A conjecture of May states that there is an up-to-adjunction strictification of symmetric bimonoidal functors between bipermutative categories. The main result of this paper proves a weaker form of May's conjecture that starts with multiplicatively strong symmetric bimonoidal functors. As the main application, for May's multiplicative infinite loop space machine from bipermutative categories to either E-infinity ring spaces or E-infinity ring spectra, multiplicatively strong symmetric bimonoidal functors can be replaced by strict symmetric bimonoidal functors.
{"title":"May's Conjecture on Bimonoidal Functors and Multiplicative Infinite Loop Space Theory","authors":"Donald Yau","doi":"arxiv-2405.10834","DOIUrl":"https://doi.org/arxiv-2405.10834","url":null,"abstract":"A conjecture of May states that there is an up-to-adjunction strictification\u0000of symmetric bimonoidal functors between bipermutative categories. The main\u0000result of this paper proves a weaker form of May's conjecture that starts with\u0000multiplicatively strong symmetric bimonoidal functors. As the main application,\u0000for May's multiplicative infinite loop space machine from bipermutative\u0000categories to either E-infinity ring spaces or E-infinity ring spectra,\u0000multiplicatively strong symmetric bimonoidal functors can be replaced by strict\u0000symmetric bimonoidal functors.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148653","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 article we study the low dimensional homology of the projective linear group $textrm{PGL}_2(A)$ over a $textrm{GE}_2$-ring $A$. In particular, we prove a Bloch-Wigner type exact sequence over local domains. As applications we prove that $H_2(textrm{PGL}_2(A),mathbb{Z}left[frac{1}{2}right])simeq K_2(A)left[frac{1}{2}right]$ and $H_3(textrm{PGL}_2(A),mathbb{Z}left[frac{1}{2}right])simeq K_3^{textrm{ind}}(A)left[frac{1}{2}right]$.
本文研究了$textrm{GE}_2$环$A$上的投影线性群$textrm{PGL}_2(A)$的低维同源性。特别是，我们证明了在局部域上的布洛赫-维格纳型精确序列。Asapplications we prove that$H_2(textrm{PGL}_2(A),mathbb{Z}left[frac{1}{2}right])simeqK_2(A)left[frac{1}{2}right]$ and$H_3(textrm{PGL}_2(A),mathbb{Z}left[frac{1}{2}right])simeqK_3^{textrm{ind}}(A)left[frac{1}{2}right]$.
{"title":"The Low Dimensional Homology of Projective Linear Group of Rank Two","authors":"Behrooz Mirzaii, Elvis Torres Pérez","doi":"arxiv-2405.08950","DOIUrl":"https://doi.org/arxiv-2405.08950","url":null,"abstract":"In this article we study the low dimensional homology of the projective\u0000linear group $textrm{PGL}_2(A)$ over a $textrm{GE}_2$-ring $A$. In\u0000particular, we prove a Bloch-Wigner type exact sequence over local domains. As\u0000applications we prove that\u0000$H_2(textrm{PGL}_2(A),mathbb{Z}left[frac{1}{2}right])simeq\u0000K_2(A)left[frac{1}{2}right]$ and\u0000$H_3(textrm{PGL}_2(A),mathbb{Z}left[frac{1}{2}right])simeq\u0000K_3^{textrm{ind}}(A)left[frac{1}{2}right]$.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141063982","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}
We prove a new kind of homological stability theorem for automorphism groups of finitely-generated projective modules over Dedekind domains, which takes into account all possible stabilisation maps between these, rather than only stabilisation by the free module of rank 1. We show the same kind of stability holds for Clausen and Jansen's reductive Borel--Serre spaces.
{"title":"Homological stability for general linear groups over Dedekind domains","authors":"Oscar Randal-Williams","doi":"arxiv-2405.07566","DOIUrl":"https://doi.org/arxiv-2405.07566","url":null,"abstract":"We prove a new kind of homological stability theorem for automorphism groups\u0000of finitely-generated projective modules over Dedekind domains, which takes\u0000into account all possible stabilisation maps between these, rather than only\u0000stabilisation by the free module of rank 1. We show the same kind of stability\u0000holds for Clausen and Jansen's reductive Borel--Serre spaces.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"78 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925305","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}
We develop the formalism of derived divided power algebras, and revisit the theory of derived De Rham and crystalline cohomology in this framework. We characterize derived De Rham cohomology of a derived commutative ring $A$, together with the Hodge filtration on it, in terms of a universal property as the largest filtered divided power thickening of $A$. We show that our approach agrees with A.Raksit's. Along the way, we develop some fundamentals of square-zero extensions and derivations in derived algebraic geometry in connection with derived De Rham cohomology.
