We provide a novel proof of the homological excess intersection formula for local complete intersections. The novelty is that the proof makes use of global morphisms comparing the intersections to a self intersection.
{"title":"A global proof of the homological excess intersection formula","authors":"Oscar Finegan","doi":"arxiv-2406.17485","DOIUrl":"https://doi.org/arxiv-2406.17485","url":null,"abstract":"We provide a novel proof of the homological excess intersection formula for\u0000local complete intersections. The novelty is that the proof makes use of global\u0000morphisms comparing the intersections to a self intersection.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505644","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 the feasibility of investigating the theory of $mathrm{Mod}_{mathbb{H}mathrm{k}}$-enriched $infty$-categories, where $mathbb{H}mathrm{k}$ is the Eilenberg-Maclane Spectrum associated with a commutative and unitary ring $k$, through the framework of $mathcal{S}p$-enriched $infty$-category theory. In particular, we prove that the $infty$-category of $mathrm{Mod}_{mathbb{H}mathrm{k}}$-enriched $infty$-categories $mathcal{C}at_{infty}^{mathrm{Mod}_{mathbb{H}mathrm{k}}}$, $infty$-category of left $mathbb{H}mathrm{k}$-module objects of the $infty$-category of $mathcal{S}p$-enriched $infty$-categories $mathcal{C}at_{infty}^{mathcal{S}p}$ $mathrm{LMod}_{mathbb{H}mathrm{k}}(mathcal{C}at_{infty}^{mathcal{S}p})$ and the $infty$-category of $mathcal{C}at_{infty}^{mathcal{S}p}$-enriched $infty$-functors $Fun^{mathcal{C}at_{infty}^{mathcal{S}p}}(underline{underline{mathbb{H}mathrm{k}}},mathcal{C}at_{infty}^{mathcal{S}p})$ are equivalent.
{"title":"$mathrm{Mod}_{mathbb{H}mathrm{k}}$-enriched $infty$-categories are left $mathbb{H}mathrm{k}$-module objects of $mathcal{C}at_{infty}^{mathcal{S}p}$ and $mathcal{C}at_{infty}^{mathcal{S}p}$-enriched $infty$-functors","authors":"Matteo Doni","doi":"arxiv-2406.15884","DOIUrl":"https://doi.org/arxiv-2406.15884","url":null,"abstract":"We establish the feasibility of investigating the theory of\u0000$mathrm{Mod}_{mathbb{H}mathrm{k}}$-enriched $infty$-categories, where\u0000$mathbb{H}mathrm{k}$ is the Eilenberg-Maclane Spectrum associated with a\u0000commutative and unitary ring $k$, through the framework of\u0000$mathcal{S}p$-enriched $infty$-category theory. In particular, we prove that\u0000the $infty$-category of $mathrm{Mod}_{mathbb{H}mathrm{k}}$-enriched\u0000$infty$-categories\u0000$mathcal{C}at_{infty}^{mathrm{Mod}_{mathbb{H}mathrm{k}}}$,\u0000$infty$-category of left $mathbb{H}mathrm{k}$-module objects of the\u0000$infty$-category of $mathcal{S}p$-enriched $infty$-categories\u0000$mathcal{C}at_{infty}^{mathcal{S}p}$\u0000$mathrm{LMod}_{mathbb{H}mathrm{k}}(mathcal{C}at_{infty}^{mathcal{S}p})$\u0000and the $infty$-category of $mathcal{C}at_{infty}^{mathcal{S}p}$-enriched\u0000$infty$-functors\u0000$Fun^{mathcal{C}at_{infty}^{mathcal{S}p}}(underline{underline{mathbb{H}mathrm{k}}},mathcal{C}at_{infty}^{mathcal{S}p})$\u0000are equivalent.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"167 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505646","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 the feasibility of investigating the theory of $Rtext{-}mathrm{Mod}$-enriched categories, for any commutative and unitary ring $R$, through the framework of $mathbb{A}mathrm{b}$-enriched category theory. In particular, we prove that the category of $R$-$mathrm{Mod}$-enriched categories, $Cat(R$-$mathrm{Mod})$, the category of $underline{R}$-modules inside $Cat(mathbb{A}mathrm{b})$, $mathrm{LMod}_{underline{R}}(Cat(mathbb{A}mathrm{b}))$, and the category of $Cat(mathbb{A}mathrm{b})$-enriched functors, $Fun^{Cat(mathbb{A}mathrm{b})}(underline{underline{R}},Cat(mathbb{A}mathrm{b}))$ are equivalent.
