We construct a natural morphism $rho$ from the nerve $text{MC}_bullet(L) = text{MC}(Omega_bullet widehat{otimes} L)$ of a pronilpotent curved L${}_infty$-algebra $L$ to the simplicial subset $gamma_bullet(L) = text{MC}(Omega_bullet widehat{otimes} L,s_bullet)$ of Maurer--Cartan element satisfying the Dupont gauge condition. This morphism equals the identity on the image of the inclusion $gamma_bullet(L) hookrightarrow text{MC}_bullet(L)$. The proof uses the extension of Berglund's homotopical perturbation theory for L${}_infty$-algebras to curved L${}_infty$-algebras. The morphism $rho$ equals the holonomy for nilpotent Lie algebras. In a sequel to this paper, we use a cubical analogue $rho^square$ of $rho$ to identify $rho$ with higher holonomy for semiabelian curved Linf-algebras.
{"title":"Higher holonomy for curved L${}_infty$-algebras 1: simplicial methods","authors":"Ezra GetzlerNorthwestern University","doi":"arxiv-2408.11157","DOIUrl":"https://doi.org/arxiv-2408.11157","url":null,"abstract":"We construct a natural morphism $rho$ from the nerve $text{MC}_bullet(L) =\u0000text{MC}(Omega_bullet widehat{otimes} L)$ of a pronilpotent curved\u0000L${}_infty$-algebra $L$ to the simplicial subset $gamma_bullet(L) =\u0000text{MC}(Omega_bullet widehat{otimes} L,s_bullet)$ of Maurer--Cartan\u0000element satisfying the Dupont gauge condition. This morphism equals the\u0000identity on the image of the inclusion $gamma_bullet(L) hookrightarrow\u0000text{MC}_bullet(L)$. The proof uses the extension of Berglund's homotopical\u0000perturbation theory for L${}_infty$-algebras to curved L${}_infty$-algebras.\u0000The morphism $rho$ equals the holonomy for nilpotent Lie algebras. In a sequel\u0000to this paper, we use a cubical analogue $rho^square$ of $rho$ to identify\u0000$rho$ with higher holonomy for semiabelian curved Linf-algebras.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199179","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 introduce necklicial nerve functors from enriched categories to simplicial sets, which include Cordier's homotopy coherent, Lurie's differential graded and Le Grignou's cubical nerves. It is shown that every necklicial nerve can be lifted to the templicial objects of arXiv:2302.02484v2. Building on the work of Dugger and Spivak, we give sufficient conditions under which the left-adjoint of a necklicial nerve can be described more explicitly. As an application, we obtain novel and simple expressions for the left-adjoints of the dg-nerve and cubical nerve.
{"title":"Nerves of enriched categories via necklaces","authors":"Arne Mertens","doi":"arxiv-2408.10049","DOIUrl":"https://doi.org/arxiv-2408.10049","url":null,"abstract":"We introduce necklicial nerve functors from enriched categories to simplicial\u0000sets, which include Cordier's homotopy coherent, Lurie's differential graded\u0000and Le Grignou's cubical nerves. It is shown that every necklicial nerve can be\u0000lifted to the templicial objects of arXiv:2302.02484v2. Building on the work of\u0000Dugger and Spivak, we give sufficient conditions under which the left-adjoint\u0000of a necklicial nerve can be described more explicitly. As an application, we\u0000obtain novel and simple expressions for the left-adjoints of the dg-nerve and\u0000cubical nerve.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199181","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 introduce a novel concept of action for unitary magmas, facilitating the classification of various split extensions within this algebraic structure. Our method expands upon the recent study of split extensions and semidirect products of unitary magmas conducted by Gran, Janelidze, and Sobral. Building on their research, we explore split extensions in which the middle object does not necessarily maintain a bijective correspondence with the Cartesian product of its end objects. Although this phenomenon is not observed in groups or any associative semiabelian variety of universal algebra, it shares similarities with instances found in monoids through weakly Schreier extensions and certain exotic non-associative algebras, such as semi-left-loops. Our work seeks to contribute to the comprehension of split extensions in unitary magmas and may offer valuable insights for potential abstractions of categorical properties in more general contexts.
{"title":"Unitary magma actions","authors":"Nelson Martins-Ferreira","doi":"arxiv-2408.08721","DOIUrl":"https://doi.org/arxiv-2408.08721","url":null,"abstract":"We introduce a novel concept of action for unitary magmas, facilitating the\u0000classification of various split extensions within this algebraic structure. Our\u0000method expands upon the recent study of split extensions and semidirect\u0000products of unitary magmas conducted by Gran, Janelidze, and Sobral. Building\u0000on their research, we explore split extensions in which the middle object does\u0000not necessarily maintain a bijective correspondence with the Cartesian product\u0000of its end objects. Although this phenomenon is not observed in groups or any\u0000associative semiabelian variety of universal algebra, it shares similarities\u0000with instances found in monoids through weakly Schreier extensions and certain\u0000exotic non-associative algebras, such as semi-left-loops. Our work seeks to\u0000contribute to the comprehension of split extensions in unitary magmas and may\u0000offer valuable insights for potential abstractions of categorical properties in\u0000more general contexts.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"164 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199192","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}
Over the topos of sets, the notion of Lawvere theory is infinite countably-sorted algebraic but not one-sorted algebraic. Shifting viewpoint over the object-classifier topos, a finite algebraic presentation of Lawvere theories is considered.
