论文围绕 test category 推广 Dold–Kan 对应，定义 Whitehead 范畴并构造模型结构，证明同调函子在局部化后给出范畴等价。
摘要：
本工作源于 Grothendieck 的手稿《Pursuing Stacks》的第 V 章和第 VII 章，该手稿包含一系列问题，以及一种此前未被探索的、关于 test category（测试范畴）概念与同调之间相互作用的形式体系。本论文的主要目标是在 test category 的背景下展示同伦 Dold–Kan 对应。更具体地说，我们遵循 Grothendieck 的思路，引入一个推广单纯同调的函子，它从任意小范畴上的交换预层范畴，映向同调次数非负的交换群导出范畴。随后，我们寻找一些条件，以确保在经由那样一类态射——其在导出范畴中的像是同构——局部化之后，该函子诱导出范畴等价。一般而言，交换预层范畴上还存在第二类弱等价，它来自 test category 理论；我们把这两类态射重合的小范畴称为 Whitehead 范畴，从而推广 Δ 的情形。我们证明 test category 的若干重要例子都是 Whitehead 范畴，尤其是 Joyal 的范畴 Θ。对于任意 Whitehead 局部 test category，我们在其交换预层范畴上构造了一个模型范畴结构，其弱等价正是上述弱等价。随后我们证明，对于任意 Whitehead test category，同调函子确实诱导局部化范畴之间的等价。由此，我们得到许多同伦 Dold–Kan 对应的例子，其中尤其包括范畴 Θ。
英文摘要：
This work originates from chapters V and VII of Grothendieck's manuscript Pursuing Stacks, which contains a series of questions, as well as a previously unexplored formalism, concerning the interactions between the notion of test categories and homology. The main objective of this thesis is to exhibit homotopical Dold-Kan correspondences in the context of test categories. More precisely, we introduce, following Grothendieck, a functor generalizing simplicial homology, from the category of abelian presheaves over any small category to the derived category of abelian groups in non-negative homological degree. We then look for conditions ensuring that this functor induces an equivalence of categories, after localization by the class of morphisms whose image in the derived category is an isomorphism. Generally, there exists a second class of weak equivalences, arising from the theory of test categories, on the category of abelian presheaves, and we call Whitehead Categories those small categories for which these two classes coincide, generalizing the case of $Δ$. We show that important examples of test categories are Whitehead categories, notably Joyal's category $Θ$. We construct, for any Whitehead local test category, a model category structure on its category of abelian presheaves with the weak equivalences mentioned above. We then prove that for any Whitehead test category, the homology functor does induce an equivalence between the localized categories. We obtain this way many examples of homotopical Dold-Kan correspondences, including, among others, the category $Θ$.
We establish a large class of homotopy coherent Morita-equivalences of Dold-Kan type relating diagrams with values in any weakly idempotent complete additive $\infty$-category; the guiding example is an $\infty$-categorical Dold-Kan correspondence between the $\infty$-categories of simplicial objects and connective coherent chain complexes. Our results generalize many known 1-categorical equivalences such as the classical Dold-Kan correspondence, Pirashvili's Dold-Kan type theorem for abelian $Γ$-groups and, more generally, the combinatorial categorical equivalences of Lack and Street.
In this work, we establish a categorification of the classical Dold-Kan correspondence in the form of an equivalence between suitably defined $\infty$-categories of simplicial stable $\infty$-categories and connective chain complexes of stable $\infty$-categories. The result may be regarded as a contribution to the foundations of an emerging subject that could be termed categorified homological algebra.
We prove a Dold-Kan type correspondence between the category of dendroidal abelian groups and a suitably constructed category of dendroidal complexes. Our result naturally extends the classical Dold-Kan correspondence between the category of simplicial abelian groups and the category of non-negatively graded chain complexes.
