We show that discrete and classical homotopy theories are equivalent after localizing at n-equivalences for any non-negative integer n. By constructing an explicit homotopy inverse to the graph nerve functor associating an n-fibrant cubical set to a graph, we are also able to give explicit computations of several previously unknown discrete homotopy groups of boundaries of cubes and suspensions of cycles.
We prove that symmetric monoidal weak n-groupoids in the Tamsamani model provide a model for stable n-types. Moreover, we recover the classical statement that Picard categories model stable 1-types.
We construct a "diagonal" cofibrantly generated model structre on the category of simplicial objects in the category of topological categories sCat_{Top}, which is the category of diagrams [Δ^{op}, Cat_{Top}]. Moreover, we prove that the diagonal model structures is left proper and cellular. We also prove that the category of \infty-groupoids (the full subcategory of topological categories) has a cofibrantly generated model structure and is Quillen equivalent to the model category of simplicial sets, which proves the Grothendieck's homotopy hypothesis.
We introduce and study a notion of cylinder coherator similar to the notion of Grothendieck coherator which define more flexible notion of weak infinity groupoids. We show that each such cylinder coherator produces a combinatorial semi-model category of weak infinity groupoids, whose objects are all fibrant and which is in a precise sense "freely generated by an object". We show that all those semi model categories are Quillen equivalent together and Quillen to the model category of spaces. A general procedure is given to produce such coherator, and several explicit examples are presented: one…
We explicitly construct an SO(2)-action on a skeletal version of the 2-dimensional framed bordism bicategory. By the 2-dimensional Cobordism Hypothesis for framed manifolds, we obtain an SO(2)-action on the core of fully-dualizable objects of the target bicategory. This action is shown to coincide with the one given by the Serre automorphism. We give an explicit description of the bicategory of homotopy fixed points of this action, and discuss its relation to the classification of oriented 2d topological quantum field theories.
Freyd's Generating Hypothesis is an important problem in topology with deep structural consequences for finite stable homotopy. Due to its complexity some recent work has examined analogous questions in various other triangulated categories. In this short note we analyze the question in noncommutative stable homotopy, which is a canonical generalization of finite stable homotopy. Along the way we also discuss Spanier--Whitehead duality in this extended setup.
We show that conically smooth stratified spaces embed fully faithfully into $\infty$-categories. This articulates a stratified generalization of the homotopy hypothesis proposed by Grothendieck. As such, each $\infty$-category defines a stack on conically smooth stratified spaces, and we identify the descent conditions it satisfies. These include $\mathbb{R}^1$-invariance and descent for open covers and blow-ups, analogous to sheaves for the h-topology in $\mathbb{A}^1$-homotopy theory. In this way, we identify $\infty$-categories as striation sheaves, which are those sheaves on conically smoo…
This paper studies a discrete homotopy theory for graphs introduced by Barcelo et al. We prove two main results. First we show that if $G$ is a graph containing no 3- or 4-cycles, then the $n$th discrete homotopy group $A_n(G)$ is trivial for all $n\geq 2$. Second we exhibit for each $n\geq 1$ a natural homomorphism $ψ:A_n(G)\to \mathcal{H}_n(G)$, where $\mathcal{H}_n(G)$ is the $n$th discrete cubical singular homology group, and an infinite family of graphs $G$ for which $\mathcal{H}_n(G)$ is nontrivial and $ψ$ is surjective. It follows that for each $n\geq 1$ there are graphs $G$ for which $…
In this paper, we determine the homotopy type of the Morse complex of certain collections of simplicial complexes by studying dominating vertices or strong collapses. We show that if $K$ contains two leaves that share a common vertex, then the Morse complex is strongly collapsible and hence has the homotopy type of a point. We also show that the pure Morse complex of a tree is strongly collapsible, thereby recovering as a corollary a result of Ayala et al. In addition, we prove that the Morse complex of a disjoint union $K\sqcup L$ is the Morse complex of the join $K*L$. This result is used to…
