We compute a lower bound for the number of simplices that are needed to triangulate the Grassmann manifold $G_k(mathbb{R}^n)$. In particular, we show that the number of top-dimensional simplices grows exponentially with $n$. More precise estimates are given for $k=2,3,4$. Our method can be used to estimate the minimal size of triangulations for other spaces, like Lie groups, flag manifolds, Stiefel manifolds etc.
{"title":"How many simplices are needed to triangulate a Grassmannian?","authors":"Dejan Govc, W. Marzantowicz, Petar Pavešić","doi":"10.12775/tmna.2020.027","DOIUrl":"https://doi.org/10.12775/tmna.2020.027","url":null,"abstract":"We compute a lower bound for the number of simplices that are needed to triangulate the Grassmann manifold $G_k(mathbb{R}^n)$. In particular, we show that the number of top-dimensional simplices grows exponentially with $n$. More precise estimates are given for $k=2,3,4$. Our method can be used to estimate the minimal size of triangulations for other spaces, like Lie groups, flag manifolds, Stiefel manifolds etc.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88577916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-18DOI: 10.4310/hha.2021.v23.n1.a11
J. Williamson
We provide an enhancement of Shipley's algebraicization theorem which behaves better in the context of commutative algebras. This involves defining flat model structures as in Shipley and Pavlov-Scholbach, and showing that the functors still provide Quillen equivalences in this refined context. The use of flat model structures allows one to identify the algebraic counterparts of change of groups functors, as demonstrated in forthcoming work of the author.
{"title":"Flatness and Shipley’s algebraicization theorem","authors":"J. Williamson","doi":"10.4310/hha.2021.v23.n1.a11","DOIUrl":"https://doi.org/10.4310/hha.2021.v23.n1.a11","url":null,"abstract":"We provide an enhancement of Shipley's algebraicization theorem which behaves better in the context of commutative algebras. This involves defining flat model structures as in Shipley and Pavlov-Scholbach, and showing that the functors still provide Quillen equivalences in this refined context. The use of flat model structures allows one to identify the algebraic counterparts of change of groups functors, as demonstrated in forthcoming work of the author.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89548402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $f_1,...,f_k:Mto N$ be maps between closed manifolds, $N(f_1,...,f_k)$ and $R(f_1,...,f_k)$ be the Nielsen and the Reideimeister coincidence numbers respectively. In this note, we relate $R(f_1,...,f_k)$ with $R(f_1,f_2),...,R(f_1,f_k)$. When $N$ is a torus or a nilmanifold, we compute $R(f_1,...,f_k)$ which, in these cases, is equal to $N(f_1,...,f_k)$.
让$ f,…，f_k:M到N$是闭流形之间的映射，$N(f_1，…，f_k)$和$R(f_1，…，f_k)$分别是Nielsen和Reideimeister符合数。在这个报告中,我们与$ R (f,…,f_k) $ $ R (f, f₂)…,R (f, f_k) $。当$N$是环面或零流形时，我们计算$R(f_1，…，f_k)$，在这种情况下，它等于$N(f_1，…，f_k)$。
{"title":"Computation of Nielsen and Reidemeister coincidence numbers for multiple maps","authors":"Tha'is F. M. Monis, P. Wong","doi":"10.12775/tmna.2020.002","DOIUrl":"https://doi.org/10.12775/tmna.2020.002","url":null,"abstract":"Let $f_1,...,f_k:Mto N$ be maps between closed manifolds, $N(f_1,...,f_k)$ and $R(f_1,...,f_k)$ be the Nielsen and the Reideimeister coincidence numbers respectively. In this note, we relate $R(f_1,...,f_k)$ with $R(f_1,f_2),...,R(f_1,f_k)$. When $N$ is a torus or a nilmanifold, we compute $R(f_1,...,f_k)$ which, in these cases, is equal to $N(f_1,...,f_k)$.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"359 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76412015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Coherent strings of composable morphisms play an important role in various important constructions in abstract stable homotopy theory (for example algebraic K-theory or higher Toda brackets) and in the representation theory of finite dimensional algebras (as representations of Dynkin quivers of type A). In a first step we will prove a strong comparison result relating composable strings of morphisms and coherent diagrams on cubes with support on a path from the initial to the final object. We observe that both structures are equivalent (by passing to higher analogues of mesh categories) to distinguished coherent diagrams on special classes of morphism objects in the 2-category of parasimplices. Furthermore, we show that the notion of distinguished coherent diagrams generalizes well to arbitrary morphism objects in this 2-category. The resulting categories of coherent diagrams lead to higher versions of the $mathsf{S}_{bullet}$-construction and are closely related to representations of higher Auslander algebras of Dynkin quivers of type A. Understanding these categories and the functors relating them in general will require a detailed analysis of the 2-category of parasimplices as well as basic results from abstract cubical homotopy theory (since subcubes of distinguished diagrams very often turn out to be bicartesian). Finally, we show that the previous comparison result extends to a duality theorem on general categories of distinguished coherent diagrams, as a special case leading to some new derived equivalences between higher Auslander algebras.
