E. Macías-Virgós, D. Mosquera-Lois
We introduce a notion of categorical homotopic distance between functors by adapting the notion of homotopic distance in topological spaces, recently defined by the authors to the context of small categories. Moreover, this notion generalizes the work on categorical LS-category of small categories by Tanaka.
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The concept of homotopic distance and its higher analog are introduced in [6]. In this paper we introduce some important properties of higher homotopic distance, investigate the conditions under which $\cat$, $\secat$ and higher dimensional topological complexity are equal to the higher homotopic distance, and give alternative proofs, using higher homotopic distance, to some $\TC_n$-related theorems.
We first study the higher version of the relative topological complexity by using the homotopic distance. We also introduced the generalized version of the relative topological complexity of a topological pair on both the Schwarz genus and the homotopic distance. With these concepts, we give some inequalities including the topological complexity and the Lusternik-Schnirelmann category, the most important parts of the study of robot motion planning in topology. Finally, by defining the parametrised topological complexity via the homotopic distance, we present some estimates on the higher settin…
We show that both Lusternik-Schnirelmann category and topological complexity are particular cases of a more general notion, that we call homotopic distance between two maps. As a consequence, several properties of those invariants can be proved in a unified way and new results arise.
We prove that the homotopic distance between two maps defined on a manifold is bounded above by the sum of their subspace distances on the critical submanifol of any Morse-Bott function. This generalizes the Lusternik-Schnirelmann theorem (for Morse functions), and a similar result by Farber for the topological complexity. Analogously, we prove that, for analytic manifolds, the homotopic distance is bounded by the sum of the subspace distances on any submanifold and its cut locus. As an application, we show how navigation functions can be used to solve a generalized motion planning problem.
Homotopic distance $\D$ as introduced in \cite{MVML} can be realized as a pseudometric on $\mathrm{Map}(X,Y)$. In this paper, we study the topology induced by the pseudometric $\D$. In particular, we consider the space $\mathrm{Map}(S^1,S^1)$ and show that homotopic distance between any two maps in this space is 1. Moreover, while a general proof of the non-compactness of the space $\mathrm{Map}(X,Y)$ is still an open problem, it can be shown that $\mathrm{Map}(S^1,S^1)$ is not compact.
Homotopic distances and group-like spaces
研究映射间不同版本的同伦距离，在类群空间和 CW H - 空间上展开，发展概率和对角化版本理论并比较性质，证明在相关空间中同伦距离可由Lusternik - Schnirelmann 范畴精确计算。
$m-$homotopic Distances in Digital Images
本文定义了数字图像中的$m-$同伦距离及其高阶版本，探讨了其与$m-$范畴和$m-$复杂度的关联，并考察了相关概念的同伦不变性。
Discrete homotopic distance between Lipschitz maps
本文提出两个Lipschitz映射间的离散同伦距离$D_r$，通过指定步长$r$有效忽略较小孔洞以实现分类，并证明其推广了离散LS范畴与拓扑复杂度且在离散同伦下不变。
Parametrized homotopic distance
本文提出参数化同伦距离，建立其与纤维截面范畴的关系及上下界，并推广拓扑复杂度与 LS 范畴相关定理。
Weak and Strong Fibrations of Functors
本文提出了一种针对小范畴的同调框架，扩展了代数拓扑的经典不变量到范畴设置。通过构造真正的路径范畴，定义了函子的强和弱纤维化，并引入了纤维化替代和扩展了如Svarc genus和sectional category等同调不变量到小范畴。最后，将该框架应用于小范畴中的运动规划，提供了Farber拓扑复杂度的范畴类比，并消除了现有方法中常见的有限性约束。
Which spaces look like an n-sphere through the eyes of the n-th Postnikov section functor and the n-connected cover functor? The answer is what we call the Postnikov genus of the n-sphere. We define in fact the notion of localization genus for any homotopical localization functor in the sense of Bousfield and Dror Farjoun. This includes exotic genus notions related for example to Neisendorfer localization, or the classical Mislin genus, which corresponds to rationalization.
