Orientals as free weak ω-categories

The orientals are the free strict ω-categories on the simplices introduced by Street. The aim of this paper is to show that they are also the free weak ω-categories on the same generating data. More precisely, we exhibit the complicial nerves of the orientals as fibrant replacements of the simplices in Verity's model structure for weak complicial sets.     

A folk model structure on omega-cat

The primary aim of this work is an intrinsic homotopy theory of strict ω-categories. We establish a model structure on ωCat, the category of strict ω-categories. The constructions leading to the model structure in question are expressed entirely within the scope of ωCat, building on a set of generating cofibrations and a class of weak equivalences as basic items. All objects are fibrant while free objects are cofibrant. We further exhibit model structures of this type on n-categories for arbitrary n∈N, as specializations of the ω-categorical one along right adjoints. In particular, known cases for n=1 and n=2 nicely fit into the scheme.     

Ordinal subdivision and special pasting in quasicategories

Quasicategories are simplicial sets with properties generalising those of the nerve of a category. They model weak (ω,1)-categories. Using a combinatorially defined ordinal subdivision, we examine composition rules for certain special pasting diagrams in quasicategories. The subdivision is of combinatorial interest in its own right and is linked with various combinatorial constructions.     

Multiple Categories: The Equivalence of a Globular and a Cubical Approach

We show the equivalence of two kinds of strict multiple category, namely the well-known globular ω-categories, and the cubical ω-categories with connections.     

Accessible aspects of 2-category theory

Categorical structures and their pseudomaps rarely form locally presentable 2-categories in the sense of Cat-enriched category theory. However, we show that if the categorical structure in question is sufficiently weak (such as the structure of monoidal, but not strict monoidal, categories) then the 2-category in question is accessible. Furthermore, we explore the flexible limits that such 2-categories possess and their interaction with filtered colimits.     

Nerves of bicategories as stratified simplicial sets

In this paper, we aim to move towards a definition of weak n-category akin to Street’s definition of weak ω-category. This will be accomplished in dimension 1 directly and in dimension 2 by comparison with work of Duskin. In particular, we discuss the relationship between certain weak complicial sets and Duskin’s n-dimensional Postnikov complexes.     

A homotopy theory for stacks

We give a homotopy theoretic characterization of stacks on a site C as the homotopy sheaves of groupoids on C. We use this characterization to construct a model category in which stacks are the fibrant objects. We compare different definitions of stacks and show that they lead to Quillen equivalent model categories. In addition, we show that these model structures are Quillen equivalent to the S ²-nullification of Jardine’s model structure on sheaves of simplicial sets on C.     

2-nerves for bicategories

We describe a Cat-valued nerve of bicategories, which associates to every bicategory a simplicial object in Cat, called the 2-nerve. This becomes the object part of a 2-functor N : NHom → [Δ^op,Cat], where NHom is a 2-category whose objects are bicategories and whose 1-cells are normal homomorphisms of bicategories. The 2-functor N is fully faithful and has a left biadjoint, and we characterize its image. The 2-nerve of a bicategory is always a weak 2-category in the sense of Tamsamani, and we show that NHom is biequivalent to a certain 2-category whose objects are Tamsamani weak 2-categories. The hospitality of Macquarie University and the support of the Australian Research Council are gratefully acknowledged by S. Lack. The support of the Australian Research Council is gratefully acknowledged by S. Paoli.     

A characterization of fibrant Segal categories

In this note we prove that Reedy fibrant Segal categories are fibrant objects in the model category structure . Combining this result with a previous one, we thus have that the fibrant objects are precisely the Reedy fibrant Segal categories. We also show that the analogous result holds for Segal categories that are fibrant in the projective model structure on simplicial spaces, considered as objects in the model structure .

     

A categorical characterization of strong Steiner ω-categories

Strong Steiner ω-categories are a class of ω-categories that admit algebraic models in the form of chain complexes, whose formalism allows for several explicit computations. The conditions defining strong Steiner ω-categories are traditionally expressed in terms of the associated chain complex, making them somewhat disconnected from the ω-categorical intuition. The purpose of this paper is to characterize this class as the class of ω-categories generated by polygraphs that satisfy a loop-freeness condition that does not make explicit use of the associated chain complex and instead relies on the categorical features of ω-categories.     