{"title":"A finite algebraic presentation of Lawvere theories in the object-classifier topos","authors":"Marcelo Fiore, Sanjiv Ranchod","doi":"arxiv-2408.08980","DOIUrl":"https://doi.org/arxiv-2408.08980","url":null,"abstract":"Over the topos of sets, the notion of Lawvere theory is infinite\u0000countably-sorted algebraic but not one-sorted algebraic. Shifting viewpoint\u0000over the object-classifier topos, a finite algebraic presentation of Lawvere\u0000theories is considered.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199180","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 study various acyclicity conditions on higher-categorical pasting diagrams in the combinatorial framework of regular directed complexes. We present an apparently weakest acyclicity condition under which the $omega$-category presented by a diagram shape is freely generated in the sense of polygraphs. We then consider stronger conditions under which this $omega$-category is equivalent to one obtained from an augmented directed chain complex in the sense of Steiner, or consists only of subsets of cells in the diagram. Finally, we study the stability of these conditions under the operations of pasting, suspensions, Gray products, joins and duals.
{"title":"Acyclicity conditions on pasting diagrams","authors":"Amar Hadzihasanovic, Diana Kessler","doi":"arxiv-2408.16775","DOIUrl":"https://doi.org/arxiv-2408.16775","url":null,"abstract":"We study various acyclicity conditions on higher-categorical pasting diagrams\u0000in the combinatorial framework of regular directed complexes. We present an\u0000apparently weakest acyclicity condition under which the $omega$-category\u0000presented by a diagram shape is freely generated in the sense of polygraphs. We\u0000then consider stronger conditions under which this $omega$-category is\u0000equivalent to one obtained from an augmented directed chain complex in the\u0000sense of Steiner, or consists only of subsets of cells in the diagram. Finally,\u0000we study the stability of these conditions under the operations of pasting,\u0000suspensions, Gray products, joins and duals.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225583","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 that there exist Hopf algebras with surjective, non-bijective antipode which admit no non-trivial morphisms from Hopf algebras with bijective antipode; in particular, they are not quotients of such. This answers a question left open in prior work, and contrasts with the dual setup whereby a Hopf algebra has injective antipode precisely when it embeds into one with bijective antipode. The examples rely on the broader phenomenon of realizing pre-specified subspace lattices as comodule lattices: for a finite-dimensional vector space $V$ and a sequence $(mathcal{L}_r)_r$ of successively finer lattices of subspaces thereof, assuming the minimal subquotients of the supremum $bigvee_r mathcal{L}_r$ are all at least 2-dimensional, there is a Hopf algebra equipping $V$ with a comodule structure in such a fashion that the lattice of comodules of the $r^{th}$ dual comodule $V^{r*}$ is precisely the given $mathcal{L}_r$.
{"title":"Prescribed duality dynamics in comodule categories","authors":"Alexandru Chirvasitu","doi":"arxiv-2408.08167","DOIUrl":"https://doi.org/arxiv-2408.08167","url":null,"abstract":"We prove that there exist Hopf algebras with surjective, non-bijective\u0000antipode which admit no non-trivial morphisms from Hopf algebras with bijective\u0000antipode; in particular, they are not quotients of such. This answers a\u0000question left open in prior work, and contrasts with the dual setup whereby a\u0000Hopf algebra has injective antipode precisely when it embeds into one with\u0000bijective antipode. The examples rely on the broader phenomenon of realizing\u0000pre-specified subspace lattices as comodule lattices: for a finite-dimensional\u0000vector space $V$ and a sequence $(mathcal{L}_r)_r$ of successively finer\u0000lattices of subspaces thereof, assuming the minimal subquotients of the\u0000supremum $bigvee_r mathcal{L}_r$ are all at least 2-dimensional, there is a\u0000Hopf algebra equipping $V$ with a comodule structure in such a fashion that the\u0000lattice of comodules of the $r^{th}$ dual comodule $V^{r*}$ is precisely the\u0000given $mathcal{L}_r$.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199182","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 prove the existence of an algebraic model for quasi-coherent sheaves on certain non-connective geometric stacks arising in stable homotopy theory and spectral algebraic geometry using the machinery of adapted homology theories.
{"title":"Algebraic Models for Quasi-Coherent Sheaves in Spectral Algebraic Geometry","authors":"Adam Pratt","doi":"arxiv-2408.07972","DOIUrl":"https://doi.org/arxiv-2408.07972","url":null,"abstract":"In this paper we prove the existence of an algebraic model for quasi-coherent\u0000sheaves on certain non-connective geometric stacks arising in stable homotopy\u0000theory and spectral algebraic geometry using the machinery of adapted homology\u0000theories.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199183","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}
The coefficient categories of six functor formalisms are often locally rigid, and when this is the case, the exceptional pushforward and pullback adjunctions may be defined formally. In this short note it is shown that for f a proper map resp. an open embedding the well known formulas f_! = f_* resp. f_! = f_# may likewise be deduced formally.