Axiomatic Sectional category, Topological complexity, and Homotopic distance
本文通过三个公理构建了截面类别、拓扑复杂度和同调距离的公理化框架，并引入了统一这些概念的提升类别概念。
Equivariant homotopic distance
本文引入等变同伦距离D_G，统一了等变Lusternik-Schnirelmann范畴与拓扑复杂度，并建立其性质、界及在Hopf G-空间和等变纤维化中的应用。
New Bounds on Distributional Sectional Category and Applications to Distributional Homotopic Distance
本文针对分布截面范畴（dsecat）建立新界限，证明Jauhari猜想并得到不同系数下的下界，应用于分布同伦距离，推导相关等价表述与新不等式。
Homotopical Dold-Kan Correspondences
论文围绕 test category 推广 Dold–Kan 对应，定义 Whitehead 范畴并构造模型结构，证明同调函子在局部化后给出范畴等价。
On integral extensions between the abelianization functor and its symmetric powers
本文研究有限生成自由群范畴上阿贝尔化函子与其对称幂的积分扩张群，利用Arone构造的有界复形计算了部分扩张群，并与相关结果作了比较。
A double $(\infty,1)$-categorical nerve for double categories
本文构造双重范畴到双重 $(\infty,1)$-范畴的神经，证明其右 Quillen 且同伦全忠实，并沿嵌入限制得到 Lack 模型结构的右诱导关系。
We study linear Batalin-Vilkovisky (BV) quantization, which is a derived and shifted version of the Weyl quantization of symplectic vector spaces. Using a variety of homotopical machinery, we implement this construction as a symmetric monoidal functor of $\infty$-categories. We also show that this construction has a number of pleasant properties: It has a natural extension to derived algebraic geometry, it can be fed into the higher Morita category of $E_n$-algebras to produce a "higher BV quantization" functor, and when restricted to formal moduli problems, it behaves like a determinant. Alon…
This paper is a fundamental study of comodules and contramodules over a comonoid in a symmetric closed monoidal category. We study both algebraic and homotopical aspects of them. Algebraically, we enrich the comodule and contramodule categories over the original category, construct enriched functors between them and enriched adjunctions between the functors. Homotopically, for simplicial sets and topological spaces, we investigate the categories of comodules and contramodules and the relations between them.
In this paper we will introduce and give topological properties of a new concept named simplicial distance which is the simplicial analog of the homotopic distance (in the sense of Marcias-Virgos and Mosquera-Lois in their paper [6]). According to our definition of simplicial distance, simplicial complexity is a particular case of this new concept.
A model structure for weakly horizontally invariant double categories
本文在双范畴范畴上构造模型结构，其纤维对象为弱水平不变双范畴，并证明该结构沿水平嵌入的变体右诱导出2-范畴的Lack模型结构，且关于Gray张量积是幺半的，同时给出Whitehead定理刻画弱等价。
In the homotopical study of spaces of smooth embeddings, the functor calculus method (Goodwillie-Klein-Weiss manifold calculus) has opened up important connections to operad theory. Using this and a few simplifying observations, we arrive at an operadic description of the obstructions to deforming smooth immersions into smooth embeddings. We give an application which in some respects improves on recent results of Arone-Turchin and Dwyer-Hess concerning high-dimensional variants of spaces of long knots.
