We use Klyachko's methods [A funny property of sphere and equations over groups, Comm. in Alg. 21 (1993) 2555--2575] (see also Fenn-Rourke, L'Enseignment Math. 42 (1996) 49--74 and math.GR/9810184 and Cohen-Rourke, math.GR/0009101) to prove that, if a 1-cell and a 2-cell are added to a complex with torsion-free fundamental group, and with the 2-cell attached by an amenable t-shape, then pi_2 changes by extension of scalars. It then follows using a result of Bogley and Pride, Proc. Edinburgh Math. Soc. 35 (1992) 1--39, that the resulting fundamental group is also torsion free. We also prove that the normal closure of the attaching word contains no words of smaller complexity.
Let p be a fibration over a finite simplicial complex, whose fibers have the homotopy type of finite simplicial complexes. Then p is equivalent to an approximate fibration whose total space is a compact ENR. The proof uses homotopy coherent diagrams and their homotopy colimits. We also comment on the simple homotopy type of the total space and give an application to the fibering of Hilbert cube manifolds.
There exists a simplified Bar-Natan Khovanov complex for open 2-braids. The Khovanov cohomology of a knot diagram made by gluing tangles of this type is therefore often amenable to calculation. We lift this idea to the level of the Lipshitz-Sarkar stable homotopy type and use it to make new computations. Similarly, there exists a simplified Khovanov-Rozansky sl_n complex for open 2-braids with oppositely oriented strands and an even number of crossings. Diagrams made by gluing tangles of this type are called matched diagrams, and knots admitting matched diagrams are called bipartite knots. To …
We use a classical result of McCord and reduction methods of finite spaces to prove a generalization of Thomason's theorem on homotopy colimits over posets. In particular this allows us to characterize the homotopy colimits of diagrams of simplicial complexes in terms of the Grothendieck construction on the diagrams of their face posets. We also derive analogues of well known results on homotopy colimits in the combinatorial setting, including a cofinality theorem and a generalization of Quillen's Theorem A for posets.
We study the homotopy theory of a certain type of diagram categories whose vertices are in variable categories with a functorial path, leading to a good calculation of the homotopy category in terms of cofibrant objects. The theory is applied to the category of mixed Hodge diagrams of differential graded algebras. Using Sullivan's minimal models, we prove a multiplicative version of Beilinson's Theorem on mixed Hodge complexes. As a consequence, we obtain functoriality for the mixed Hodge structures on the rational homotopy type of complex algebraic varieties. In this context, the mixed Hodge …
Twisted diagrams are "diagrams" with components in different categories. Structure maps are defined using auxiliary data which consists of functors relating the various categories to each other. Prime examples of the construction are spectra (in the sense of homotopy theory) and quasi-coherent sheaves on schemes. We develop the basic theory of twisted diagrams, and establish various model structures (which are well-known in special cases). We also introduce a notion of homotopy sheaves, a collection of local data which is compatible up to weak equivalence, and study basic properties of such ob…
In [math.AT/9907138] we proved that strongly homotopy algebras are homotopy invariant concepts in the category of chain complexes. Our arguments were based on the fact that strongly homotopy algebras are algebras over minimal cofibrant operads and on the principle that algebras over cofibrant operads are homotopy invariant. In our approach, algebraic models for colored operads describing diagrams of homomorphisms played an important role. The aim of this paper is to give an explicit description of these models. A possible application is an appropriate formulation of the `ideal' homological per…
Let $D$ be a large category which is cocomplete. We construct a model structure (in the sense of Quillen) on the category of small functors from $D$ to simplicial sets. As an application we construct homotopy localization functors on the category of simplicial sets which satisfy a stronger universal property than the customary homotopy localization functors do.
The main objective of this paper is to show that the homotopy colimit of a diagram of quasi-categories and indexed by a small category is a localization of Lurie's higher Grothendieck construction of the diagram. We thereby generalize Thomason's classical result which states that the homotopy colimit of a diagram of categories has the homotopy type of (the classifying space of) the Grothendieck construction of the diagram of categories.