{"title":"The bivariant parasimplicial $mathsf{S}_{bullet}$-construction","authors":"F. Beckert","doi":"10.25926/kj82-kr40","DOIUrl":"https://doi.org/10.25926/kj82-kr40","url":null,"abstract":"Coherent strings of composable morphisms play an important role in various important constructions in abstract stable homotopy theory (for example algebraic K-theory or higher Toda brackets) and in the representation theory of finite dimensional algebras (as representations of Dynkin quivers of type A). In a first step we will prove a strong comparison result relating composable strings of morphisms and coherent diagrams on cubes with support on a path from the initial to the final object. \u0000We observe that both structures are equivalent (by passing to higher analogues of mesh categories) to distinguished coherent diagrams on special classes of morphism objects in the 2-category of parasimplices. Furthermore, we show that the notion of distinguished coherent diagrams generalizes well to arbitrary morphism objects in this 2-category. The resulting categories of coherent diagrams lead to higher versions of the $mathsf{S}_{bullet}$-construction and are closely related to representations of higher Auslander algebras of Dynkin quivers of type A. \u0000Understanding these categories and the functors relating them in general will require a detailed analysis of the 2-category of parasimplices as well as basic results from abstract cubical homotopy theory (since subcubes of distinguished diagrams very often turn out to be bicartesian). Finally, we show that the previous comparison result extends to a duality theorem on general categories of distinguished coherent diagrams, as a special case leading to some new derived equivalences between higher Auslander algebras.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75319230","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For $G$ a finite group, a normalized 2-cocycle $alphain Z^{2}(G,mathbb{S}^{1})$ and $X$ a $G$-space on which a normal subgroup $A$ acts trivially, we show that the $alpha$-twisted $G$-equivariant $K$-theory of $X$ decomposes as a direct sum of twisted equivariant $K$-theories of $X$ parametrized by the orbits of an action of $G$ on the set of irreducible $alpha$-projective representations of $A$. This generalizes the decomposition obtained by Gomez and Uribe for equivariant $K$-theory. We also explore some examples of this decomposition for the particular case of the dihedral groups $D_{2n}$ with $nge 1$ an even integer.
{"title":"A Decomposition of Twisted Equivariant K-Theory","authors":"J. M. G'omez, J. Ram'irez","doi":"10.3842/SIGMA.2021.041","DOIUrl":"https://doi.org/10.3842/SIGMA.2021.041","url":null,"abstract":"For $G$ a finite group, a normalized 2-cocycle $alphain Z^{2}(G,mathbb{S}^{1})$ and $X$ a $G$-space on which a normal subgroup $A$ acts trivially, we show that the $alpha$-twisted $G$-equivariant $K$-theory of $X$ decomposes as a direct sum of twisted equivariant $K$-theories of $X$ parametrized by the orbits of an action of $G$ on the set of irreducible $alpha$-projective representations of $A$. This generalizes the decomposition obtained by Gomez and Uribe for equivariant $K$-theory. We also explore some examples of this decomposition for the particular case of the dihedral groups $D_{2n}$ with $nge 1$ an even integer.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91092804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is a chapter of the Handbook of Homotopy Theory, that surveys the classifications of thick tensor-ideals.
