In this paper we continue Prasma's homotopical group theory program by considering homotopy normal maps in arbitrary $infty$-topoi. We show that maps of group objects equipped with normality data, in Prasma's sense, are algebras for a "normal closure" monad in a way which generalizes the standard loops-suspension monad. We generalize a result of Prasma by showing that monoidal functors of $infty$-topoi preserve normal maps or, equivalently, that monoidal functors of $infty$-topoi preserve the property of "being a fiber" for morphisms between connected objects. We also formulate Noether's Isomorphism Theorems in this setting, prove the first of them, and provide counterexamples to the other two. Accomplishing these goals requires us to spend substantial time synthesizing existing work of Lurie so that we may rigorously talk about group objects in $infty$-topoi in the "usual way." One nice result of this labor is the formulation and proof of an Orbit-Stabilizer Theorem for group actions in $infty$-topoi.
{"title":"Higher Groups and Higher Normality","authors":"Jonathan Beardsley, Landon Fox","doi":"arxiv-2407.21210","DOIUrl":"https://doi.org/arxiv-2407.21210","url":null,"abstract":"In this paper we continue Prasma's homotopical group theory program by\u0000considering homotopy normal maps in arbitrary $infty$-topoi. We show that maps\u0000of group objects equipped with normality data, in Prasma's sense, are algebras\u0000for a \"normal closure\" monad in a way which generalizes the standard\u0000loops-suspension monad. We generalize a result of Prasma by showing that\u0000monoidal functors of $infty$-topoi preserve normal maps or, equivalently, that\u0000monoidal functors of $infty$-topoi preserve the property of \"being a fiber\"\u0000for morphisms between connected objects. We also formulate Noether's\u0000Isomorphism Theorems in this setting, prove the first of them, and provide\u0000counterexamples to the other two. Accomplishing these goals requires us to\u0000spend substantial time synthesizing existing work of Lurie so that we may\u0000rigorously talk about group objects in $infty$-topoi in the \"usual way.\" One\u0000nice result of this labor is the formulation and proof of an Orbit-Stabilizer\u0000Theorem for group actions in $infty$-topoi.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869298","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 establish a calculus of differences for taut endofunctors of the category of sets, analogous to the classical calculus of finite differences for real valued functions. We study how the difference operator interacts with limits and colimits as categorical versions of the usual product and sum rules. The first main result is a lax chain rule which has no counterpart for mere functions. We also show that many important classes of functors (polynomials, analytic functors, reduced powers, ...) are taut, and calculate explicit formulas for their differences. Covariant Dirichlet series are introduced and studied. The second main result is a Newton summation formula expressed as an adjoint to the difference operator.
{"title":"Taut functors and the difference operator","authors":"Robert Paré","doi":"arxiv-2407.21129","DOIUrl":"https://doi.org/arxiv-2407.21129","url":null,"abstract":"We establish a calculus of differences for taut endofunctors of the category\u0000of sets, analogous to the classical calculus of finite differences for real\u0000valued functions. We study how the difference operator interacts with limits\u0000and colimits as categorical versions of the usual product and sum rules. The\u0000first main result is a lax chain rule which has no counterpart for mere\u0000functions. We also show that many important classes of functors (polynomials,\u0000analytic functors, reduced powers, ...) are taut, and calculate explicit\u0000formulas for their differences. Covariant Dirichlet series are introduced and\u0000studied. The second main result is a Newton summation formula expressed as an\u0000adjoint to the difference operator.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869297","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 $k$ be a field that is finitely generated over its prime field. In Grothendieck's anabelian letter to Faltings, he conjectured that sending a $k$-scheme to its '{e}tale topos defines a fully faithful functor from the localization of the category of finite type $k$-schemes at the universal homeomorphisms to a category of topoi. We prove Grothendieck's conjecture for infinite fields of arbitrary characteristic. In characteristic $0$, this shows that seminormal finite type $k$-schemes can be reconstructed from their '{e}tale topoi, generalizing work of Voevodsky. In positive characteristic, this shows that perfections of finite type $k$-schemes can be reconstructed from their '{e}tale topoi.
