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Carlos Simpson

Publications and source records attributed to Carlos Simpson.

36 records · Page 2Linked to original sources
arXiv2001

Some properties of the theory of n-categories

Let $L_n$ denote the Dwyer-Kan localization of the category of weak n-categories divided by the n-equivalences. We propose a list of properties that this simplicial category is likely to have, and conjecture that these properties characterize $L_n$ up to equivalence. We show, using these properties, how to obtain the morphism $n-1$-categories between two points in an object of $L_n$ and how to obtain the composition map between the morphism objects.

math.CT↗
arXiv2001

Descente pour les n-champs (Descent for n-stacks)

We develop the theory of n-stacks (or more generally Segal n-stacks which are $\infty$-stacks such that the morphisms are invertible above degree n). This is done by systematically using the theory of closed model categories (cmc). Our main results are: a definition of n-stacks in terms of limits, which should be perfectly general for stacks of any type of objects; several other characterizations of n-stacks in terms of ``effectivity of descent data''; construction of the stack associated to an n-prestack; a strictification result saying that any ``weak'' n-stack is equivalent to a (strict) n-stack; and a descent result saying that the (n+1)-prestack of n-stacks (on a site) is an (n+1)-stack. As for other examples, we start from a ``left Quillen presheaf'' of cmc's and introduce the associated Segal 1-prestack. For this situation, we prove a general descent result, giving sufficient conditions for this prestack to be a stack. This applies to the case of complexes, saying how complexes of sheaves of $\Oo$-modules can be glued together via quasi-isomorphisms. This was the problem that originally motivated us.

math.AG↗
arXiv2000

Calculating maps between n-categories

We give an explicit way of calculating the set of homotopy classes of morphisms from a Tamsamani n-category A to another one B. This calculation uses a Reedy-cofibrant cosimplicial resolution of A, using a new notion of ``free cofibration'' of n-precats. The free cofibrations of n-precats seem to be the analogue for n-categories of the Bousfield-Kan cofibrations in the theory of diagrams.

math.CT↗
arXiv2000

Nonabelian mixed Hodge structures

We propose a definition of ``nonabelian mixed Hodge structure'' together with a construction associating to a smooth projective variety $X$ and to a nonabelian mixed Hodge structure $V$, the ``nonabelian cohomology of $X$ with coefficients in $V$'' which is a (pre-)nonabelian mixed Hodge structure denoted $H=Hom(X_M, V)$. We describe the basic definitions and then give some conjectures saying what is supposed to happen. At the end we compute an example: the case where $V$ has underlying homotopy type the complexified 2-sphere, and mixed Hodge structure coming from its identification with $\pp ^1$. For this example we show that $Hom (X_M,V)$ is a namhs for any smooth projective variety $X$.

math.AG↗
arXiv2000

A closed model structure for $n$-categories, internal $Hom$, $n$-stacks and generalized Seifert-Van Kampen

We define a closed model category containing the $n$-nerves defined by Tamsamani, and admitting internal $Hom$. This allows us to construct the $n+1$-category $nCAT$ by taking the internal $Hom$ for fibrant objects. We prove a generalized Seifert-Van Kampen theorem for Tamsamani's Poincaré $n$-groupoid of a topological space. We give a still-speculative discussion of $n$-stacks, and similarly of comparison with other possible definitions of $n$-category.

alg-geom↗
arXiv1999

A Giraud-type characterization of the simplicial categories associated to closed model categories as $\infty$-pretopoi

Theorem (after Giraud, SGA 4): Suppose $A$ is a simplicial category. The following conditions are equivalent: (i) There is a cofibrantly generated closed model category $M$ such that $A$ is equivalent to the Dwyer-Kan simplicial localization $L(M)$; (ii) $A$ admits all small homotopy colimits, and there is a small subset of objects of $A$ which are $A$-small, and which generate $A$ by homotopy colimits; (iii) There exists a small 1-category $C$ and a morphism $g:C\to A$ sending objects of $C$ to $A$-small objects, which induces a fully faithful inclusion $i:A\to \hat{C}$, such that $i$ admits a left homotopy-adjoint $ψ$. We call a Segal category $A$ which satisfies these equivalent conditions, an $\infty$-pretopos. Note that (i) implies that $A$ admits all small homotopy limits too. If furthermore there exists $C\to A$ as in (iii) such that the adjoint $ψ$ preserves finite homotopy limits, then we say that $A$ is an ``$\infty$-topos''.