六个函数形式的系数范畴通常是局部刚性的，在这种情况下，可以正式定义特殊的前推和回拉谓词。在这篇短文中，我们将证明，对于 f 一个适当的映射对应于一个开放的嵌入，众所周知的公式 f_！= f_* resp！= f_#同样可以正式推导出来。
{"title":"Local Rigidity and Six Functor Formalisms","authors":"Adrian Clough","doi":"arxiv-2408.07564","DOIUrl":"https://doi.org/arxiv-2408.07564","url":null,"abstract":"The coefficient categories of six functor formalisms are often locally rigid,\u0000and when this is the case, the exceptional pushforward and pullback adjunctions\u0000may be defined formally. In this short note it is shown that for f a proper map\u0000resp. an open embedding the well known formulas f_! = f_* resp. f_! = f_# may\u0000likewise be deduced formally.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199186","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}
Compact metric spaces form an important class of metric spaces, but the category that they define lacks many important properties such as completeness and cocompleteness. In recent studies of "metric domain theory" and Stone-type dualities, the more general notion of a (separated) metric compact Hausdorff space emerged as a metric counterpart of Nachbin's compact ordered spaces. Roughly speaking, a metric compact Hausdorff space is a metric space equipped with a emph{compatible} compact Hausdorff topology (which does not need to be the induced topology). These spaces maintain many important features of compact metric spaces, and, notably, the resulting category is much better behaved. Moreover, one can use inspiration from the theory of Nachbin's compact ordered spaces to solve problems for metric structures. In this paper we continue this line of research: in the category of separated metric compact Hausdorff spaces we characterise the regular monomorphisms as the embeddings and the epimorphisms as the surjective morphisms. Moreover, we show that epimorphisms out of an object $X$ can be encoded internally on $X$ by their kernel metrics, which are characterised as the continuous metrics below the metric on $X$; this gives a convenient way to represent quotient objects. Finally, as the main result, we prove that its dual category has an algebraic flavour: it is Barr-exact. While we show that it cannot be a variety of finitary algebras, it remains open whether it is an infinitary variety.
{"title":"Barr-coexactness for metric compact Hausdorff spaces","authors":"Marco Abbadini, Dirk Hofmann","doi":"arxiv-2408.07039","DOIUrl":"https://doi.org/arxiv-2408.07039","url":null,"abstract":"Compact metric spaces form an important class of metric spaces, but the\u0000category that they define lacks many important properties such as completeness\u0000and cocompleteness. In recent studies of \"metric domain theory\" and Stone-type\u0000dualities, the more general notion of a (separated) metric compact Hausdorff\u0000space emerged as a metric counterpart of Nachbin's compact ordered spaces.\u0000Roughly speaking, a metric compact Hausdorff space is a metric space equipped\u0000with a emph{compatible} compact Hausdorff topology (which does not need to be\u0000the induced topology). These spaces maintain many important features of compact\u0000metric spaces, and, notably, the resulting category is much better behaved.\u0000Moreover, one can use inspiration from the theory of Nachbin's compact ordered\u0000spaces to solve problems for metric structures. In this paper we continue this line of research: in the category of separated\u0000metric compact Hausdorff spaces we characterise the regular monomorphisms as\u0000the embeddings and the epimorphisms as the surjective morphisms. Moreover, we\u0000show that epimorphisms out of an object $X$ can be encoded internally on $X$ by\u0000their kernel metrics, which are characterised as the continuous metrics below\u0000the metric on $X$; this gives a convenient way to represent quotient objects.\u0000Finally, as the main result, we prove that its dual category has an algebraic\u0000flavour: it is Barr-exact. While we show that it cannot be a variety of\u0000finitary algebras, it remains open whether it is an infinitary variety.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199184","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 connectivity bound for maps of $infty$-operads of the form $mathbb{A}_{k_1} otimes cdots otimes mathbb{A}_{k_n} to mathbb{E}_n$, and as a consequence, give an inductive way to construct $mathbb{E}_n$-algebras in $m$-categories. The result follows from a version of Eckmann-Hilton argument that takes into account both connectivity and arity of $infty$-operads. Along the way, we prove a technical Blakers-Massey type statement for algebras of coherent $infty$-operads.
{"title":"$mathbb{E}_n$-algebras in m-categories","authors":"Yu Leon Liu","doi":"arxiv-2408.05607","DOIUrl":"https://doi.org/arxiv-2408.05607","url":null,"abstract":"We prove a connectivity bound for maps of $infty$-operads of the form\u0000$mathbb{A}_{k_1} otimes cdots otimes mathbb{A}_{k_n} to mathbb{E}_n$,\u0000and as a consequence, give an inductive way to construct\u0000$mathbb{E}_n$-algebras in $m$-categories. The result follows from a version of\u0000Eckmann-Hilton argument that takes into account both connectivity and arity of\u0000$infty$-operads. Along the way, we prove a technical Blakers-Massey type\u0000statement for algebras of coherent $infty$-operads.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199185","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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