我们发展了派生分权代数的形式主义，并在此框架内重温了派生德拉姆与晶体同调的理论。我们用衍生交换环 $A$ 的最大滤波除幂增厚这一普遍性质来描述衍生交换环 $A$ 的衍生 De Rham 同调及其上的霍奇滤波。我们证明了我们的方法与 A.Raksit 的方法一致。在此过程中，我们发展了派生代数几何中与派生德拉姆同调相关的平方零扩展和派生的一些基本原理。
{"title":"Divided Powers and Derived De Rham Cohomology","authors":"Kirill Magidson","doi":"arxiv-2405.05153","DOIUrl":"https://doi.org/arxiv-2405.05153","url":null,"abstract":"We develop the formalism of derived divided power algebras, and revisit the\u0000theory of derived De Rham and crystalline cohomology in this framework. We\u0000characterize derived De Rham cohomology of a derived commutative ring $A$,\u0000together with the Hodge filtration on it, in terms of a universal property as\u0000the largest filtered divided power thickening of $A$. We show that our approach\u0000agrees with A.Raksit's. Along the way, we develop some fundamentals of\u0000square-zero extensions and derivations in derived algebraic geometry in\u0000connection with derived De Rham cohomology.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"16 4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925373","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}
We present the magnitude homology of a finite digraph $G$ as a certain subquotient of its path algebra. We use this to prove that the second magnitude homology group ${rm MH}_{2,ell}(G,mathbb{Z})$ is a free abelian group for any $ell$, and to describe its rank. This allows us to give a condition, denoted by $(mathcal{V}_2)$, equivalent to vanishing of ${rm MH}_{2,ell}(G,mathbb{Z})$ for $ell>2.$ Recall that a digraph is called diagonal, if its magnitude homology is concentrated in diagonal degrees. Using the condition $(mathcal V_2),$ we show that the GLMY-fundamental group of a diagonal (undirected) graph is trivial. In other words, the two-dimensional CW-complex obtained from a diagonal graph by attaching 2-cells to all squares and triangles of the graph is simply connected. We also give an interpretation of diagonality in terms of Koszul algebras: a digraph $G$ is diagonal if and only if the distance algebra $sigma G$ is Koszul for any ground field; and if and only if $G$ satisfies $(mathcal{V}_2)$ and the path cochain algebra $Omega^bullet(G)$ is Koszul for any ground field. Besides, we show that the path cochain algebra $Omega^bullet(G)$ is quadratic for any $G.$ To obtain a source of examples of (non-)diagonal digraphs, we study the extended Hasse diagram $hat G_K$ of a simplicial complex $K$. For a combinatorial triangulation $K$ of a piecewise-linear manifold $M,$ we express the non-diagonal part of the magnitude homology of $hat G_K$ via the homology of $M$. As a corollary we obtain that, if $K$ is a combinatorial triangulation of a closed piecewise-linear manifold $M$, then $hat G_K$ is diagonal if and only if $M$ is a homology sphere.