If K is a discrete group and Z is a K-spectrum, then the homotopy fixed point spectrum Z^{hK} is Map_*(EK_+, Z)^K, the fixed points of a familiar expression. Similarly, if G is a profinite group and X is a discrete G-spectrum, then X^{hG} is often given by (H_{G,X})^G, where H_{G,X} is a certain explicit construction given by a homotopy limit in the category of discrete G-spectra. Thus, in each of two common equivariant settings, the homotopy fixed point spectrum is equal to the fixed points of an explicit object in the ambient equivariant category. We enrich this pattern by proving in a preci…
We show that the categories PsTop and Lim of pseudotopological spaces and limit spaces, respectively, admit cofibration category structures, and that PsTop admits a model category structure, giving several ways to simultaneously study the homotopy theory of classical topological spaces, combinatorial spaces such as graphs and matroids, and metric spaces endowed with a privileged scale, in addition to spaces of maps between them. In the process, we give a sufficient condition for a topological construct which contains compactly generated Hausdorff spaces as a subcategory to admit an $I$-categor…
In this paper we show that both the homotopy category of strict $n$-categories, $1\leqslant n \leqslant \infty$, and the homotopy category of Steiner's augmented directed complexes are equivalent to the category of homotopy types. In order to do so, we first prove an abstract result, based on a strategy of Fritsch and Latch, giving sufficient conditions for a nerve functor with values in simplicial sets to induce an equivalence at the level of homotopy categories. We then apply this result to strict $n$-categories and augmented directed complexes, for which the hypothesis of our theorem were e…
We construct a discrete model of the homotopy theory of $S^1$-spaces. We define a category $\sP$ with objects composed of a simplicial set and a cyclic set along with suitable compatibility data. $\sP$ inherits a model structure from the model structures on the categories of simplicial sets and cyclic sets. We then show that there is a Quillen equivalence between $\sP$ and the model category of $S^1$-spaces in which weak equivalences and fibrations are maps inducing weak equivalences and fibrations on passage to all fixed point sets.
The aim of this paper is to develop a refinement of Forman's discrete Morse theory. To an acyclic partial matching $μ$ on a finite regular CW complex $X$, Forman introduced a discrete analogue of gradient flows. Although Forman's gradient flow has been proved to be useful in practical computations of homology groups, it is not sufficient to recover the homotopy type of $X$. Forman also proved the existence of a CW complex which is homotopy equivalent to $X$ and whose cells are in one-to-one correspondence with the critical cells of $μ$, but the construction is ad hoc and does not have a combin…
Let G be a profinite group with finite virtual cohomological dimension and let X be a discrete G-spectrum. If H and K are closed subgroups of G, with H normal in K, then, in general, the K/H-spectrum X^{hH} is not known to be a continuous K/H-spectrum, so that it is not known (in general) how to define the iterated homotopy fixed point spectrum (X^{hH})^{hK/H}. To address this situation, we define homotopy fixed points for delta-discrete G-spectra and show that the setting of delta-discrete G-spectra gives a good framework within which to work. In particular, we show that by using delta-discre…
In classical homotopy theory, two spaces are homotopy equivalent if one space can be continuously deformed into the other. This theory, however, does not respect the discrete nature of graphs. For this reason, a discrete homotopy theory that recognizes the difference between the vertices and edges of a graph was invented, called A-homotopy theory [1-5]. In classical homotopy theory, covering spaces and lifting properties are often used to compute the fundamental group of the circle. In this paper, we develop the lifting properties for A-homotopy theory. Using a covering graph and these lifting…
We use discrete Morse theory to provide another proof of Bernini, Ferrari, and Steingrimson's formula for the Mobius function of the consecutive pattern poset. In addition, we are able to determine the homotopy type of this poset. Earlier, Bjorner determined the Mobius function and homotopy type of factor order and the results are remarkably similar to those in the pattern case. In his thesis, Willenbring used discrete Morse theory to give an illuminating proof of Bjorner's result. Since our proof parallels Willenbring's, we also consider the relationship between the two posets.
Let $G$ be a discrete group. We prove that the category of $G$-posets admits a model structure that is Quillen equivalent to the standard model structure on $G$-spaces. As is already true nonequivariantly, the three classes of maps defining the model structure are not well understood calculationally. To illustrate, we exhibit some examples of cofibrant and fibrant posets and an example of a non-cofibrant finite poset.