这是《同伦理论手册》的一章，研究了厚张量理想的分类。
{"title":"A guide to tensor-triangular classification","authors":"Paul Balmer","doi":"10.1201/9781351251624-4","DOIUrl":"https://doi.org/10.1201/9781351251624-4","url":null,"abstract":"This is a chapter of the Handbook of Homotopy Theory, that surveys the classifications of thick tensor-ideals.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"36 1","pages":"145-162"},"PeriodicalIF":0.0,"publicationDate":"2019-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84884840","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-09DOI: 10.25537/dm.2021v26.1423-1464
Sanath K. Devalapurkar, Peter J. Haine
This note provides modern proofs of some classical results in algebraic topology, such as the James Splitting, the Hilton-Milnor Splitting, and the metastable EHP sequence. We prove fundamental splitting results begin{equation*} Sigma Omega Sigma X simeq Sigma X vee (Xwedge SigmaOmega Sigma X) quad text{and} quad Omega(X vee Y) simeq Omega Xtimes Omega Ytimes Omega Sigma(Omega X wedge Omega Y) end{equation*} in the maximal generality of an $infty$-category with finite limits and pushouts in which pushouts squares remain pushouts after basechange along an arbitrary morphism (i.e., Mather's Second Cube Lemma holds). For connected objects, these imply the classical James and Hilton-Milnor Splittings. Moreover, working in this generality shows that the James and Hilton-Milnor splittings hold in many new contexts, for example in: elementary $infty$-topoi, profinite spaces, and motivic spaces over arbitrary base schemes. The splitting result in this last context extend Wickelgren and Williams' splitting result for motivic spaces over a perfect field. We also give two proofs of the metastable EHP sequence in the setting of $infty$-topoi: the first is a new, non-computational proof that only utilizes basic connectedness estimates involving the James filtration and the Blakers-Massey Theorem, while the second reduces to the classical computational proof.
本文给出了代数拓扑中一些经典结果的现代证明，如James分裂、Hilton-Milnor分裂和亚稳态EHP序列。我们证明了一个具有有限极限的$infty$ -范畴的极大一般性中的基本分裂结果begin{equation*} Sigma Omega Sigma X simeq Sigma X vee (Xwedge SigmaOmega Sigma X) quad text{and} quad Omega(X vee Y) simeq Omega Xtimes Omega Ytimes Omega Sigma(Omega X wedge Omega Y) end{equation*}，其中推入的平方在基沿任意态射变换后仍然是推入的(即Mather的第二立方引理成立)。对于连接对象，这意味着经典的詹姆斯分裂和希尔顿-米尔诺分裂。此外，在这种一般性下的工作表明，James和Hilton-Milnor分裂在许多新的情况下都成立，例如:初等$infty$ -拓扑、无限空间和任意基方案上的动机空间。最后一种情况下的分裂结果扩展了Wickelgren和Williams在理想场上的动力空间的分裂结果。我们还给出了$infty$ -拓扑下亚稳态EHP序列的两个证明:第一个证明是一种新的非计算证明，它只利用了James过滤和Blakers-Massey定理的基本连通性估计，而第二个证明则简化为经典的计算证明。
{"title":"On the James and Hilton-Milnor Splittings, & the metastable EHP sequence.","authors":"Sanath K. Devalapurkar, Peter J. Haine","doi":"10.25537/dm.2021v26.1423-1464","DOIUrl":"https://doi.org/10.25537/dm.2021v26.1423-1464","url":null,"abstract":"This note provides modern proofs of some classical results in algebraic topology, such as the James Splitting, the Hilton-Milnor Splitting, and the metastable EHP sequence. We prove fundamental splitting results begin{equation*} Sigma Omega Sigma X simeq Sigma X vee (Xwedge SigmaOmega Sigma X) quad text{and} quad Omega(X vee Y) simeq Omega Xtimes Omega Ytimes Omega Sigma(Omega X wedge Omega Y) end{equation*} in the maximal generality of an $infty$-category with finite limits and pushouts in which pushouts squares remain pushouts after basechange along an arbitrary morphism (i.e., Mather's Second Cube Lemma holds). For connected