{"title":"On diagonal digraphs, Koszul algebras and triangulations of homology spheres","authors":"Sergei O. Ivanov, Lev Mukoseev","doi":"arxiv-2405.04748","DOIUrl":"https://doi.org/arxiv-2405.04748","url":null,"abstract":"We present the magnitude homology of a finite digraph $G$ as a certain\u0000subquotient of its path algebra. We use this to prove that the second magnitude\u0000homology group ${rm MH}_{2,ell}(G,mathbb{Z})$ is a free abelian group for\u0000any $ell$, and to describe its rank. This allows us to give a condition,\u0000denoted by $(mathcal{V}_2)$, equivalent to vanishing of ${rm\u0000MH}_{2,ell}(G,mathbb{Z})$ for $ell>2.$ Recall that a digraph is called\u0000diagonal, if its magnitude homology is concentrated in diagonal degrees. Using\u0000the condition $(mathcal V_2),$ we show that the GLMY-fundamental group of a\u0000diagonal (undirected) graph is trivial. In other words, the two-dimensional\u0000CW-complex obtained from a diagonal graph by attaching 2-cells to all squares\u0000and triangles of the graph is simply connected. We also give an interpretation\u0000of diagonality in terms of Koszul algebras: a digraph $G$ is diagonal if and\u0000only if the distance algebra $sigma G$ is Koszul for any ground field; and if\u0000and only if $G$ satisfies $(mathcal{V}_2)$ and the path cochain algebra\u0000$Omega^bullet(G)$ is Koszul for any ground field. Besides, we show that the\u0000path cochain algebra $Omega^bullet(G)$ is quadratic for any $G.$ To obtain a\u0000source of examples of (non-)diagonal digraphs, we study the extended Hasse\u0000diagram $hat G_K$ of a simplicial complex $K$. For a combinatorial\u0000triangulation $K$ of a piecewise-linear manifold $M,$ we express the\u0000non-diagonal part of the magnitude homology of $hat G_K$ via the homology of\u0000$M$. As a corollary we obtain that, if $K$ is a combinatorial triangulation of\u0000a closed piecewise-linear manifold $M$, then $hat G_K$ is diagonal if and only\u0000if $M$ is a homology sphere.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925306","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}
We give an explicit algebraic description, based on prismatic cohomology, of the algebraic K-groups of rings of the form $O_K/I$ where $K$ is a p-adic field and $I$ is a non-trivial ideal in the ring of integers $O_K$; this class includes the rings $mathbf{Z}/p^n$ where $p$ is a prime. The algebraic description allows us to describe a practical algorithm to compute individual K-groups as well as to obtain several theoretical results: the vanishing of the even K-groups in high degrees, the determination of the orders of the odd K-groups in high degrees, and the degree of nilpotence of $v_1$ acting on the mod $p$ syntomic cohomology of $mathbf{Z}/p^n$.
我们基于棱柱同调，对形式为 $O_K/I$ 的环的代数 K 群给出了明确的代数描述，其中 $K$ 是 p-adic 场，$I$ 是整数环 $O_K$ 中的非三重理想；这类环包括 $mathbf{Z}/p^n$ 环，其中 $p$ 是素数。通过代数描述，我们描述了计算单个 K 群的实用算法，并得到了几个理论结果：高度数中偶数 K 群的消失、高度数中奇数 K 群的阶的确定，以及作用于 $mathbf{Z}/p^n$ 的 mod $p$ 合成同调上的 $v_1$ 的无穷度。
{"title":"On the $K$-theory of $mathbf{Z}/p^n$","authors":"Benjamin Antieau, Achim Krause, Thomas Nikolaus","doi":"arxiv-2405.04329","DOIUrl":"https://doi.org/arxiv-2405.04329","url":null,"abstract":"We give an explicit algebraic description, based on prismatic cohomology, of\u0000the algebraic K-groups of rings of the form $O_K/I$ where $K$ is a p-adic field\u0000and $I$ is a non-trivial ideal in the ring of integers $O_K$; this class\u0000includes the rings $mathbf{Z}/p^n$ where $p$ is a prime. The algebraic description allows us to describe a practical algorithm to\u0000compute individual K-groups as well as to obtain several theoretical results:\u0000the vanishing of the even K-groups in high degrees, the determination of the\u0000orders of the odd K-groups in high degrees, and the degree of nilpotence of\u0000$v_1$ acting on the mod $p$ syntomic cohomology of $mathbf{Z}/p^n$.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"79 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925415","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 paper, we establish a theorem that proves a condition when an inclusion morphism between simplicial sets becomes a weak homotopy equivalence. Additionally, we present two applications of this result. The first application demonstrates that cofinal full inclusion functors of (infty)-categories are weak homotopy equivalences. For our second application, we provide an alternative proof of Barwick's cofinality theorem of algebraic (K)-theory.