{"title":"Reconstruction of schemes from their étale topoi","authors":"Magnus Carlson, Peter J. Haine, Sebastian Wolf","doi":"arxiv-2407.19920","DOIUrl":"https://doi.org/arxiv-2407.19920","url":null,"abstract":"Let $k$ be a field that is finitely generated over its prime field. In\u0000Grothendieck's anabelian letter to Faltings, he conjectured that sending a\u0000$k$-scheme to its '{e}tale topos defines a fully faithful functor from the\u0000localization of the category of finite type $k$-schemes at the universal\u0000homeomorphisms to a category of topoi. We prove Grothendieck's conjecture for\u0000infinite fields of arbitrary characteristic. In characteristic $0$, this shows\u0000that seminormal finite type $k$-schemes can be reconstructed from their\u0000'{e}tale topoi, generalizing work of Voevodsky. In positive characteristic,\u0000this shows that perfections of finite type $k$-schemes can be reconstructed\u0000from their '{e}tale topoi.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"361 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869299","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 exponentiable functors in the context of synthetic $infty$-categories. We do this within the framework of simplicial Homotopy Type Theory of Riehl and Shulman. Our main result characterizes exponentiable functors. In order to achieve this, we explore Segal type completions. Moreover, we verify that our result is semantically sound.
{"title":"Exponentiable functors between synthetic $infty$-categories","authors":"César Bardomiano-Martínez","doi":"arxiv-2407.18072","DOIUrl":"https://doi.org/arxiv-2407.18072","url":null,"abstract":"We study exponentiable functors in the context of synthetic\u0000$infty$-categories. We do this within the framework of simplicial Homotopy\u0000Type Theory of Riehl and Shulman. Our main result characterizes exponentiable\u0000functors. In order to achieve this, we explore Segal type completions.\u0000Moreover, we verify that our result is semantically sound.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"350 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771989","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 $mathbf{C}$ be a Cauchy-complete category. The subtoposes of $[mathbf{C}^{mathrm{op}}, mathbf{Set}]$ are sometimes all of the form $[mathbf{D}^{mathrm{op}}, mathbf{Set}]$ where $mathbf{D}$ is a full Cauchy-complete subcategory of $mathbf{C}$. This is the case for instance when $mathbf{C}$ is finite, an Artinian poset, or the simplex category. In order to unify these situations, we give two formulations of a sufficient condition. The first formulation involves a two-player game, and the second formulation combines two "local" properties of $mathbf{C}$.
{"title":"A Criterion for Categories on which every Grothendieck Topology is Rigid","authors":"Jérémie Marquès","doi":"arxiv-2407.18417","DOIUrl":"https://doi.org/arxiv-2407.18417","url":null,"abstract":"Let $mathbf{C}$ be a Cauchy-complete category. The subtoposes of\u0000$[mathbf{C}^{mathrm{op}}, mathbf{Set}]$ are sometimes all of the form\u0000$[mathbf{D}^{mathrm{op}}, mathbf{Set}]$ where $mathbf{D}$ is a full\u0000Cauchy-complete subcategory of $mathbf{C}$. This is the case for instance when\u0000$mathbf{C}$ is finite, an Artinian poset, or the simplex category. In order to\u0000unify these situations, we give two formulations of a sufficient condition. The\u0000first formulation involves a two-player game, and the second formulation\u0000combines two \"local\" properties of $mathbf{C}$.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"85 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869300","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 unital $infty$-operads by their arity restrictions. Given $k geq 1$, we develop a model for unital $k$-restricted $infty$-operads, which are variants of $infty$-operads which has only $(leq k)$-arity morphisms, as complete Segal presheaves on closed $k$-dendroidal trees, which are closed trees build from corollas with valences $leq k$. Furthermore, we prove that the restriction functors from unital $infty$-operads to unital $k$-restricted $infty$-operads admit fully faithful left and right adjoints by showing that the left and right Kan extensions preserve complete Segal objects. Varying $k$, the left and right adjoints give a filtration and a co-filtration for any unital $infty$-operads by $k$-restricted $infty$-operads.