math.AT↗
arXiv1999

Algebraic aspects of higher nonabelian Hodge theory

We look more closely at the higher nonabelian de Rham cohomology of a smooth projective variety or family of varieties that had been defined in some previous papers. We formalize using $n$-stacks the notion of shape underlying this nonabelian cohomology. A generalization of the de Rham construction to any appropriate formal category or family of formal categories, yields various algebraic aspects of Hodge theory for the de Rham shape, such as the Hodge filtration, the Gauss-Manin connection, Griffiths transversality, an extension of the Gauss-Manin connection with regular singularities across singular points, etc. Along the way, we develop a little bit more technology for $n$-categories, such as a canonical fibrant replacement for the $n+1$-category $nCAT$; and we pose the following general type of question: what are the properties of the nonabelian cohomology $n$-stack $Hom(X,T)$ as a function of the properties of the coefficient $n$-stack $T$ and the domain $n$-stack $X$?

math.AG↗
arXiv1998

Homotopy types of strict 3-groupoids

We look at strict $n$-groupoids and show that if $\Re$ is any realization functor from the category of strict $n$-groupoids to the category of spaces satisfying a minimal property of compatibility with homotopy groups, then there is no strict $n$-groupoid $G$ such that $\Re (G)$ is the $n$-type of $S^2$ (for $n\geq 3$). At the end we speculate on how one might fix this problem by introducing a notion of ``snucategory'', a strictly associative $n$-category with only weak units.

math.CT↗
arXiv1998

Flexible sheaves

We look at homotopy-coherent diagrams of spaces (after Segal, Leitch, Vogt, Mather, Cordier) over a Grothendieck site; we call these ``flexible presheaves''. After some preliminary materiel, we define the ``flexible sheaf'' condition. This descent condition (known to Thomason) is the same as what Jardine called being ``flasque'' with respect to the presheaves representable by objects in the site; and it is more recently known as the condition of being an $n$-stack. We construct the flexible sheaf associated to a flexible presheaf in the $n$-truncated case, as an application of a certain natural operation $n+2$ times. We prove an analogue of Vogt's theorem for the case where the Grothendieck topology is nontrivial, identifying the set of morphisms in Illusie's derived category as the set of homotopy classes of homotopy-coherent morphisms between flexible sheaves. The homotopy-coherent point of view allows one easily to define the flexible mapping sheaf $Hom (R,T)$ between two flexible sheaves. This revision fills major gaps in the bibliography. References to the additional items are inserted in the text. A new introduction and abstract are added (the old ones are retained as comments in the source file). A few other minor changes in the exposition include arrangement of internal references.

q-alg↗
arXiv1998

Secondary Kodaira-Spencer classes and nonabelian Dolbeault cohomology

If $X$ is a smooth projective variety moving in a family, we define a secondary Kodaira-Spencer class for nonabelian Dolbeault cohomology $Hom(X_{Dol}, T)$ of $X$ with coefficients in the complexified 2-sphere $T=S^2\otimes \cc$ (which is a 3-stack on $Sch /\cc$). Let $Z$ be a simply connected projective surface with $h^{2,0}\neq 0$, and let $X$ be the blow-up of $Z$ at a point $P$. As $P$ moves in $Z$, the blow-up $X$ moves in a family and we show that the secondary Kodaira-Spencer class is nontrivial. This contrasts with the fact that the variations of mixed Hodge structures on the homotopy groups of $X$ are constant. We discuss various surrounding notions, including two appendices where we give some details about the Breen calculations in characteristic zero and representability of simply connected complex shapes.

alg-geom↗
arXiv1997

Effective generalized Seifert-Van Kampen: how to calculate $ΩX$

Suppose $X$ is a 1-connected simplicial set with finitely many nondegenerate simplices. We give an effective algorithm to calculate a simplicial set with the $n$-type of the loop space $ΩX$. Iterating gives an algorithm to calculate the $π_i(X)$, different from the algorithms already known due to E. Brown and Kan-Curtis. The method is an effective version of the generalized Seifert-Van Kampen theorem of alg-geom/9704006. This can be viewed as a Van Kampen statement concerning the loop space $ΩX$ with its delooping structure. We use Segal's delooping machinery but at the end we speculate on extensions to other delooping machinery.

q-alg↗
arXiv1997

Limits in $n$-categories

We define notions of direct and inverse limits in an $n$-category. We prove that the $n+1$-category $nCAT'$ of fibrant $n$-categories admits direct and inverse limits. At the end we speculate (without proofs) on some applications of the notion of limit, including homotopy fiber product and homotopy coproduct for $n$-categories, the notion of $n$-stack, representable functors, and finally on a somewhat different note, a notion of relative Malcev completion of the higher homotopy at a representation of the fundamental group.