{"title":"Cofinality Theorems of Infinity Categories and Algebraic K-Theory","authors":"Hisato Matsukawa","doi":"arxiv-2405.03498","DOIUrl":"https://doi.org/arxiv-2405.03498","url":null,"abstract":"In this paper, we establish a theorem that proves a condition when an\u0000inclusion morphism between simplicial sets becomes a weak homotopy equivalence.\u0000Additionally, we present two applications of this result. The first application\u0000demonstrates that cofinal full inclusion functors of (infty)-categories are\u0000weak homotopy equivalences. For our second application, we provide an\u0000alternative proof of Barwick's cofinality theorem of algebraic (K)-theory.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140882746","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}
We give a characterization of locally standard, $mathbb{Z}$-equivariantly formal manifolds in general position. In particular, we show that for dimension $2n$ at least $10$, to every such manifold with labeled GKM graph $Gamma$ there is an equivariantly formal torus manifold such that the restriction of the $T^n$-action to a certain $T^{n-1}$-action yields the same labeled graph $Gamma$, thus showing that the (equivariant) cohomology with $mathbb{Z}$-coefficients of those manifolds has the same description as that of equivariantly formal torus manifolds.
{"title":"Characterization of locally standard, $mathbb{Z}$-equivariantly formal manifolds in general position","authors":"Nikolas Wardenski","doi":"arxiv-2405.03319","DOIUrl":"https://doi.org/arxiv-2405.03319","url":null,"abstract":"We give a characterization of locally standard, $mathbb{Z}$-equivariantly\u0000formal manifolds in general position. In particular, we show that for dimension\u0000$2n$ at least $10$, to every such manifold with labeled GKM graph $Gamma$\u0000there is an equivariantly formal torus manifold such that the restriction of\u0000the $T^n$-action to a certain $T^{n-1}$-action yields the same labeled graph\u0000$Gamma$, thus showing that the (equivariant) cohomology with\u0000$mathbb{Z}$-coefficients of those manifolds has the same description as that\u0000of equivariantly formal torus manifolds.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140882588","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}
Sergei O. Ivanov, Roman Mikhailov, Fedor Pavutnitskiy
In this paper, we study operations on functors in the category of abelian groups simplar to the derivation in the sense of Dold-Puppe. They are defined as derived limits of a functor applied to the relation subgroup over a category of free presentations of the group. The integral homology of the Eilenberg-Maclane space $K(mathbb Z,3)$ appears as a part of description of these operations applied to symmetric powers.
{"title":"Limits via relations","authors":"Sergei O. Ivanov, Roman Mikhailov, Fedor Pavutnitskiy","doi":"arxiv-2405.03175","DOIUrl":"https://doi.org/arxiv-2405.03175","url":null,"abstract":"In this paper, we study operations on functors in the category of abelian\u0000groups simplar to the derivation in the sense of Dold-Puppe. They are defined\u0000as derived limits of a functor applied to the relation subgroup over a category\u0000of free presentations of the group. The integral homology of the\u0000Eilenberg-Maclane space $K(mathbb Z,3)$ appears as a part of description of\u0000these operations applied to symmetric powers.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"2012 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140882763","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}
We undertake a systematic study of the Hochschild homology, i.e. (the geometric realization of) the cyclic nerve, of $(infty,1)$-categories (and more generally of category-objects in an $infty$-category), as a version of factorization homology. In order to do this, we codify $(infty,1)$-categories in terms of quiver representations in them. By examining a universal instance of such Hochschild homology, we explicitly identify its natural symmetries, and construct a non-stable version of the cyclotomic trace map. Along the way we give a unified account of the cyclic, paracyclic, and epicyclic categories. We also prove that this gives a combinatorial description of the $n=1$ case of factorization homology as presented in [AFR18], which parametrizes $(infty,1)$-categories by solidly 1-framed stratified spaces.
{"title":"Symmetries of the cyclic nerve","authors":"David Ayala, Aaron Mazel-Gee, Nick Rozenblyum","doi":"arxiv-2405.03897","DOIUrl":"https://doi.org/arxiv-2405.03897","url":null,"abstract":"We undertake a systematic study of the Hochschild homology, i.e. (the\u0000geometric realization of) the cyclic nerve, of $(infty,1)$-categories (and\u0000more generally of category-objects in an $infty$-category), as a version of\u0000factorization homology. In order to do this, we codify $(infty,1)$-categories\u0000in terms of quiver representations in them. By examining a universal instance\u0000of such Hochschild homology, we explicitly identify its natural symmetries, and\u0000construct a non-stable version of the cyclotomic trace map. Along the way we\u0000give a unified account of the cyclic, paracyclic, and epicyclic categories. We\u0000also prove that this gives a combinatorial description of the $n=1$ case of\u0000factorization homology as presented in [AFR18], which parametrizes\u0000$(infty,1)$-categories by solidly 1-framed stratified spaces.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925424","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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