We extend the nonabelian Dold-Kan decomposition for simplicial groups of Carrasco and Cegarra in two ways. First, we show that the total order of the subgroups in their decomposition belongs to a family of total orders all giving rise to Dold-Kan decompositions. We exhibit a particular partial order such that the family is characterized as consisting of all total orders extending the partial order. Second, we consider symmetric-simplicial groups and show that, by using a specially chosen presentation of the category of symmetric-simplicial operators, new Dold-Kan decompositions exist which are…
We introduce a bialgebra axiom for a pair $(c,\ell)$ of a colax-monoidal and a lax-monoidal structures on a functor $F\colon \mathscr{M}_1\to \mathscr{M}_2$ between two (strict) symmetric monoidal categories. This axiom can be regarded as a weakening of the property of $F$ to be a strict symmetric monoidal functor. We show that this axiom transforms well when passing to the adjoint functor or to the categories of monoids. Rather unexpectedly, this axiom holds for the Alexander-Whitney colax-monoidal and the Eilenberg-MacLane lax-monoidal structures on the normalized chain complex functor in th…
We show that weak monoidal Quillen equivalences induce equivalences of symmetric monoidal $\infty$-categories with respect to the Dwyer-Kan localization of the symmetric monoidal model categories. The result will induce a Dold-Kan correspondence of coalgebras in $\infty$-categories. Moreover, it shows that Shipley's zig-zag of Quillen equivalences lifts to an explicit symmetric monoidal equivalence of $\infty$-categories for the stable Dold-Kan correspondence. We study homotopy coherent coalgebras associated to a monoidal monoidal category. We show examples when these coalgebras cannot be rigi…
The (dual) Dold-Kan correspondence says that there is an equivalence of categories $K:\cha\to \Ab^Δ$ between nonnegatively graded cochain complexes and cosimplicial abelian groups, which is inverse to the normalization functor. We show that the restriction of $K$ to $DG$-rings can be equipped with an associative product and that the resulting functor $DGR^*\to\ass^Δ$, although not itself an equivalence, does induce one at the level of homotopy categories. The dual of this result for chain $DG$ and simplicial rings was obtained independently by S. Schwede and B. Shipley through different method…
The classical Dold-Kan correspondence is known to admit a categorification in the form of an equivalence between the $\infty$-categories of $2$-simplicial stable $\infty$-categories and connective chain complexes of stable $\infty$-categories. In this work, we extend these concepts to give a categorification of the classical Dwyer-Kan correspondence by showing that the $\infty$-category of $2$-duplicial stable $\infty$-categories is equivalent to the $\infty$-category of conective chain complexes with right adjoints.
The purpose of this note is to point out that simplicial methods and the well-known Dold-Kan construction in simplicial homotopy theory can be fruitfully applied to convert link homology theories into homotopy theories. Dold and Kan prove that there is a functor from the category of chain complexes over a commutative ring with unit to the category of simplicial objects over that ring such that chain homotopic maps go to homotopic maps in the simplicial category. Furthermore, this is an equivalence of categories. In this way, given a link homology theory, we construct a mapping taking link diag…
This chapter, written for "Stable categories and structured ring spectra," edited by Andrew J. Blumberg, Teena Gerhardt, and Michael A. Hill, surveys the history of homotopical categories, from Gabriel and Zisman's categories of fractions to Quillen's model categories, through Dwyer and Kan's simplicial localizations and culminating in $(\infty,1)$-categories, first introduced through concrete models and later re-conceptualized in a model-independent framework. This reader is not presumed to have prior acquaintance with any of these concepts. Suggested exercises are included to fertilize intui…
We consider the category whose objects are filtered, or complete, $L_\infty$-algebras and whose morphisms are $\infty$-morphisms which respect the filtrations. We then discuss the homotopical properties of the Getzler-Hinich simplicial Maurer-Cartan functor which associates to each filtered $L_\infty$-algebra a Kan simplicial set, or $\infty$-groupoid. In previous work with V. Dolgushev, we showed that this functor sends weak equivalences of filtered $L_\infty$-algebras to weak homotopy equivalences of simplicial sets. Here we sketch a proof of the fact that this functor also sends fibrations …
We establish that a category of fibrant objects (in the sense of Brown) admits a Dwyer-Kan homotopical calculus of right fractions. This is done using a homotopical calculus of cocycles, which is an auxiliary structure that can be defined on every category of fibrant objects. As an application, we deduce some non-abelian versions of the Verdier hypercovering theorem.
