Daniel Dugger
We show that every combinatorial model category can be obtained, up to Quillen equivalence, by localizing a model category of diagrams of simplicial sets. This says that any combinatorial model category can be built up from a category of `generators' and a set of `relations'---that is, any combinatorial model category has a presentation.
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Presentations of configuration categories
该研究将流形的构型范畴通过神经构造视为Segal空间，证明了当流形为紧流形内部时，经适当截断的无序构型范畴可作为完全Segal空间具有有限表示。
A model category is called combinatorial if it is cofibrantly generated and its underlying category is locally presentable. As shown in recent years, homotopy categories of combinatorial model categories share useful properties, such as being well generated and satisfying a very general form of Ohkawa's theorem.
We prove existence results for small presentations of model categories generalizing a theorem of D. Dugger from combinatorial model categories to more general model categories. Some of these results are shown under the assumption of Vopěnka's principle. Our main theorem applies in particular to cofibrantly generated model categories where the domains of the generating cofibrations satisfy a slightly stronger smallness condition. As a consequence, assuming Vopěnka's principle, such a cofibrantly generated model category is Quillen equivalent to a combinatorial model category. Moreover, if there…
We propose a notion of weak (n+k,n)-category, which we call (n+k,n)-Theta-spaces. The (n+k,n)-Theta-spaces are precisely the fibrant objects of a certain model category structure on the category of presheaves of simplicial sets on Joyal's category Theta_n. This notion is a generalization of that of complete Segal spaces (which are precisely the (infty,1)-Theta-spaces). Our main result is that the above model category is cartesian.
We obtain combinatorial model categories of parametrised spectra, together with systems of base change Quillen adjunctions associated to maps of parameter spaces. We work with simplicial objects and use Hovey's sequential and symmetric stabilisation machines. By means of a Grothendieck construction for model categories, we produce combinatorial model categories controlling the totality of parametrised stable homotopy theory. The global model category of parametrised symmetric spectra is equipped with a symmetric monoidal model structure (the external smash product) inducing pairings in twisted…
A multifiltration is a functor indexed by $\mathbb{N}^r$ that maps any morphism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural $\mathbb{N}^r$-graded $R[x_1,\ldots, x_r]$-module structure on the homology of a multifiltration of simplicial complexes. To do that we study multifiltrations of sets and vector spaces. We prove in particular that the $\mathbb{N}^r$-graded $R[x_1,\ldots, x_r]$-modules that can occur as $R$-spans of multifiltrations of sets are the direct sums of monomial ideals.
We extend the framework of combinatorial model categories, so that the category of small presheaves over large indexing categories and ind-categories would be embraced by the new machinery called class-combinatorial model categories. The definition of the new class of model categories is based on the corresponding extension of the theory of locally presentable and accessible categories developed in the companion paper [arXiv:1110.0605], where we introduced the concepts of locally class-presentable and class-accessible categories. In this work we prove that the category of weak equivalences of …
Using full images of accessible functors, we prove some results about combinatorial and accessible model categories. In particular, we give an example of a weak factorization system on a locally presentable category which is not accessible.
For a diagram of simplicial combinatorial model categories, we show that the associated lax limit, endowed with the projective model structure, is a presentation of the lax limit of the underlying $\infty$-categories. Our approach can also allow for the indexing category to be simplicial, as long as the diagram factors through its homotopy category. Analogous results for the associated homotopy limit (and other intermediate limits) directly follow.
Presentation of finite Reedy categories as localizations of finite direct categories
本文通过构造直范畴 Down(C) 及其到 Reedy 范畴 C 的函子，将有限 Reedy 范畴呈现为有限直范畴的 (∞,1)-范畴局部化，从而为同伦类型论的有限截断单纯类型元理论提供工具。
The Whitehead asphericity problem, regarded as a problem of combinatorial group theory, asks whether any subpresentation of an aspherical group presentation is also aspherical. This is a long standing open problem which has attracted a lot of attention. Related to it, throughout the years there have been given several useful characterizations of asphericity which are either combinatorial or topological in nature. The aim of this paper is two fold. First, it brings in methods from semigroup theory to give a new combinatorial characterization of asphericity in terms of what we define here to be …
Freely generated $n$-categories, coinserters and presentations of low dimensional categories
研究自由生成n-范畴等相关问题，核心方法是通过上插入器和上均衡器给出计算表示的统一描述，主要贡献为从计算体恢复自由严格n-范畴，满足双范畴泛性质，给出群胚亏格等相关结论及同调下界等。
We prove existence results a la Jeff Smith for left-induced model category structures, of which the injective model structure on a diagram category is an important example. We further develop the notions of fibrant generation and Postnikov presentation from Hess, which are dual to a weak form of cofibrant generation and cellular presentation. As examples, for k a field and H a differential graded Hopf algebra over k, we produce a left-induced model structure on augmented H-comodule algebras and show that the category of bounded below chain complexes of finite-dimensional k-vector spaces has a …
In a previous work, we have introduced a weakening of Quillen model categories called weak model categories. They still allow all the usual constructions of model category theory, but are easier to construct and are in some sense better behaved. In this paper we continue to develop their general theory by introducing combinatorial and accessible weak model categories. We give simple necessary and sufficient conditions under which such a weak model category can be extended into a left and/or right semi-model category. As an application, we recover Cisinski-Olschok theory and generalize it to we…
G. Raptis has recently proved that, assuming Vopěnka's principle, every cofibrantly generated model category is Quillen equivalent to a combinatorial one. His result remains true for a slightly more general concept of a cofibrantly generated model category. We show that Vopěnka's principle is equivalent to this claim. The set-theoretical status of the original Raptis' result is open.
