arXiv subjects
Publications and source records attributed to Lukáš Vokřínek.
The paper presents an enriched categorical account of homological perturbation theory, including the formulation, proof and functoriality properties of the homological perturbation lemma.
We provide a generalization of the construction of a spectrum of a commutative ring as a locally ringed space, applicable to cone injectivity classes in general contexts, especially in locally finitely presentable categories. In its full generality, the spectrum functor fails to be fully faithful and we study reasonable sufficient conditions under which it is. Further, assuming the full faithfulness, we introduce a generalization of another concept from algebraic geometry -- the functor of points -- and prove equivalence of the two resulting notions of schemes.
We prove an adjoint functor theorem in the setting of categories enriched in a monoidal model category $\mathcal V$ admitting certain limits. When $\mathcal V$ is equipped with the trivial model structure this recaptures the enriched version of Freyd's adjoint functor theorem. For non-trivial model structures, we obtain new adjoint functor theorems of a homotopical flavour - in particular, when $\mathcal V$ is the category of simplical sets we obtain a homotopical adjoint functor theorem appropriate to the $\infty$-cosmoi of Riehl and Verity. We also investigate accessibility in the enriched setting, in particular obtaining homotopical cocompleteness results for accessible $\infty$-cosmoi.
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 obtained as a special case of a more general result: For finite diagrams of simplicial sets $X$, $A$, $Y$, i.e. functors $\mathcal{I}^\mathrm{op} \to \mathsf{sSet}$, in the stable range $\operatorname{dim} X \leq 2 \operatorname{conn} Y$ and $\operatorname{conn} Y > 1$, we give an algorithm that computes the set $[X, Y]^A$ of homotopy classes of maps of diagrams $\ell \colon X \to Y$ extending a given $f \colon A \to Y$. Again, for fixed $n = \dim X$, the running time of the algorithm is polynomial. The algorithm can be utilized to compute homotopy invariants in the equivariant setting -- for example, one can algorithmically compute equivariant stable homotopy groups. Further, one can apply the result to solve problems from computational topology, which we showcase on the following Tverberg-type problem: Given a $k$-dimensional simplicial complex $K$, is there a map $K \to \mathbb{R}^{d}$ without $r$-tuple intersection points? In the metastable range of dimensions, $rd \geq (r+1)k +3$, the result of Mabillard and Wagner shows this problem equivalent to the existence of a particular equivariant map. In this range, our algorithm is applicable and, thus, the $r$-Tverberg problem is algorithmically decidable (in polynomial time when $k$, $d$ and $r$ are fixed).
Let $X$ and $Y$ be finite simplicial sets (e.g. finite simplicial complexes), both equipped with a free simplicial action of a finite group $G$. Assuming that $Y$ is $d$-connected and $\dim X\le 2d$, for some $d\geq 1$, we provide an algorithm that computes the set of all equivariant homotopy classes of equivariant continuous maps $|X|\to|Y|$; the existence of such a map can be decided even for $\dim X\leq 2d+1$. For fixed $G$ and $d$, the algorithm runs in polynomial time. This yields the first algorithm for deciding topological embeddability of a $k$-dimensional finite simplicial complex into $\mathbb{R}^n$ under the conditions $k\leq\frac 23 n-1$. More generally, we present an algorithm that, given a lifting-extension problem satisfying an appropriate stability assumption, computes the set of all homotopy classes of solutions. This result is new even in the non-equivariant situation.
Cofibration categories are a formalization of homotopy theory useful for dealing with homotopy colimits that exist on the level of models as colimits of cofibrant diagrams. In this paper, we deal with their enriched version. Our main result claims that the category $[\mathcal{C},\mathcal{M}]$ of enriched diagrams equipped with the projective structure inherits a structure of a cofibration category whenever $\mathcal{C}$ is locally cofibrant (or, more generally, locally flat).
Given topological spaces X and Y, a fundamental problem of algebraic topology is understanding the structure of all continuous maps X -> Y . We consider a computational version, where X, Y are given as finite simplicial complexes, and the goal is to compute [X,Y], i.e., all homotopy classes of such maps. We solve this problem in the stable range, where for some d >= 2, we have dim X <= 2d - 2 and Y is (d - 1)-connected; in particular, Y can be the d-dimensional sphere S^d. The algorithm combines classical tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and simplicial sets) with algorithmic tools from effective algebraic topology (locally effective simplicial sets and objects with effective homology). In contrast, [X,Y] is known to be uncomputable for general X,Y, since for X = S^1 it includes a well known undecidable problem: testing triviality of the fundamental group of Y. In follow-up papers, the algorithm is shown to run in polynomial time for d fixed, and extended to other problems, such as the extension problem, where we are given a subspace A of X and a map A -> Y and ask whether it extends to a map X -> Y, or computing the Z_2-index---everything in the stable range. Outside the stable range, the extension problem is undecidable.