{"title":"Unital k-Restricted Infinity-Operads","authors":"Amartya Shekhar Dubey, Yu Leon Liu","doi":"arxiv-2407.17444","DOIUrl":"https://doi.org/arxiv-2407.17444","url":null,"abstract":"We study unital $infty$-operads by their arity restrictions. Given $k geq\u00001$, we develop a model for unital $k$-restricted $infty$-operads, which are\u0000variants of $infty$-operads which has only $(leq k)$-arity morphisms, as\u0000complete Segal presheaves on closed $k$-dendroidal trees, which are closed\u0000trees build from corollas with valences $leq k$. Furthermore, we prove that\u0000the restriction functors from unital $infty$-operads to unital $k$-restricted\u0000$infty$-operads admit fully faithful left and right adjoints by showing that\u0000the left and right Kan extensions preserve complete Segal objects. Varying $k$,\u0000the left and right adjoints give a filtration and a co-filtration for any\u0000unital $infty$-operads by $k$-restricted $infty$-operads.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771825","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 paper solves the first problem of the open problems in topos theory posted by William Lawvere, which asks the existence of a Grothendieck topos that has a proper class many quotient topoi. This paper concretely constructs such Grothendieck topoi, including the presheaf topos of the free monoid generated by countably infinite elements $mathbf{PSh}(M_omega)$. Utilizing the combinatorics of the classifying topos of the theory of inhabited objects and considering pairing functions, the problem is reduced to making rigid relational structures. This is accomplished by using Kunen's theorem on elementary embeddings in set theory.
{"title":"A solution to the first Lawvere's problem A Grothendieck topos that has a proper class many quotient topoi","authors":"Yuhi Kamio, Ryuya Hora","doi":"arxiv-2407.17105","DOIUrl":"https://doi.org/arxiv-2407.17105","url":null,"abstract":"This paper solves the first problem of the open problems in topos theory\u0000posted by William Lawvere, which asks the existence of a Grothendieck topos\u0000that has a proper class many quotient topoi. This paper concretely constructs\u0000such Grothendieck topoi, including the presheaf topos of the free monoid\u0000generated by countably infinite elements $mathbf{PSh}(M_omega)$. Utilizing\u0000the combinatorics of the classifying topos of the theory of inhabited objects\u0000and considering pairing functions, the problem is reduced to making rigid\u0000relational structures. This is accomplished by using Kunen's theorem on\u0000elementary embeddings in set theory.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771822","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 $mathcal C$ be a Grothendieck category and $U$ be a monad on $mathcal C$ that is exact and preserves colimits. In this article, we prove that every hereditary torsion theory on the Eilenberg-Moore category $EM_U$ of modules over a monad $U$ is differential. Further, if $delta:Ulongrightarrow U$ denotes a derivation on a monad $U$, then we show that every $delta$-derivation on a $U$-module $M$ extends uniquely to a $delta$-derivation on the module of quotients of $M$.
{"title":"Differential torsion theories on Eilenberg-Moore categories of monads","authors":"Divya Ahuja, Surjeet Kour","doi":"arxiv-2407.16782","DOIUrl":"https://doi.org/arxiv-2407.16782","url":null,"abstract":"Let $mathcal C$ be a Grothendieck category and $U$ be a monad on $mathcal\u0000C$ that is exact and preserves colimits. In this article, we prove that every\u0000hereditary torsion theory on the Eilenberg-Moore category $EM_U$ of modules\u0000over a monad $U$ is differential. Further, if $delta:Ulongrightarrow U$\u0000denotes a derivation on a monad $U$, then we show that every\u0000$delta$-derivation on a $U$-module $M$ extends uniquely to a\u0000$delta$-derivation on the module of quotients of $M$.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771824","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}
Davide Trotta, Jonathan Weinberger, Valeria de Paiva
Grothendieck fibrations are fundamental in capturing the concept of dependency, notably in categorical semantics of type theory and programming languages. A relevant instance are Dialectica fibrations which generalise G"odel's Dialectica proof interpretation and have been widely studied in recent years. We characterise when a given fibration is a generalised, dependent Dialectica fibration, namely an iterated completion of a fibration by dependent products and sums (along a given class of display maps). From a technical perspective, we complement the work of Hofstra on Dialectica fibrations by an internal viewpoint, categorifying the classical notion of quantifier-freeness. We also generalise both Hofstra's and Trotta et al.'s work on G"odel fibrations to the dependent case, replacing the class of cartesian projections in the base category by arbitrary display maps. We discuss how this recovers a range of relevant examples in categorical logic and proof theory. Moreover, as another instance, we introduce Hilbert fibrations, providing a categorical understanding of Hilbert's $epsilon$- and $tau$-operators well-known from proof theory.