alg-geom↗
arXiv1997

Mixed twistor structures

The purpose of this paper is to introduce the notion of mixed twistor structure, a generalization of the notion of mixed Hodge structure. The utility of this notion is to make possible a theory of weights for various things surrounding arbitrary representations of the fundamental group of a smooth projective variety. We give some examples of generalizations of classical results for variations of mixed Hodge structure, to the twistor setting. This supports a ``meta-theorem'' (which we state but don't prove) that one can everywhere replace the word ``Hodge'' by the word ``twistor''. We show that the jet spaces of hyperkähler or more generally hypercomplex manifolds have natural mixed twistor structures which determine the hypercomplex structure in a formal neighborhood of a point.

alg-geom↗
arXiv1996

Algebraic (geometric) $n$-stacks

We propose a generalization of Artin's definition of algebraic stack, which we call {\em geometric $n$-stack}. The main observation is that there is an inductive structure to the definition whereby the ingredients for the definition of geometric $n$-stack involve only $n-1$-stacks and so are already previously defined. We use this inductive structure to obtain some basic properties. We look at maps from a projective variety into certain such $n$-stacks, and obtain an interpretation of the Brill-Noether locus as the set of points of a geometric $n$-stack. At the end we explain how this provides a context for looking at de Rham theory for higher nonabelian cohomology, how one can define the Hodge filtration and so on.

alg-geom↗
arXiv1996

The topological realization of a simplicial presheaf

The purpose of this article is to define the topological realization of a simplicial presheaf and to prove (under appropriate conditions) that it is homotopy-invariant under Illusie weak equivalence. In particular this applies to the site of schemes over $Spec (\cc)$ with the etale or Zariski topologies. As an application we show how to calculate the topological realization of a Deligne-Mumford stack. At the end we speculate on how to extend this to the case of $n$-topoi.

q-alg↗
arXiv1996

A relative notion of algebraic Lie group and applications to $n$-stacks

If $S$ is a scheme of finite type over $k=\cc $, let $\Xx /S$ denote the big etale site of schemes over $S$. We introduce {\em presentable group sheaves}, a full subcategory of the category of sheaves of groups on $\Xx /S$ which is closed under kernel, quotient, and extension. Group sheaves which are representable by group schemes of finite type over $S$ are presentable; pullback and finite direct image preserve the notions of presentable group sheaves; over $S=Spec (k)$ then presentable group sheaves are just group schemes of finite type over $Spec(k)$; there is a notion of connectedness extending the usual notion over $Spec(k)$; and a presentable group sheaf $G$ has a Lie algebra object $Lie(G $. If $G$ is a connected presentable group sheaf then $G/Z(G)$ is determined up to isomorphism by the Lie algebra sheaf $Lie (G)$. We envision the category of presentable group sheaves as a generalisation relative to an arbitrary base scheme $S$, of the category of algebraic Lie groups over $Spec (k)$. The notion of presentable group sheaf is used in order to define {\em presentable $n$-stacks} over $\Xx$. Roughly, an $n$-stack is presentable if there is a surjection from a scheme of finite type to its $π_0$ (the actual condition on $π_0$ is slightly more subtle), and if its $π_i$ (which are sheaves on various $\Xx /S$) are presentable group sheaves. The notion of presentable $n$-stack is closed under homotopy fiber product and truncation. We propose the notion of presentable $n$-stack as an answer in characteristic zero for A. Grothendieck's search for what he called ``schematization of homotopy types''.

alg-geom↗
arXiv1996

The Hodge filtration on nonabelian cohomology

This is partly a survey article on nonabelian Hodge theory, but we also give proofs of results that have only been announced elsewhere. In the introduction we discuss a wide range of recent work on this subject and give some references. In the body of the paper, we discuss Corlette's nonabelian Hodge theorem, Hitchin's quaternionic structure on the moduli space of representations of $π_1(X)$ (for a compact Kähler manifold $X$), and Deligne's construction of the resulting twistor space. We then mention the interpretation of Deligne's space of $λ$-connections as the Hodge filtration of the nonabelian de Rham cohomology $M_{DR}(X,G)= H^1(X,G)$. We go on to prove several properties of this Hodge filtration, such as Griffiths transversality and regularity of the Gauss-Manin connection; a local triviality property coming from Goldman and Millson's analysis of the singularities; and a weight property coming from Langton's theory. We construct a compactification of $M_{DR}$ using the Hodge filtration with its weight property. At the end, we define a nonabelian version of the Noether-Lefschetz locus, and prove that it is algebraic if the base of the fibration is compact. We pose the open problem of studying degenerations of nonabelian Hodge structure sufficiently well to be able to prove algebraicity of the Noether-Lefshetz locus even when the base is quasiprojective.

alg-geom↗