The 2-category theory of quasi-categories

In this paper we re-develop the foundations of the category theory of quasi-categories (also called ∞-categories) using 2-category theory. We show that Joyal's strict 2-category of quasi-categories admits certain weak 2-limits, among them weak comma objects. We use these comma quasi-categories to encode universal properties relevant to limits, colimits, and adjunctions and prove the expected theorems relating these notions. These universal properties have an alternate form as absolute lifting diagrams in the 2-category, which we show are determined pointwise by the existence of certain initial or terminal vertices, allowing for the easy production of examples.     

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)].     

The Boardman-Vogt resolution of operads in monoidal model categories

We extend the -construction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for well-pointed Σ-cofibrant operads. The standard simplicial resolution of Godement as well as the cobar-bar chain resolution are shown to be particular instances of this generalised -construction.     

Using equivariant obstruction theory in combinatorial geometry

A significant group of problems coming from the realm of combinatorial geometry can be approached fruitfully through the use of Algebraic Topology. From the first such application to Kneser's problem in 1978 by Lovász [L. Lovász, Knester's conjecture, chromatic number of distance graphs on the sphere, Acta. Sci. Math (Szeged) 45 (1983) 317-323] through the solution of the Lovász conjecture [E. Babson, D. Kozlov, Proof of Lovasz conjecture, Annals of Mathematics (2) (2004), submitted for publication; C. Schultz, A short proof of for all n and a graph colouring theorem by Babson and Kozlov, 2005, arXiv: math.AT/0507346v2], many methods from Algebraic Topology have been developed. Specifically, it appears that the understanding of equivariant theories is of the most importance. The solution of many problems depends on the existence of an elegantly constructed equivariant map. A variety of results from algebraic topology were applied in solving these problems. The methods used ranged from well known theorems like Borsuk-Ulam and Dold theorem to the integer/ideal-valued index theories. Recently equivariant obstruction theory has provided answers where the previous methods failed. For example, in papers [R.T. ?ivaljevi?, Equipartitions of measures in R⁴, arXiv: math.0412483, Trans. Amer. Math. Soc., submitted for publication] and [P. Blagojevi?, A. Dimitrijevi? Blagojevi?, Topology of partition of measures by fans and the second obstruction, arXiv: math.CO/0402400, 2004] obstruction theory was used to prove the existence of different mass partitions. In this paper we extract the essence of the equivariant obstruction theory in order to obtain an effective general position map scheme for analyzing the problem of existence of equivariant maps. The fact that this scheme is useful is demonstrated in this paper with three applications:

(A)

a “half-page” proof of the Lovász conjecture due to Babson and Kozlov [E. Babson, D. Kozlov, Proof of Lovasz conjecture, Annals of Mathematics (2) (2004), submitted for publication] (one of two key ingredients is Schultz' map [C. Schultz, A short proof of for all n and a graph colouring theorem by Babson and Kozlov, 2005, arXiv: math.AT/0507346v2]),

(B)

a generalization of the result of V. Makeev [V.V. Makeev, Equipartitions of continuous mass distributions on the sphere an in the space, Zap. Nauchn. Sem. S.-Petersburg (POMI) 252 (1998) 187-196 (in Russian)] about the sphere S² measure partition by 3-planes (Section 3), and

(C)

the new (a,b,a), class of 3-fan 2-measures partitions (Section 3).

These three results, sorted by complexity, share the spirit of analyzing equivariant maps from spheres to complements of arrangements of subspaces.     

Obstruction theory in model categories

Many examples of obstruction theory can be formulated as the study of when a lift exists in a commutative square. Typically, one of the maps is a cofibration of some sort and the opposite map is a fibration, and there is a functorial obstruction class that determines whether a lift exists. Working in an arbitrary pointed proper model category, we classify the cofibrations that have such an obstruction theory with respect to all fibrations. Up to weak equivalence, retract, and cobase change, they are the cofibrations with weakly contractible target. Equivalently, they are the retracts of principal cofibrations. Without properness, the same classification holds for cofibrations with cofibrant source. Our results dualize to give a classification of fibrations that have an obstruction theory.     