{"title":"Skolem, Gödel, and Hilbert fibrations","authors":"Davide Trotta, Jonathan Weinberger, Valeria de Paiva","doi":"arxiv-2407.15765","DOIUrl":"https://doi.org/arxiv-2407.15765","url":null,"abstract":"Grothendieck fibrations are fundamental in capturing the concept of\u0000dependency, notably in categorical semantics of type theory and programming\u0000languages. A relevant instance are Dialectica fibrations which generalise\u0000G\"odel's Dialectica proof interpretation and have been widely studied in\u0000recent years. We characterise when a given fibration is a generalised, dependent Dialectica\u0000fibration, namely an iterated completion of a fibration by dependent products\u0000and sums (along a given class of display maps). From a technical perspective,\u0000we complement the work of Hofstra on Dialectica fibrations by an internal\u0000viewpoint, categorifying the classical notion of quantifier-freeness. We also\u0000generalise both Hofstra's and Trotta et al.'s work on G\"odel fibrations to the\u0000dependent case, replacing the class of cartesian projections in the base\u0000category by arbitrary display maps. We discuss how this recovers a range of\u0000relevant examples in categorical logic and proof theory. Moreover, as another\u0000instance, we introduce Hilbert fibrations, providing a categorical\u0000understanding of Hilbert's $epsilon$- and $tau$-operators well-known from\u0000proof theory.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771823","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}
Pseudocolimits are formal gluing constructions that combine objects in a category indexed by a pseudofunctor. When the objects are categories and the domain of the pseudofunctor is small and filtered it has been known since Exppose 6 in SGA4 that the pseudocolimit can be computed by taking the Grothendieck construction of the pseudofunctor and inverting the class of cartesian arrows with respect to the canonical fibration. This paper is a reformatted version of a MSc thesis submitted and defended at Dalhousie University in August 2022. The first part presents a set of conditions for defining an internal category of elements of a diagram of internal categories and proves it is the oplax colimit. The second part presents a set of conditions on an ambient category and an internal category with an object of weak-equivalences that allows an internal description of the axioms for a category of (right) fractions and a definition of the internal category of (right) fractions when all the conditions and axioms are satisfied. These are combined to present a suitable context for computing the pseudocolimit of a small filtered diagram of internal categories.
{"title":"Pseudocolimits of Small Filtered Diagrams of Internal Categories","authors":"Deni Salja","doi":"arxiv-2407.18971","DOIUrl":"https://doi.org/arxiv-2407.18971","url":null,"abstract":"Pseudocolimits are formal gluing constructions that combine objects in a\u0000category indexed by a pseudofunctor. When the objects are categories and the\u0000domain of the pseudofunctor is small and filtered it has been known since\u0000Exppose 6 in SGA4 that the pseudocolimit can be computed by taking the\u0000Grothendieck construction of the pseudofunctor and inverting the class of\u0000cartesian arrows with respect to the canonical fibration. This paper is a\u0000reformatted version of a MSc thesis submitted and defended at Dalhousie\u0000University in August 2022. The first part presents a set of conditions for\u0000defining an internal category of elements of a diagram of internal categories\u0000and proves it is the oplax colimit. The second part presents a set of\u0000conditions on an ambient category and an internal category with an object of\u0000weak-equivalences that allows an internal description of the axioms for a\u0000category of (right) fractions and a definition of the internal category of\u0000(right) fractions when all the conditions and axioms are satisfied. These are\u0000combined to present a suitable context for computing the pseudocolimit of a\u0000small filtered diagram of internal categories.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869301","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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