Viewing Kan complexes as $\infty$-groupoids implies that pointed and connected Kan complexes are to be viewed as $\infty$-groups. A fundamental question is then: to what extent can one "do group theory" with these objects? In this paper we develop a notion of a finite $\infty$-group: an $\infty$-group with finitely many non-trivial homotopy groups which are all finite. We prove a homotopical analog of the Sylow theorems for finite $\infty$-groups. We derive two corollaries: the first is a homotopical analog of the Burnside's fixed point lemma for $p$-groups and the second is a "group-theoretic…
Working in the Arone-Ching framework for homotopical descent, it follows that the Bousfield-Kan completion map with respect to integral homology is the unit of a derived adjunction. We prove that this derived adjunction, comparing spaces with coalgebra complexes over the associated integral homology comonad, via integral chains, can be turned into a derived equivalence by replacing spaces with the full subcategory of simply connected spaces. In particular, this provides an integral chains characterization of the homotopy type of simply connected spaces.
For a cofibrantly generated Quillen model category, we show that the cofibrant replacement functor constructed using the small object argument admits a cotriple structure. If all acyclic cofibrations are monomorphisms, the fibrant replacement functor constructed using the small object argument admits a triple structure. For a triple in the base category, the associated cosimplicial resolution is not necessarily homotopy invariant. However using a mix of the triple with the cofibrant replacement cotriple we construct a 'homotopically correct' version of the cosimplicial resolution of the triple…
We show that the Rezk classification diagram of a relative category admitting a homotopical version of the two-sided calculus of fractions is a Segal space up to Reedy-fibrant replacement. This generalizes the result of Rezk and Bergner on the classification diagram of a closed model category, as well as the result of Barwick and Kan on the classification diagram of a partial model category.
This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts (for part II, see math.AG/0404373). In this first part we investigate a notion of higher topos. For this, we use S-categories (i.e. simplicially enriched categories) as models for certain kind of \infty-categories, and we develop the notions of S-topologies, S-sites and stacks over them. We prove in particular, that for an S-category T endowed with an S-topology, there exists a model category of stacks over T, generalizing the model category structure on s…
We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established model categories of monoids, modules and algebras [Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000), 491-511]. As an application we extend the Dold-Kan equivalence to show that the model categories of simplicial rings, modules and algebras are Quillen equivalent to the associated model categories of connected differential g…
We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as bundles, connections and spin structures. Using right Kan extensions, we can assign to any such theory an ordinary quantum field theory defined on the category of spacetimes and we shall clarify under which conditions it satisfies the axioms of locally covariant quantum field theory. The same constructions can be performed i…
We study the *homotopy theory* of $\infty$-categories enriched in the $\infty$-category $sS$ of simplicial spaces. That is, we consider $sS$-enriched $\infty$-categories as presentations of ordinary $\infty$-categories by means of a "local" geometric realization functor $Cat_{sS} \to Cat_\infty$, and we prove that their homotopy theory presents the $\infty$-category of $\infty$-categories, i.e. that this functor induces an equivalence $Cat_{sS} [[ W_{DK}^{-1} ]] \xrightarrow{\sim} Cat_\infty$ from a localization of the $\infty$-category of $sS$-enriched $\infty$-categories. Following Dwyer--Ka…
We describe an $A_\infty$-quasi-equivalence of dg-categories between the first authors' $\mathcal{P}_{\mathcal{A}}$ ---the category of category of prefect $A^0$-modules with flat $\Z$-connection, corresponding to the de Rham dga $\mathcal{A}$ of a compact manifold $M$--- and the dg-category of \emph{infinity-local systems} on $M$ ---homotopy coherent representations of the smooth singular simplicial set of $M$, $\Pinf$. We understand this as a generalization of the Riemann--Hilbert correspondence to $\Z$-connections ($\Z$-graded superconnections in some circles). In one formulation an infinity…
We generalize the construction of reflection functors from classical representation theory of quivers to arbitrary small categories with freely attached sinks or sources. These reflection morphisms are shown to induce equivalences between the corresponding representation theories with values in arbitrary stable homotopy theories, including representations over fields, rings or schemes as well as differential-graded and spectral representations. Specializing to representations over a field and to specific shapes, this recovers derived equivalences of Happel for finite, acyclic quivers. However,…
In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterize its n-arrows using a graphical calculus that we develop here. The hom-spaces are appropriately fibrant, indeed are nerves of categories, which indicates that all of the expected coherence equations in each dimension are present. To justify our terminology, we prove that any adjunction of quasi-categories extends to a homotopy coherent adjunction and furthermore that these extensions are homotopically unique in the sense that the relevant spaces of extensions are contr…