In this paper we develop homotopy theoretical methods for studying diagrams. In particular we explain how to construct homotopy colimits and limits in an arbitrary model category. The key concept we introduce is that of a model approximation. Our key result says that if a category admits a model approximation then so does any diagram category with values in this category. From the homotopy theoretical point of view categories with model approximations have similar properties to those of model categories. They admit homotopy categories (localizations with respect to weak equivalences). They als…
Let $G$ be a finite group acting on a small category $I$. We study functors $X \colon I \to \mathscr{C}$ equipped with families of compatible natural transformations that give a kind of generalized $G$-action on $X$. Such objects are called $G$-diagrams. When $\mathscr{C}$ is a sufficiently nice model category we define a model structure on the category of $G$-diagrams in $\mathscr{C}$. There are natural $G$-actions on Bousfield-Kan style homotopy limits and colimits of $G$-diagrams. We prove that weak equivalences between point-wise (co)fibrant $G$-diagrams induce weak $G$-equivalences on hom…
Given a diagram of rings, one may consider the category of modules over them. We are interested in the homotopy theory of categories of this type: given a suitable diagram of model categories M(s) (as s runs through the diagram), we consider the category of diagrams where the object X(s) at s comes from M(s). We develop model structures on such categories of diagrams, and Quillen adjunctions that relate categories based on different diagram shapes. Under certain conditions, cellularizations (or right Bousfield localizations) of these adjunctions induce Quillen equivalences. As an application w…
This paper contains two results on how homotopy limits of topological spaces interact with connectivity. The first is a formula for the connectivity of the homotopy limit of diagrams shaped over suitably finite categories, in terms of the connectivity of the spaces in the diagram. The second result shows that the homotopy fiber of the restriction map induced on homotopy limits by a functor of indexing categories is itself a homotopy limit, indexed over a Grothendieck construction.
Let $I$ be a small category with finite dimensional nerve, and $X\colon I\to Cat$ a diagram of small categories. We show that, under a "Reedy quasi-fibrancy condition", the homotopy limit of the geometric realization of $X$ is itself the geometric realization of a category. This categorical model for the homotopy limit is defined explicitly, as a category of natural transformations of diagrams. For the poset $\bullet\to\bullet\leftarrow\bullet$ we recover the model for homotopy pullbacks provided by Quillen's Theorem $B$ (specifically Barwick and Kan's version of Quillen's Theorem $B_2$). For …
Naturally occurring diagrams in algebraic topology are commutative up to homotopy, but not on the nose. It was quickly realized that very little can be done with this information. Homotopy coherent category theory arose out of a desire to catalog the higher homotopical information required to restore constructibility (or more precisely, functoriality) in such "up to homotopy" settings. These notes provide a three-part introduction to homotopy coherent category theory. The first part surveys the classical theory of homotopy coherent diagrams of topological spaces. The second part introduces the…
We give a generalization of Fukaya's Morse homotopy theoretic approach for 2-loop Chern--Simons perturbation theory to 3-valent graphs with arbitrary number of loops at least 2. We construct a sequence of invariants of integral homology 3-spheres with values in a space of 3-valent graphs (Jacobi diagrams or Feynman diagrams) by counting graphs in an integral homology 3-sphere satisfying certain condition described by a set of ordinary differential equations.
We show that the homotopy colimit construction for diagrams of categories with an operad action, recently introduced by Fiedorowicz, Stelzer and Vogt, has the desired homotopy type for diagrams of weak braided monoidal categories. This provides a more flexible way to realize E-2 spaces categorically.
We study the homotopy type of the harmonic compactification of the moduli space of a 2-cobordism S with one outgoing boundary component, or equivalently of the space of Sullivan diagrams of type S on one circle. Our results are of two types: vanishing and non-vanishing. In our vanishing results we are able to show that the connectivity of the harmonic compactification increases with the number of incoming boundary components. Moreover, we extend the genus stabilization maps of moduli spaces to the harmonic compactification and show that the connectivity of these maps increases with the genus a…
We extend the notion of simplicial set with effective homology to diagrams of simplicial sets. Further, for a given finite diagram of simplicial sets $X \colon \mathcal{I} \to \mathsf{sSet}$ such that each simplicial set $X(i)$ has effective homology, we present an algorithm computing the homotopy colimit $\mathsf{hocolim} X$ as a simplicial set with effective homology. We also give an algorithm computing the cofibrant replacement $X^\mathsf{cof}$ of $X$ as a diagram with effective homology. This is applied to computing of equivariant cohomology operations.