We study convergent (terminating and confluent) presentations of n-categories. Using the notion of polygraph (or computad), we introduce the homotopical property of finite derivation type for n-categories, generalizing the one introduced by Squier for word rewriting systems. We characterize this property by using the notion of critical branching. In particular, we define sufficient conditions for an n-category to have finite derivation type. Through examples, we present several techniques based on derivations of 2-categories to study convergent presentations by 3-polygraphs.
Combinatorial model categories are equivalent to presentable quasicategories
本文通过建立组合模型范畴、可表示拟范畴等局部可表示（无穷,1）-范畴模型的相对范畴间的Dwyer-Kan等价，证明了其基础拟范畴等价，为相关范畴研究提供了等价性依据。
We show that $\mathit{ann}$-categories admit a presentation by crossed bimodules, and prove that morphisms between them can be expressed by special kinds spans between the presentations. More precisely, we prove the groupoid of morphisms between two $\mathit{ann}$-categories is equivalent to that of bimodule butterflies between the presentations. A bimodule butterfly is a specialization of a butterfly, i.e. a special kind of span or fraction, between the underlying complexes
We show that the category of algebraically cofibrant objects in a combinatorial and simplicial model category A has a model structure that is left-induced from that on A. In particular it follows that any presentable model category is Quillen equivalent (via a single Quillen equivalence) to one in which all objects are cofibrant.
Framed Combinatorial Topology with Labels in $\infty$-Categories
本文扩展框架组合拓扑，引入$\infty$-范畴中的标签系统，构造分类网格和桁架的$\infty$-范畴，为高阶范畴模型和广义纽结研究奠定基础。
In his book on model categories, Hovey asked whether the 2-category $\mathbf{Mod}$ of model categories admits a "model 2-category structure" whose weak equivalences are the Quillen equivalences. We show that $\mathbf{Mod}$ does not have pullbacks and so cannot form a model 2-category. This lack of pullbacks can be traced to the two-out-of-three axiom on the weak equivalences of a model category. Accordingly, we define a premodel category to be a complete and cocomplete category equipped with two nested weak factorization systems. Combinatorial premodel categories form a complete and cocomplete…
A combinatorial genesis of the right-angled relations in Artin's classical braid groups
本文通过组合方法研究Artin经典辫形群的右角关系，利用UC(n,p×q)空间的特性，推导出其基本群与Artin群的近似关系，并计算了相关空间的范畴和拓扑复杂度。
Operads were originally defined as V-operads, that is, enriched in a symmetric or braided monoidal category V. The symmetry or braiding in V is required in order to describe the associativity axiom the operads must obey, as well as the associativity that must be a property of the action of an operad on any of its algebras. After a review of the role of operads in loop space theory and higher categories we go over definitions of iterated monoidal categories and introduce a large family of simple examples. Then we generalize the definition of operad by defining n-fold operads and their algebras …
We construct model category structures on various types of (marked) *-categories. These structures are used to present the infinity categories of (marked) *-categories obtained by inverting (marked) unitary equivalences. We use this presentation to explicitly calculate the \infty-categorical G-fixed points and G-orbits for G-equivariant (marked) *-categories.