Levels in the toposes of simplicial sets and cubical sets

The essential subtoposes of a fixed topos form a complete lattice, which gives rise to the notion of a level in a topos. In the familiar example of simplicial sets, levels coincide with dimensions and give rise to the usual notions of n-skeletal and n-coskeletal simplicial sets. In addition to the obvious ordering, the levels provide a stricter means of comparing the complexity of objects, which is determined by the answer to the following question posed by Bill Lawvere: when does n-skeletal imply k-coskeletal? This paper, which subsumes earlier unpublished work of some of the authors, answers this question for several toposes of interest to homotopy theory and higher category theory: simplicial sets, cubical sets, and reflexive globular sets. For the latter, n-skeletal implies (n+1)-coskeletal but for the other two examples the situation is considerably more complicated: n-skeletal implies (2n−1)-coskeletal for simplicial sets and 2n-coskeletal for cubical sets, but nothing stronger. In a discussion of further applications, we prove that n-skeletal cyclic sets are necessarily (2n+1)-coskeletal.     

Representability theorems for presheaves of spectra

Suppose that M is a simplicial model category and that F is a contravariant simplicial functor defined on M which takes values in pointed simplicial sets. This note displays conditions on the simplicial model category M and the functor F such that F is representable up to weak equivalence. The conditions on F are homotopy coherent versions of the classical conditions for Brown representability, while M should have the fundamental properties of the stable model structure for presheaves of spectra on a Grothendieck site.     

Splitting and conjoining objects in monotopological categories

Power-sets are defined for any concrete category (over Set) with finite concrete products, and their structure described for monotopological categories. These sets are used to define the notions of splitting object and of conjoining object. Characterizations of the existence of these objects in monotopological categories are given. It is proved that no proper monotopological category can be concretely cartesian closed. Most well-known monotopological categories with splitting objects are topological or are c-categories, but it is shown that there are many proper monotopological categories which are not c-categories, and yet have splitting objects, and may even be cartesian closed. One of the characterizations of the existence of splitting objects is used to prove that a monotopological category with splitting objects is cartesian closed iff the largest initial completion in which it is epireflective is cartesian closed iff its MacNeille completion is cartesian closed.     

A Quillen Model Structure for 2-Categories

We describe a cofibrantly generated Quillen model structure on the locally finitely presentable category 2-Cat of (small) 2-categories and 2-functors; the weak equivalences are the biequivalences, and the homotopy relation on 2-functors is just pseudonatural equivalence. The model structure is proper, and is compatible with the monoidal structure given by the Gray tensor product. It is not compatible with the Cartesian closed structure, in which the tensor product is the product.The model structure restricts to a model structure on the full subcategory PsGpd of 2-Cat, consisting of those 2-categories in which every arrow is an equivalence and every 2-cell is invertible. The model structure on PsGpd is once again proper, and compatible with the monoidal structure given by the Gray tensor product.     

Homotopical algebraic geometry I: topos theory

This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. 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 ∞-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 simplicial presheaves over a Grothendieck site of Joyal and Jardine. We also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves, by proving that there exists a one-to-one correspondence between S-topologies on an S-category T, and certain left exact Bousfield localizations of the model category of pre-stacks on T. Based on the above results, we study the notion of model topos introduced by Rezk, and we relate it to our model categories of stacks over S-sites.In the second part of the paper, we present a parallel theory where S-categories, S-topologies and S-sites are replaced by model categories, model topologies and model sites. We prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over S-sites is provided by the simplicial localization construction of Dwyer and Kan. As an example of application, we propose a definition of étale K-theory of ring spectra, extending the étale K-theory of commutative rings.     

Calculus of functors and model categories

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure [B. Chorny, W.G. Dwyer, Homotopy theory of small diagrams over large categories, preprint, 2005]. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for a classification of polynomial and homogeneous functors. In the n-homogeneous model structure, the nth derivative is a Quillen functor to the category of spectra with Σn-action. After taking into account only finitary functors—which may be done in two different ways—the above Quillen map becomes a Quillen equivalence. This improves the classification of finitary homogeneous functors by T.G. Goodwillie [T.G. Goodwillie, Calculus. III. Taylor series, Geom. Topol. 7 (2003) 645-711 (electronic)].