In a paper by Ford, it is claimed that to any pushout square of categories with all involved functors injective, there is associated an exact "Mayer--Vietoris" sequence of derived (co)limits. We provide a counter-example to this general statement. Further, we construct the Mayer--Vietoris sequence under some restrictions that cover the well-known case of a pushout square of group monomorphisms.
In a recent paper, it was shown that the problem of existence of a continuous map $X \to Y$ extending a given map $A \to Y$ defined on a subspace $A \subseteq X$ is undecidable, even for $Y$ an even-dimensional sphere. In the present paper, we prove that the same problem for $Y$ an odd-dimensional sphere is decidable. More generally, the same holds for any $d$-connected target space $Y$ whose homotopy groups $π_k Y$ are finite for $k>2d$.
This paper presents two algorithms. In their simplest form, the first algorithm decides the existence of a pointed homotopy between given simplicial maps f, g from X to Y and the second computes the group $[ΣX,Y]^*$ of pointed homotopy classes of maps from a suspension; in both cases, the target Y is assumed simply connected and the algorithms run in polynomial time when the dimension of X is fixed. More generally, these algorithms work relative to a subspace A of X, fibrewise over a simply connected B and also equivariantly when all spaces are equipped with a free action of a fixed finite group G.
An algorithmic computation of the set of unpointed stable homotopy classes of equivariant fibrewise maps was described in a recent paper of the author and his collaborators. In the present paper, we describe a simplification of this computation that uses an abelian heap structure on this set that was observed in another paper of the author. A heap is essentially a group without a choice of its neutral element; in addition, we allow it to be empty.
In this paper, we show how certain ``stability phenomena'' in unpointed model categories provide the sets of homotopy classes with the structure of abelian heaps, i.e. abelian groups without a choice of a zero. In contrast with the classical situation of stable (pointed) model categories, these sets can be empty.
Good colimits introduced by J. Lurie generalize transfinite composites and provide an important tool for understanding cofibrant generation in locally presentable categories. We will explore the relation of good colimits to transfinite composites further and show, in particular, how they eliminate the use of large objects in the usual small object argument.
We describe a general method for algorithmic construction of G-equivariant chain homotopy equivalences from non-equivariant ones. As a consequence, we obtain an algorithm for computing equivariant (co)homology of Eilenberg-MacLane spaces K(pi,n), where pi is a finitely generated ZG-module. This result will be used in a forthcoming paper to construct equivariant Postnikov towers of simply connected spaces with free actions of a finite group G and further to compute stable equivariant homotopy classes of maps between such spaces. The methods of this paper work for modules over any non-negatively graded differential graded algebra, whose underlying graded abelian group is free with 1 as one of the generators.
Let V be a cofibrantly generated closed symmetric monoidal model category and M a model V-category. We say that a weighted colimit W*D of a diagram D weighted by W is a homotopy weighted colimit if the diagram D is pointwise cofibrant and the weight W is cofibrant in the projective model structure on [C^op,V]. We then proceed to describe such homotopy weighted colimits through homotopy tensors and ordinary (conical) homotopy colimits. This is a homotopy version of the well known isomorphism W*D=\int^C(W\tensor D). After proving this homotopy decomposition in general we study in some detail a few special cases. For simplicial sets tensors may be replaced up to weak equivalence by conical homotopy colimits and thus the weighted homotopy colimits have no added value. The situation is completely different for model dg-categories where the desuspension cannot be constructed from conical homotopy colimits. In the last section we characterize those V-functors inducing a Quillen equivalence on the enriched presheaf categories.
The paper is concerned with defining a topology on the set of ideals of codimension d of the algebra C^\infty(M,R) with M being a compact smooth manifold. Its main property is that it is compact Hausdorff and it contains as a subspace the configuration space of d distinct unordered points in M and therefore provides a "compactification" of this configuration space. It naturally forms a space over the symmetric product SP_d(M) and as such has a very rich structure. Moreover it is covered by a (naturally defined) set of charts in which it is semialgebraic.
We prove a generalization of Thom's transversality theorem. It gives conditions under which the jet map $f_*|_Y:Y\subseteq J^r(D,M)\ra J^r(D,N)$ is generically (for $f:M\ra N$) transverse to a submanifold $Z\subseteq J^r(D,N)$. We apply this to study transversality properties of a restriction of a fixed map $g:M\ra P$ to the preimage $(j^sf)^{-1}(A)$ of a submanifold $A\subseteq J^s(M,N)$ in terms of transversality properties of the original map $f$. Our main result is that for a reasonable class of submanifolds $A$ and a generic map $f$ the restriction $g|_{(j^sf)^{-1}(A)}$ is also generic. We also present an example of $A$ where the theorem fails.
We gather conditions on a class H of continuous maps of topological spaces that allow a reasonable theory of fibrations up to an equivalence (a map from this class) which we call H-fibrations. The weak homotopy equivalences recover quasifibrations and homology equivalences yield homology fibrations. We study local H-fibrations that behave nicely with respect to homotopy colimits together with universal H-fibrations that behave nicely with respect to pullbacks. We then proceed to classify H-fibrations up to a natural notion of equivalence.