We state the generating hypothesis in the homotopy category of G-spectra for a compact Lie group G, and prove that if G is finite, then the generating hypothesis implies the strong generating hypothesis, just as in the non-equivariant case. We also give an explicit counterexample to the generating hypothesis in the category of rational S^1-equivariant spectra.
Motivated by constructions in topological data analysis and algebraic combinatorics, we study homotopy theory on the category of Cech closure spaces $\mathbf{Cl}$, the category whose objects are sets endowed with a Cech closure operator and whose morphisms are the continuous maps between them. We introduce new classes of Cech closure structures on metric spaces, graphs, and simplicial complexes, and we show how each of these cases gives rise to an interesting homotopy theory. In particular, we show that there exists a natural family of Cech closure structures on metric spaces which produces a …
When G is a profinite group and H and K are closed subgroups, with H normal in K, it is not known, in general, how to form the iterated homotopy fixed point spectrum (Z^{hH})^{hK/H}, where Z is a continuous G-spectrum and all group actions are to be continuous. However, we show that, if G=G_n, the extended Morava stabilizer group, and Z=L_{K(n)}(E_n \wedge X), where L_{K(n)} is Bousfield localization with respect to Morava K-theory, E_n is the Lubin-Tate spectrum, and X is any spectrum with trivial G_n-action, then the iterated homotopy fixed point spectrum can always be constructed. Also, we …
Incidence relations among the cells of a regular CW complex produce a poset-enriched category of entrance paths whose classifying space is homotopy-equivalent to that complex. We show here that each acyclic partial matching (in the sense of discrete Morse theory) on the cells corresponds precisely to a homotopy-preserving localization of the associated entrance path category. Restricting attention further to the full localized subcategory spanned by critical cells, we obtain the discrete flow category whose classifying space is also shown to lie in the homotopy class of the original CW complex…
For a profinite group $G$, let $(\text{-})^{hG}$, $(\text{-})^{h_dG}$, and $(\text{-})^{h'G}$ denote continuous homotopy fixed points for profinite $G$-spectra, discrete $G$-spectra, and continuous $G$-spectra (coming from towers of discrete $G$-spectra), respectively. We establish some connections between the first two notions, and by using Postnikov towers, for $K \vartriangleleft_c G$ (a closed normal subgroup), give various conditions for when the iterated homotopy fixed points $(X^{hK})^{hG/K}$ exist and are $X^{hG}$. For the Lubin-Tate spectrum $E_n$ and $G <_c G_n$, the extended Morava …
We define a partial ordering on the set Q = Q(M) of pairs of topes of an oriented matroid M, and show the geometric realization |Q| of the order complex of Q has the same homotopy type as the Salvetti complex of M. For any element e of the ground set, the complex |Qe| associated to the rank-one oriented matroid on {e} has the homotopy type of the circle. There is a natural free simplicial action of Z4 on |Q|, with orbit space isomorphic to the order complex of the poset Q(M,e) associated to the pointed (or affine) oriented matroid (M,e). If M is the oriented matroid of an arrangement A of line…
The Conley index for flows is a topological invariant describing the behavior around an isolated invariant set $S$. It is defined as the homotopy type of a quotient space $N/L$, where $(N,L)$ is an index pair for $S$. In the case of a discrete dynamical system, i.e., a continuous self-map $f\colon X\to X$, the definition is similar. But one needs to consider the index map $f_{(N,L)}\colon N/L\to N/L$ induced by $f$. The Conley index in this situation is defined as the homotopy class $[f_{(N,L)}]$ modulo shift equivalence. The shift equivalence relation is rarely used outside this context and n…
Nilpotency for discrete groups can be defined in terms of central extensions. In this paper, the analogous definition for spaces is stated in terms of principal fibrations having infinite loop spaces as fibers, yielding a new invariant we compare with classical cocategory, but also with the more recent notion of homotopy nilpotency introduced by Biedermann and Dwyer. This allows us to characterize finite homotopy nilpotent loop spaces in the spirit of Hubbuck's Torus Theorem, and corresponding results for $p$-compact groups and $p$-Noetherian groups.