objects, these imply the classical James and Hilton-Milnor Splittings. Moreover, working in this generality shows that the James and Hilton-Milnor splittings hold in many new contexts, for example in: elementary $infty$-topoi, profinite spaces, and motivic spaces over arbitrary base schemes. The splitting result in this last context extend Wickelgren and Williams' splitting result for motivic spaces over a perfect field. We also give two proofs of the metastable EHP sequence in the setting of $infty$-topoi: the first is a new, non-computational proof that only utilizes basic connectedness estimates involving the James filtration and the Blakers-Massey Theorem, while the second reduces to the classical computational proof.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"83 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75886126","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-09DOI: 10.1017/s0004972720000167
Cesar A. IPANAQUE ZAPATA, Jes'us Gonz'alez
In robotics, a topological theory of motion planning was initiated by M. Farber. The multitasking motion planning problem is new and its theoretical part via topological complexity has hardly been developed, but the concrete implementations are still non-existent, and in fact this work takes the first step in this last direction (producing explicit algorithms.) We present optimal motion planning algorithms which can be used in designing practical systems controlling objects moving in Euclidean space without collisions between them and avoiding obstacles. Furthermore, we present the multitasking version of the algorithms.
{"title":"SEQUENTIAL COLLISION-FREE OPTIMAL MOTION PLANNING ALGORITHMS IN PUNCTURED EUCLIDEAN SPACES","authors":"Cesar A. IPANAQUE ZAPATA, Jes'us Gonz'alez","doi":"10.1017/s0004972720000167","DOIUrl":"https://doi.org/10.1017/s0004972720000167","url":null,"abstract":"In robotics, a topological theory of motion planning was initiated by M. Farber. The multitasking motion planning problem is new and its theoretical part via topological complexity has hardly been developed, but the concrete implementations are still non-existent, and in fact this work takes the first step in this last direction (producing explicit algorithms.) We present optimal motion planning algorithms which can be used in designing practical systems controlling objects moving in Euclidean space without collisions between them and avoiding obstacles. Furthermore, we present the multitasking version of the algorithms.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89900627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work we exhibit an unexpected connection between sectional category theory and the fixed point property. On the one hand, a topological space $X$ is said to have textit{the fixed point property} (FPP) if, for every continuous self-map $f$ of $X$, there is a point $x$ of $X$ such that $f(x)=x$. On the other hand, for a continuous surjection $p:Eto B$, the textit{standard sectional number} $sec_{text{op}}(p)$ is the minimal cardinality of open covers ${U_i}$ of $B$ such that each $U_i$ admits a continuous local section for $p$. Let $F(X,k)$ denote the configuration space of $k$ ordered distinct points in $X$ and consider the natural projection $pi_{k,1}:F(X,k)to X$. We demonstrate that a space $X$ has the FPP if and only if $sec_{text{op}}(pi_{2,1})=2$. This characterization connects a standard problem in fixed point theory to current research trends in topological robotics.
