-differential forms a*(M) has a bigrading given by the Hodge type; and the corresponding decomposition of the de Rham differential d = d + d gives rise to a double complex (a*'*(M),d, d). The spectral sequence corresponding to the first (holomorphic) degree is E = H<*{M, QM) =» HPp(M) with dx = d : this is the Hodge-de Rham (HdR) spectral sequence. When M is compact and Kahler, E = Eoo by Hodge theory. A folklore conjecture of about thirty years' standing says that for any compact complex manifold one should have E2 = E^. We will give an example to show that this conjecture is false. 1. There is an old observation of H. Samelson (see Wang [4]) that every compact Lie group G of even dimension (equivalently even rank) can be made into a complex manifold in such a way that all left-translations by elements g e G are holomorphic maps: we call such a structure an LICS on G (= left invariant, integrable complex structure on G). If G is in addition semisimple, then no LICS can be Kahlerian because H(G : R) = 0. Our example is a particular LICS on SO(9) (equivalently Spin(9): see §3 below), for which E2 has complex dimension 26. Since one knows that E^ has complex dimension 16, we obtain the required nondegeneration of HdR. Received by the editors November 23, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 32C10, 58A14, 55J05, 32M10. 1 Samelson's paper appeared in a Portuguese journal, which is perhaps understandably not available in Indian libraries. ©1989 American Mathematical Society 0273-0979/89 $1.00 + $.25 per page 19

Theories are a canonical way of describing categories with extra struc- ture. 2-theory-morphisms are used when discussing how one structure can be replaced with another structure. This is central to categorical coher- ence theory. We place a Quillen model category structure on the category of 2-theories and 2-theory-morphisms where the weak equivalences are biequivalences of 2-theories. A biequivalence of 2-theories (Morita equiv- alence) induces and is induced by a biequivalence of 2-categories of alge- bras. This model category structure allows one to talk of the homotopy of 2-theories and discuss the universal properties of coherence.

In this paper we put a cofibrantly generated model category struc- ture on the category of small simplicial categories. The weak equivalences are a simplicial analogue of the notion of equivalence of categories.