We call a diagram D absolutely cartesian if F(D) is homotopy cartesian for all homotopy functors F. This is a sensible notion for diagrams in categories C where Goodwillie's calculus of functors may be set up for functors with domain C. We prove a classification theorem for absolutely cartesian squares of spaces and state a conjecture of the classification for higher dimensional cubes.
We adapt the notion of an algebraic theory to work in the setting of quasicategories developed recently by Joyal and Lurie. We develop the general theory at some length. We study one extended example in detail: the theory of commutative monoids (which turns out to be essentially just a 2-category). This gives a straightforward, combinatorially explicit, and instructive notion of a commutative monoid. We prove that this definition is equivalent (in appropriate senses) both to the classical concept of an E-infinity monoid and to Lurie's concept of a commutative algebra object.
Consider a push-out diagram of spaces C <-- A --> B, construct the homotopy push-out, and then the homotopy pull-back of the diagram one gets by forgetting the initial object A. We compare the difference between A and this homotopy pull-back. This difference is measured in terms of the homotopy fibers of the original maps. Restricting our attention to the connectivity of these maps, we recover the classical Blakers-Massey Theorem.
An affine monoid is an additive monoid which is cancellative, pointed and finitely generated. An affine monoid $Λ$ has the partial order defined by $λ\le λ+ μ$. The Frobenius complex is the order complex of an open interval of $Λ$ with respect to this partial order. The reduced homology of the Frobenius complex is related to the torsion group of the monoid algebra $K[Λ]$. In this paper, we pay attention to homotopy types of Frobenius complexes, and we express the homotopy types of the Frobenius complexes of $Λ$ in terms of those of $Λ_1$ and $Λ_2$ when $Λ$ is an affine monoid obtained by gluin…
In this paper we represent the Vassiliev model for the homotopy type of the one-point compactification of subspace arrangements as a homotopy colimit of an appropriate diagram over the nerve complex of the intersection semilattice of the arrangement. Furthermore, using a generalization of simplicial collapses to diagrams of topological spaces over simplicial complexes, we construct an explicit deformation retraction from the Vassiliev model to the Ziegler-Zivaljevic model.
For $\Cc$ a $G$-category, we give a condition on a diagram of simplicial sets indexed on $\Cc$ that allows us to define a natural $G$-action on its homotopy colimit, and in some other simplicial sets and categories defined in terms of the diagram. Well-known theorems on homeomorphisms and homotopy equivalences are generalized to an equivariant version.
We construct a new model category presenting the homotopy theory of presheaves on "inverse EI $(\infty,1)$-categories", which contains universe objects that satisfy Voevodsky's univalence axiom. In addition to diagrams on ordinary inverse categories, as considered in previous work of the author, this includes a new model for equivariant algebraic topology with a compact Lie group of equivariance. Thus, it offers the potential for applications of homotopy type theory to equivariant homotopy theory.
For any finite group $G$, we define the notion of a Bredon homotopy action of $G$, modelled on the diagram of fixed point sets $(X_H)_{H\leq G}$ for a $G$-space $X$, together with a pointed homotopy action of the group $N_{G}H/H$ on $X^{H}/(\bigcup_{H<K} X^{K})$. We then describe a procedure for constructing a suitable diagram $\underline{X}:O_G^{op}\to Top$ from this data, by solving a sequence of elementary lifting problems. If successful, we obtain a $G$-space $X'$ realizing the given homotopy information, determined up to Bredon $G$-homotopy type. Such lifting methods may also be used to u…
Associated to each small category $C$, there is a category of $C$-shaped diagrams of simplicial sets and an $\infty$-category of $NC$-shaped homotopy coherent diagrams of spaces. We present a functor which exhibits the latter as the $\infty$-categorical localisation of the former at the objectwise weak homotopy equivalences. This builds on a Quillen equivalence between the projective and covariant model structures associated to $C$ due to Heuts-Moerdijk, as well as Cisinski's theory of $\infty$-categorical localisations. We use the localisation functor to give simplified proofs that the left (…
By analogy with methods of Spivak, there is a realization functor which takes a persistence diagram $Y$ in simplicial sets to an extended pseudo-metric space (or ep-metric space) $Re(Y)$. The functor $Re$ has a right adjoint, called the singular functor, which takes an ep-metric space $Z$ to a persistence diagram $S(Z)$. We give an explicit description of $Re(Y)$, and show that it depends only on the $1$-skeleton $sk_{1}Y$ of $Y$. If $X$ is a totally ordered ep-metric space, then there is an isomorphism $Re(V_{\ast}(X)) \cong X$, between the realization of the Vietoris-Rips diagram $V_{\ast}(X…
We prove an extension of the Quillen Theorem Bn for homotopy fibres to a similar result for homotopy pullbacks and use this to obtain sufficient conditions on a pullback diagram of categories to guarantee that it be a homotopy pullback.