Tanaka introduced a notion of Lusternik Schnirelmann category, denoted $\mathrm{ccat}\, \mathcal{C}$, of a small category $\mathcal{C}$. Among other properties, he proved an analog of Varadarajan's theorem for fibrations, relating the LS-categories of the total space, the base and the fiber. In this paper we recall the notion of homotopic distance $\mathrm{D}(F,G)$ between two functors $F,G\colon \mathcal{C} \to \mathcal{D}$, later introduced by us, which has $\mathrm{ccat} \mathcal{C}=\mathrm{D}(\mathrm{id}_{\mathcal{C}},\bullet)$ as a particular case. We consider another particular case, the…
We construct a model structure on the category $\mathrm{DblCat}$ of double categories and double functors. Unlike previous model structures for double categories, it recovers the homotopy theory of 2-categories through the horizontal embedding $\mathbb{H}\colon2\mathrm{Cat}\to\mathrm{DblCat}$, which is both left and right Quillen, and homotopically fully faithful. Furthermore, we show that Lack's model structure on $2\mathrm{Cat}$ is both left- and right-induced along $\mathbb{H}$ from our model structure on $\mathrm{DblCat}$. In addition, we obtain a $2\mathrm{Cat}$-enrichment of our model st…
In this paper we study the question of how to transfer homotopic structure from the category sD of simplicial objects in a fixed category D to D. To this end we use a sort of homotopy colimit s : sD --> D, which we call simple functor. For instance, the Bousfield-Kan homotopy colimit in a Quillen simplicial model category is an example of simple functor. As a remarkable example outside the setting of Quillen models we include Deligne simple of mixed Hodge complexes. We prove here that the simple functor induces an equivalence on the corresponding localized categories. We also describe a natura…
A Computational (Co)homological Approach to Contiguity Distance
本文提出了一种基于同调的计算方法，用于计算连续映射之间的距离，并通过具体计算验证了该方法的有效性。
We study the *homotopy theory* of $\infty$-categories enriched in the $\infty$-category $sS$ of simplicial spaces. That is, we consider $sS$-enriched $\infty$-categories as presentations of ordinary $\infty$-categories by means of a "local" geometric realization functor $Cat_{sS} \to Cat_\infty$, and we prove that their homotopy theory presents the $\infty$-category of $\infty$-categories, i.e. that this functor induces an equivalence $Cat_{sS} [[ W_{DK}^{-1} ]] \xrightarrow{\sim} Cat_\infty$ from a localization of the $\infty$-category of $sS$-enriched $\infty$-categories. Following Dwyer--Ka…
We introduce the higher topological complexity (TC_{n}) of a fibration in two ways: the higher homotopic distance and the Schwarz genus. Then we have some results on this notion related to TC, TC_{n} or cat of a topological space or a fibration. We also show that TC_{n} of a fibration is a fiber homotopy equivalence.
Pushouts of Dwyer maps are $(\infty,1)$-categorical
本文证明在特定函子类下，1-范畴的推出在包含进 $(\infty,1)$-范畴时得以保持，并将 Thomason 的弱同伦等价改进为弱范畴等价，同时推广了该函子类。
We show that variants of the classical reflection functors from quiver representation theory exist in any abstract stable homotopy theory, making them available for example over arbitrary ground rings, for quasi-coherent modules on schemes, in the differential-graded context, in stable homotopy theory as well as in the equivariant, motivic, and parametrized variant thereof. As an application of these equivalences we obtain abstract tilting results for trees valid in all these situations, hence generalizing a result of Happel. The main tools introduced for the construction of these reflection f…
In this paper we provide an explicit connection between level-sets persistence and derived sheaf theory over the real line. In particular we construct a functor from 2-parameter persistence modules to sheaves over $\mathbb{R}$, as well as a functor in the other direction. We also observe that the 2-parameter persistence modules arising from the level sets of Morse functions carry extra structure that we call a Mayer-Vietoris system. We prove classification, barcode decomposition, and stability theorems for these Mayer-Vietoris systems, and we show that the aforementioned functors establish a p…
Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been attempted over time, some topological in flavor, based on the vector space-valued homology functor, other combinatorial, based on arbitrary set-valued functors. To unify the study of topological and combinatorial persistence in a common categorical framework, we give axioms for a generalized rank function on objects in a target category, so that functors to that category induce persistence functions. We port the interleaving and bottleneck distances to this novel framew…
By invoking the reflection functors introduced by Bernstein, Gelfand, and Ponomarev in 1973, in this paper we define a metric on the space of all zigzag modules of a given length, which we call the reflection distance. We show that the reflection distance between two given zigzag modules of the same length is an upper bound for the $\ell^1$-bottleneck distance between their respective persistence diagrams.
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