{"title":"Sectional category and the fixed point property","authors":"C. A. I. Zapata, Jes'us Gonz'alez","doi":"10.12775/tmna.2020.033","DOIUrl":"https://doi.org/10.12775/tmna.2020.033","url":null,"abstract":"In this work we exhibit an unexpected connection between sectional category theory and the fixed point property. On the one hand, a topological space $X$ is said to have textit{the fixed point property} (FPP) if, for every continuous self-map $f$ of $X$, there is a point $x$ of $X$ such that $f(x)=x$. On the other hand, for a continuous surjection $p:Eto B$, the textit{standard sectional number} $sec_{text{op}}(p)$ is the minimal cardinality of open covers ${U_i}$ of $B$ such that each $U_i$ admits a continuous local section for $p$. Let $F(X,k)$ denote the configuration space of $k$ ordered distinct points in $X$ and consider the natural projection $pi_{k,1}:F(X,k)to X$. We demonstrate that a space $X$ has the FPP if and only if $sec_{text{op}}(pi_{2,1})=2$. This characterization connects a standard problem in fixed point theory to current research trends in topological robotics.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76007959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-01DOI: 10.32513/tbilisi/1593223222
Shoji Yokura
In 1920s R. L. Moore introduced emph{upper semicontinuous} and emph{lower semicontinuous} decompositions in studying decomposition spaces. Upper semicontinuous decompositions were studied very well by himself and later by R.H. Bing in 1950s. In this paper we consider lower semicontinuous decompositions $mathcal D$ of a topological space $X$ such that the decomposition spaces $X/mathcal D$ are Alexandroff spaces. If the associated proset (preordered set) of the decomposition space $X/mathcal D$ is a poset, then the decomposition map $pi:X to X/mathcal D$ is emph{a continuous map from the topological space $X$ to the poset $X/mathcal D$ with the associated Alexandroff topology}, which is nowadays called emph{a poset-stratified space}. As an application, we capture the face poset of a real hyperplane arrangement $mathcal A$ of $mathbb R^n$ as the associated poset of the decomposition space $mathbb R^n/mathcal D(mathcal A)$ of the decomposition $mathcal D(mathcal A)$ determined by the arrangement $mathcal A$. We also show that for any locally small category $mathcal C$ the set $hom_{mathcal C}(X,Y)$ of morphisms from $X$ to $Y$ can be considered as a poset-stratified space, and that for any objects $S, T$ (where $S$ plays as a source object and $T$ as a target object) there are a covariant functor $frak {st}^S_*: mathcal C to mathcal Strat$ and a contravariant functor $frak {st}^*_T$ $frak {st}^*_T: mathcal C to mathcal Strat$ from $mathcal C$ to the category $mathcal Strat$ of poset-stratified spaces. We also make a remark about Yoneda's Lemmas as to poset-stratified space structures of $hom_{mathcal C}(X,Y)$.
{"title":"Decomposition spaces and poset-stratified spaces","authors":"Shoji Yokura","doi":"10.32513/tbilisi/1593223222","DOIUrl":"https://doi.org/10.32513/tbilisi/1593223222","url":null,"abstract":"In 1920s R. L. Moore introduced emph{upper semicontinuous} and emph{lower semicontinuous} decompositions in studying decomposition spaces. Upper semicontinuous decompositions were studied very well by himself and later by R.H. Bing in 1950s. In this paper we consider lower semicontinuous decompositions $mathcal D$ of a topological space $X$ such that the decomposition spaces $X/mathcal D$ are Alexandroff spaces. If the associated proset (preordered set) of the decomposition space $X/mathcal D$ is a poset, then the decomposition map $pi:X to X/mathcal D$ is emph{a continuous map from the topological space $X$ to the poset $X/mathcal D$ with the associated Alexandroff topology}, which is nowadays called emph{a poset-stratified space}. As an application, we capture the face poset of a real hyperplane arrangement $mathcal A$ of $mathbb R^n$ as the associated poset of the decomposition space $mathbb R^n/mathcal D(mathcal A)$ of the decomposition $mathcal D(mathcal A)$ determined by the arrangement $mathcal A$. We also show that for any locally small category $mathcal C$ the set $hom_{mathcal C}(X,Y)$ of morphisms from $X$ to $Y$ can be considered as a poset-stratified space, and that for any objects $S, T$ (where $S$ plays as a source object and $T$ as a target object) there are a covariant functor $frak {st}^S_*: mathcal C to mathcal Strat$ and a contravariant functor $frak {st}^*_T$ $frak {st}^*_T: mathcal C to mathcal Strat$ from $mathcal C$ to the category $mathcal Strat$ of poset-stratified spaces. We also make a remark about Yoneda's Lemmas as to poset-stratified space structures of $hom_{mathcal C}(X,Y)$.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90790172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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