Cofiltered diagrams of spectra, also called pro-spectra, have arisen in diverse areas, and to date have been treated in an ad hoc manner. The purpose of this paper is to systematically develop a homotopy theory of pro-spectra and to study its relation to the usual homotopy theory of spectra, as a foundation for future applications. The surprising result we find is that our homotopy theory of pro-spectra is Quillen equivalent to the opposite of the homotopy theory of spectra. This provides a convenient duality theory for all spectra, extending the classical notion of Spanier-Whitehead duality w…
The based loop space of a configuration space of points in a Euclidean space can be viewed as a space of pure braids in a Euclidean space of one dimension higher. We continue our study of such spaces in terms of Kontsevich's CDGA of diagrams and Chen's iterated integrals. We construct a power series connection which yields a Hopf algebra isomorphism between the homology of the space of pure braids and the cobar construction on diagrams. It maps iterated Whitehead products to trivalent trees modulo the IHX relation. As an application, we establish a correspondence between Milnor invariants of B…
To every affine real arrangement of hyperplanes we associate a family of diagrams of spaces over the face poset of the arrangement. We show that any cover of the complement of the complexification of the arrangement is homotopy equivalent to the homotopy colimit of one of the diagrams. More precisely, we show that any cover of the Salvetti complex is isomorphic to the order complex of the poset limit of one of the diagrams. We thus obtain explicit simplicial models for covers of the Salvetti complex.
In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen equivalent to one another and to Rezk's complete Segal space model structure on the category of simplicial spaces. Here, we show that these results still hold if we instead use groupoid or "invertible" cases. Namely, we show that model structures on the categories of simplicial groupoids, Segal pregroupoids, and invertible simplicial spaces are all Quillen equivalent to one another and to the standard mod…
Given a diagram of Pi-algebras (graded groups equipped with an action of the primary homotopy operations), we ask whether it can be realized as the homotopy groups of a diagram of spaces. The answer given here is in the form of an obstruction theory, of somewhat wider application, formulated in terms of generalized Pi-algebras. This extends a program begun in [J. Pure Appl. Alg. 103 (1995) 167-188] and [Topology 43 (2004) 857-892] to study the realization of a single Pi-algebra. In particular, we explicitly analyze the simple case of a single map, and provide a detailed example, illustrating t…
We initiate the homotopical study of racks and quandles, two algebraic structures that govern knot theory and related braided structures in algebra and geometry. We prove analogs of Milnor's theorem on free groups for these theories and their pointed variants, identifying the homotopy types of the free racks and free quandles on spaces of generators. These results allow us to complete the stable classification of racks and quandles by identifying the ring spectra that model their stable homotopy theories. As an application, we show that the stable homotopy of a knot quandle is, in general, mor…
Given an appropriate diagram of left Quillen functors between model categories, one can define a notion of homotopy fiber product, but one might ask if it is really the correct one. Here, we show that this homotopy pullback is well-behaved with respect to translating it into the setting of more general homotopy theories, given by complete Segal spaces, where we have well-defined homotopy pullbacks.
We present an algorithm that, given finite simplicial sets $X$, $A$, $Y$ with an action of a finite group $G$, computes the set $[X,Y]^A_G$ of homotopy classes of equivariant maps $\ell \colon X \to Y$ extending a given equivariant map $f \colon A \to Y$ under the stability assumption $\dim X^H \leq 2 \operatorname{conn} Y^H$ and $\operatorname{conn} Y^H \geq 1$, for all subgroups $H\leq G$. For fixed $n = \operatorname{dim} X$, the algorithm runs in polynomial time. When the stability condition is dropped, the problem is undecidable already in the non-equivariant setting. The algorithm is obt…
Swartz proved that any matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a sphere. This was an unexpected extension from the oriented matroid case, but unfortunately the construction is not explicit. Anderson later provided an explicit construction, but had to use cell complexes of high dimensions that are homotopy equivalent to lower dimensional spheres. Using diagrams of spaces we give an explicit construction of arrangements in the right dimensions. Swartz asked if it is possible to arrange spheres of codimension two, and we provide…
We describe the second homotopy group of any CW-complex $K$ by analyzing the universal cover of a locally finite model of $K$ using the notion of $G$-coloring of a partially ordered set. As applications we prove a generalization of the Hurewicz theorem, which relates the homotopy and homology of non-necessarily simply-connected complexes, and derive new results on asphericity for two-dimensional complexes and group presentations.