A recent theorem of Dobrinskaya states that the K(π,1)-conjecture holds for an Artin group G if and only if the canonical map from BM to BG is a homotopy equivalence, where M denotes the Artin monoid associated to G. The aim of this paper is to give an alternative proof by means of discrete Morse theory and abstract homotopy theory. Moreover, we exhibit a new model for the classifying space of an Artin monoid, in the spirit of Charney, Meier and Whittlesey, and a small chain complex for computing its monoid homology, similar to the one of Squier.
The aim of this paper is to give a (discrete) Morse theoretic proof of the fact that the $k$-th skeleton of the flag complex $\mathcal{F}$, associated to the lattice of subspaces of a finite dimensional vector space, is homotopy equivalent to a wedge of spheres of dimension $\min\{k,\dim(\mathcal{F})\}$. The tight control provided by Morse theoretic methods allows us to give an explicit formula for the number of spheres appearing in each of these wedge summands.
A new approach to étale homotopy theory is presented which applies to a much broader class of objects than previously existing approaches, namely it applies not only to all schemes (without any local Noetherian hypothesis), but also to arbitrary higher stacks on the étale site of such schemes, and in particular to all algebraic stacks. This approach also produces a more refined invariant, namely a pro-object in the infinity category of spaces, rather than in the homotopy category. We prove a profinite comparison theorem at this level of generality, which states that if $\mathcal{X}$ is an arbi…
We propose an axiomatic characterization of coarse homology theories defined on the category of bornological coarse spaces. We construct a category of motivic coarse spectra. Our focus is the classification of coarse homology theories and the construction of examples. We show that if a transformation between coarse homology theories induces an equivalence on all discrete bornological coarse spaces, then it is an equivalence on bornological coarse spaces of finite asymptotic dimension. The example of coarse K-homology will be discussed in detail.
A flag complex can be defined as a simplicial complex whose simplices correspond to complete subgraphs of its 1-skeleton taken as a graph. In this article, by introducing the notion of s-dismantlability, we shall define the s-homotopy type of a graph and show in particular that two finite graphs have the same s-homotopy type if, and only if, the two flag complexes determined by these graphs have the same simplicial simple-homotopy type (Theorem 2.10, part 1). This result is closely related to similar results established by Barmak and Minian (Adv. in Math., 218 (2008), 87-104) in the framework …
Let G be a discrete group for which the classifying space for proper G-actions is finite-dimensional. We find a space W such that for any such G, the classifying space PBG for proper G-bundles has the homotopy type of the W-nullification of BG. We use this to deduce some results concerning PBG and in some cases where there is a good model for PBG we obtain information about the BZ/p-nullification of BG.
We prove that the homotopy theory of $N_\infty$ operads is equivalent to a homotopy theory of discrete operads, and we construct free and associative operadic realizations of every indexing system. This resolves a conjecture of Blumberg and Hill in the affirmative.
Let $D_{n,γ}$ be the complex of graphs on $n$ vertices and domination number at least $γ$. We prove that $D_{n,n-2}$ has the homotopy type of a finite wedge of 2-spheres. This is done by using discrete Morse theory techniques. Acyclicity of the needed matching is proved by introducing a relativized form of a well known method for constructing acyclic matchings on suitable chunks of simplices. Our approach allows us to extend our results to the realm of infinite graphs. In addition, we give evidence supporting the assertion that the homotopy equivalences $D_{n,n-1}\simeq\bigvee S^0$ and $D_{n,n…
We consider the homotopy type of maps between symplectic surface whose graphs form symplectic submanifolds of the product. We give a purely topological model for this space in terms of maps with constrained numbers of pre-images. We use this to show that the dependence of the homotopy type on the area forms of each surface is quantized- it changes only when the parameters pass certain discrete levels. When the domain is a sphere or torus, and its total area is smaller than the range, we compute the full homotopy type of the low degree components. We also give an example, showing that the homot…