Combinatorial models for stratified homotopy theory
本文证明分层单纯集上存在组合单纯模型结构，其一与Joyal模型结构的左Bousfield局部化Quillen等价，用于研究分层同伦假设。
We produce combinatorial models for configuration space in a simplicial complex, and for configurations near a single point ("local configuration space.") The model for local configuration space is built out of the poset of poset structures on a finite set. The model for global configuration space relies on a combinatorial model for a simplicial complex with a deleted subcomplex. By way of application, we study the nodal curve $y^2 z = x^3 + x^2 z$, obtaining a presentation for its two-strand braid group, a conjectural presentation for its three-strand braid group, and presentations for its tw…
We prove that every stable, combinatorial model category has a natural enrichment by symmetric spectra (or more precisely, a natural equivalence class of enrichments). This in some sense generalizes the simplicial enrichments of model categories provided by the Dwyer-Kan hammock localization. As one particular application, we are able to associate to every object of a stable, combinatorial model category a "homotopy endomorphism ring spectrum". The homotopy types of these ring spectra are preserved by Quillen equivalences, and so are an invariant for stable model categories.
Every right adjoint functor between presentable $\infty$-categories is shown to decompose canonically as a coreflection, followed by, possibly transfinitely many, monadic functors. Furthermore, the coreflection part is given a presentation in terms of a functorial iterated colimit. Background material, examples, and the relation to homology localization and completion are discussed as well.
In these notes we describe models of globular weak $(\infty,m)$-categories ($m\in\mathbb{N}$) in the Grothendieck style, i.e for each $m\in\mathbb{N}$ we define a globular coherator $Θ^{\infty}_{\mathbb{M}^m}$ whose set-models are globular weak $(\infty,m)$-categories. Then we describe the combinatorics of the small category $Θ_0$ whose objects are cubical pasting diagrams and whose morphisms are morphisms of cubical sets. This provides an accurate description of the monad, on the category of cubical sets (without degeneracies and connections), of cubical strict $\infty$-categories with connec…
A lot of well-known functors such as group homology, cyclic homology of algebras can be described as limits of certain simply defined functors over categories of presentations. In this paper, we develop technique for the description of the higher limits over categories of presentations and show that certain homological functors can be described in this way. In particular, we give a description of Hochschild homology and the derived functors of tensor, symmetric and exterior powers in the sense of Dold and Puppe as higher limits.
We present a combinatorial analogue of the nerve theorem for covers of small categories, using the Grothendieck construction. We apply our result to prove the inclusion-exclusion principle for the Euler characteristic of a finite category.
We construct a q-model structure, a h-model structure and a m-model structure on multipointed $d$-spaces and on flows. The two q-model structures are combinatorial and coincide with the combinatorial model structures already known on these categories. The four other model structures (the two m-model structures and the two h-model structures) are accessible. We give an example of multipointed $d$-space and of flow which are not cofibrant in any of the model structures. We explain why the m-model structures, Quillen equivalent to the q-model structure of the same category, are better behaved tha…
We give combinatorial models for the homotopy type of complements of elliptic arrangements (i.e., certain sets of abelian subvarieties in a product of elliptic curves). We give a presentation of the fundamental group of such spaces and, as an application, we treat the case of ordered configuration spaces of elliptic curves. Our models are finite polyhedral CW complexes, and our combinatorial tools of choice are acyclic categories (small categories without loops). As a stepping stone, we give a characterization of which acyclic categories arise as face categories of polyhedral CW complexes.
We exhibit a bridge between the theory of cellular categories, used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the combinatorial cellular categories (those where, in a precise sense, the cellular morphisms are generated by a set) are exactly those that give rise to stable independence notions. We give two applications: on the one hand, we show that the abstract elementary classes of roots of Ext studied by Baldwin-Eklof-Trlifaj are stable and tame. On the other hand, we give a simpler proof (in a special …
We introduce the notion of algebraic fibrant objects in a general model category and establish a (combinatorial) model category structure on algebraic fibrant objects. Based on this construction we propose algebraic Kan complexes as an algebraic model for oo-groupoids and algebraic quasi-categories as an algebraic model for (oo,1)-categories. We furthermore give an explicit proof of the homotopy hypothesis.
A combinatorial $K$-theory perspective on the Edge Reconstruction Conjecture in graph theory
本研究构建了基于带覆盖族的范畴$K$-理论的抽象重构问题研究框架，将其用于重新表述图论边重构猜想，并给出了该范畴$K$-理论的若干基本定理。
We give a presentation of Feynman categories from a representation--theoretical viewpoint. Feynman categories are a special type of monoidal categories and their representations are monoidal functors. They can be viewed as a far reaching generalization of groups, algebras and modules. Taking a new algebraic approach, we provide more examples and more details for several key constructions. This leads to new applications and results. The text is intended to be a self--contained basis for a crossover of more elevated constructions and results in the fields of representation theory and Feynman cat…
Matroids, Feynman categories, and Koszul duality
该研究引入并研究特定费曼范畴，将拟阵的Chow环、Orlik–Solomon代数等组合不变量组装为该范畴上的操作子结构，建立二者的Koszul型对偶以推广Getzler的结果，并给出Orlik–Solomon代数组合Leray模型的新解释。
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