Several possible presentations for the homotopy theory of (non-hypercomplete) $\infty$-stacks on a classical site S are discussed. In particular, it is shown that an elegant combinatorial description in terms of diagrams in S exists, similar to Cisinski's presentation, based on work of Quillen, Thomason and Grothendieck, of usual homotopy theory by small categories and their smallest (basic) localizer. As an application it is shown that any (local) fibered (a.k.a. algebraic) derivator over S with stable fibers extends to $\infty$-stacks in a well-defined way under mild assumptions.
Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a track functor on the track category of pointed spaces compatible with fiber sequences, suspensions and loop spaces. They also yield algebraic models of homotopy types with homotopy groups concentrated in two consecutive dimensions.
We develop a localisation theory for certain categories, yielding a 3-arrow calculus: Every morphism in the localisation is represented by a diagram of length 3, and two such diagrams represent the same morphism if and only if they can be embedded in a 3-by-3 diagram in an appropriate way. The methods to construct this localisation are similar to the Ore localisation for a 2-arrow calculus; in particular, we do not have to use zigzags of arbitrary length. Applications include the localisation of an arbitrary model category with respect to its weak equivalences as well as the localisation of it…
The primary algebraic model of a ring spectrum is the ring of homotopy groups. We introduce the secondary model which has the structure of a secondary analogue of a ring. This new algebraic model determines Massey products and cup-one squares. As an application we obtain new derivations of the homotopy ring.
This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy type of their classifying spaces. Bicategories (in particular monoidal categories) have well understood simple geometric realizations, and we here deal with homotopy types represented by lax diagrams of bicategories, that is, lax functors to the tricategory of bicategories. In this paper, it is proven that, when a certain bicategorical Grothendieck construction is performed on a lax diagram of bicategories, then the classifying space of the resulting bicategory can be though…
We describe the first stem of the stable homotopy groups of spheres by using some Puppe sequences, Thom complexes, K-Theory and Adams operations following the ideas of J. Frank Adams. We also touch upon the second and the third stems in this perspective. Neither spectral squences nor Steenrod operations are used.
We generalize two classical homotopy theory results, the Blakers-Massey Theorem and Quillen's Theorem B, to G-equivariant cubical diagrams of spaces, for a discrete group G. We show that the equivariant Freudenthal suspension Theorem for permutation representations is a direct consequence of the equivariant Blakers-Massey Theorem. We also apply this theorem to generalize to G-manifolds a result about cubes of configuration spaces from embedding calculus. Our proof of the equivariant Theorem B involves a generalization of the classical Theorem B to higher dimensional cubes, as well as a categor…
Given a link diagram L we construct spectra X^j(L) so that the Khovanov homology Kh^{i,j}(L) is isomorphic to the (reduced) singular cohomology H^i(X^j(L)). The construction of X^j(L) is combinatorial and explicit. We prove that the homotopy type of X^j(L) depends only on the isotopy class of the corresponding link.
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.
We prove that for a finitely generated group $G$, the second stable homotopy group $π_2^S(K(G,1))$ of the Eilenberg-Maclane space $K(G,1)$ is completely determined by the Schur multiplier $H_2(G)$. We also prove that the second stable homotopy group $π_2^S(K(G,1))$ is equal to the Schur multiplier $H_2(G)$ for a torsion group $G$ with no elements of order $2$ and show that for such groups, $π_2^S(K(G,1))$ is a direct factor of $π_{3}(SK(G,1))$, where $S$ denotes suspension and $π_2^S$ the second stable homotopy group. We compute $π_{3}(SK(G,1))$ and $π_2^S(K(G,1))$ for symmetric, alternating, …