The homotopy type of the polyhedral product for shifted complexes

We prove a conjecture of Bahri, Bendersky, Cohen and Gitler: if K$K$ is a shifted simplicial complex on n$n$ vertices, _X1,…,_Xn$X_1,\dots ,X_n$ are pointed connected CW$CW$-complexes and C_Xi$C{X}_i$ is the cone on  _Xi$X_i$, then the polyhedral product determined by K$K$ and the pairs (C_Xi,_Xi)$(C{X}_i,X_i)$ is homotopy equivalent to a wedge of suspensions of smashes of the _Xi$X_i$’s. Earlier work of the authors dealt with the special case where each _Xi$X_i$ is a loop space. New techniques are introduced to prove the general case. These have the advantage of simplifying the earlier results and of being sufficiently general to show that the conjecture holds for a substantially larger class of simplicial complexes. We discuss connections between polyhedral products and toric topology, combinatorics, and classical homotopy theory.     

A completion theorem for pro-G-spectra

We give a very general completion theorem for pro-spectra. We show that, if G is a compact Lie group, M[∗] is a pro-G-spectrum, and F is a family of (closed) subgroups of G, then the mapping pro-spectrum F(EF₊,M[∗]) is the F-adic completion of M[∗], in the sense that the map M[∗]→F(EF₊,M[∗]) is the universal map into an algebraically F-adically complete pro-spectrum. Here, F(EF₊,M[∗]) denotes the pro-G-spectrum , where runs over the finite subcomplexes of EF₊.     

Three models for the homotopy theory of homotopy theories

Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the “homotopy theory” of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a “homotopy theory of homotopy theories”. In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent to the homotopy theory arising from the model category structure on simplicial categories. Thus, any of these three categories with the respective weak equivalences could be considered a model for the homotopy theory of homotopy theories. One of them in particular, Rezk’s complete Segal space model category structure on the category of simplicial spaces, is much more convenient from the perspective of making calculations and therefore obtaining information about a given homotopy theory.     

The polyhedral product functor: A method of decomposition for moment-angle complexes, arrangements and related spaces

This article gives a natural decomposition of the suspension of generalized moment-angle complexes or partial product spaces which arise as polyhedral product functors described below. The geometrical decomposition presented here provides structure for the stable homotopy type of these spaces including spaces appearing in work of Goresky-MacPherson concerning complements of certain subspace arrangements, as well as Davis-Januszkiewicz and Buchstaber-Panov concerning moment-angle complexes. Since the stable decompositions here are geometric, they provide corresponding homological decompositions for generalized moment-angle complexes for any homology theory.     

A note on the homotopy theory of pro-G-spectra

This paper represents a step toward a model structure on pro-spectra in which the weak equivalences are the maps inducing pro-isomorphisms of all pro-homotopy groups. We construct a category in which these weak equivalences are inverted and show that we have not inverted “too much,” in the sense that isomorphic objects still give pro-isomorphic cohomology groups.     

A Liouville-type theorem for the p-Laplacian with potential term

In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a singular p  -Laplacian problem with a potential term, such that a nonzero subsolution of another such problem is also a ground state. Unlike in the linear case (p=2$p=2$), this condition involves comparison of both the functions and of their gradients.     

The homotopy type of the complement of a coordinate subspace arrangement

The homotopy type of the complement of a complex coordinate subspace arrangement is studied by utilising some connections between its topological and combinatorial structures. A family of arrangements for which the complement is homotopy equivalent to a wedge of spheres is described. One consequence is an application in commutative algebra: certain local rings are proved to be Golod, that is, all Massey products in their homology vanish.     

Crossed modules as homotopy normal maps

In this note we consider crossed modules of groups (N→G, G→Aut(N)), as a homotopy version of the inclusion N⊂G of a normal subgroup. Our main observation is a characterization of the underlying map N→G of a crossed module in terms of a simplicial group structure on the associated bar construction. This approach allows for “natural” generalizations to other monoidal categories, in particular we consider briefly what we call “normal maps” between simplicial groups.     

On the universal coefficients formula for shape homology

In this paper it is investigated whether various shape homology theories satisfy the Universal Coefficients Formula (UCF). It is proved that pro-homology and strong homology satisfy UCF in the class FAB of finitely generated abelian groups, while they do not satisfy UCF in the class AB of all abelian groups. Two new shape homology theories (called UCF-balanced) are constructed. It is proved that balanced pro-homology satisfies UCF in the class AB, while balanced strong homology satisfies UCF only in the class FAB.     

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.