{"title":"$Rtext{-}mathrm{Mod}$-enriched categories are left $underline{R}$-module objects of $Cat(mathbb{A}mathrm{b})$ and $Cat(mathbb{A}mathrm{b})$-enriched functors","authors":"Matteo Doni","doi":"arxiv-2406.15887","DOIUrl":"https://doi.org/arxiv-2406.15887","url":null,"abstract":"We establish the feasibility of investigating the theory of\u0000$Rtext{-}mathrm{Mod}$-enriched categories, for any commutative and unitary\u0000ring $R$, through the framework of $mathbb{A}mathrm{b}$-enriched category\u0000theory. In particular, we prove that the category of\u0000$R$-$mathrm{Mod}$-enriched categories, $Cat(R$-$mathrm{Mod})$, the category\u0000of $underline{R}$-modules inside $Cat(mathbb{A}mathrm{b})$,\u0000$mathrm{LMod}_{underline{R}}(Cat(mathbb{A}mathrm{b}))$, and the category of\u0000$Cat(mathbb{A}mathrm{b})$-enriched functors,\u0000$Fun^{Cat(mathbb{A}mathrm{b})}(underline{underline{R}},Cat(mathbb{A}mathrm{b}))$\u0000are equivalent.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505645","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 introduces a novel approach to ontology-based robot plan transfer using functorial data migrations from category theory. Functors provide structured maps between domain types and predicates which can be used to transfer plans from a source domain to a target domain without the need for replanning. Unlike methods that create models for transferring specific plans, our approach can be applied to any plan within a given domain. We demonstrate this approach by transferring a task plan from the canonical Blocksworld domain to one compatible with the AI2-THOR Kitchen environment. In addition, we discuss practical applications that may enhance the adaptability of robotic task planning in general.
{"title":"Automating Transfer of Robot Task Plans using Functorial Data Migrations","authors":"Angeline Aguinaldo, Evan Patterson, William Regli","doi":"arxiv-2406.15961","DOIUrl":"https://doi.org/arxiv-2406.15961","url":null,"abstract":"This paper introduces a novel approach to ontology-based robot plan transfer\u0000using functorial data migrations from category theory. Functors provide\u0000structured maps between domain types and predicates which can be used to\u0000transfer plans from a source domain to a target domain without the need for\u0000replanning. Unlike methods that create models for transferring specific plans,\u0000our approach can be applied to any plan within a given domain. We demonstrate\u0000this approach by transferring a task plan from the canonical Blocksworld domain\u0000to one compatible with the AI2-THOR Kitchen environment. In addition, we\u0000discuss practical applications that may enhance the adaptability of robotic\u0000task planning in general.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505647","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 technique of equipping graphs with an equivalence relation, called equality saturation, has recently proved both powerful and practical in program optimisation, particularly for satisfiability modulo theory solvers. We give a categorical semantics to these structures, called e-graphs, in terms of Cartesian categories enriched over a semilattice. We show how this semantics can be generalised to monoidal categories, which opens the door to new applications of e-graph techniques, from algebraic to monoidal theories. Finally, we present a sound and complete combinatorial representation of morphisms in such a category, based on a generalisation of hypergraphs which we call e-hypergraphs. They have the usual advantage that many of their structural equations are absorbed into a general notion of isomorphism.
{"title":"Equivalence Hypergraphs: E-Graphs for Monoidal Theories","authors":"Dan R. Ghica, Chris Barrett, Aleksei Tiurin","doi":"arxiv-2406.15882","DOIUrl":"https://doi.org/arxiv-2406.15882","url":null,"abstract":"The technique of equipping graphs with an equivalence relation, called\u0000equality saturation, has recently proved both powerful and practical in program\u0000optimisation, particularly for satisfiability modulo theory solvers. We give a\u0000categorical semantics to these structures, called e-graphs, in terms of\u0000Cartesian categories enriched over a semilattice. We show how this semantics\u0000can be generalised to monoidal categories, which opens the door to new\u0000applications of e-graph techniques, from algebraic to monoidal theories.\u0000Finally, we present a sound and complete combinatorial representation of\u0000morphisms in such a category, based on a generalisation of hypergraphs which we\u0000call e-hypergraphs. They have the usual advantage that many of their structural\u0000equations are absorbed into a general notion of isomorphism.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505649","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}
G. S. H. Cruttwell, Jean-Simon Pacaud Lemay, Elias Vandenberg
There is an abstract notion of connection in any tangent category. In this paper, we show that when applied to the tangent category of affine schemes, this recreates the classical notion of a connection on a module (and similarly, in the tangent category of schemes, this recreates the notion of connection on a quasi-coherent sheaf of modules). By contrast, we also show that in the tangent category of algebras, there are no non-trivial connections.
{"title":"A Tangent Category Perspective on Connections in Algebraic Geometry","authors":"G. S. H. Cruttwell, Jean-Simon Pacaud Lemay, Elias Vandenberg","doi":"arxiv-2406.15137","DOIUrl":"https://doi.org/arxiv-2406.15137","url":null,"abstract":"There is an abstract notion of connection in any tangent category. In this\u0000paper, we show that when applied to the tangent category of affine schemes,\u0000this recreates the classical notion of a connection on a module (and similarly,\u0000in the tangent category of schemes, this recreates the notion of connection on\u0000a quasi-coherent sheaf of modules). By contrast, we also show that in the\u0000tangent category of algebras, there are no non-trivial connections.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"115 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505648","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}
Abstract inner automorphisms can be used to promote any category into a 2-category, and we study two-dimensional limits and colimits in the resulting 2-categories. Existing connected colimits and limits in the starting category become two-dimensional colimits and limits under fairly general conditions. Under the same conditions, colimits in the underlying category can be used to build many notable two-dimensional colimits such as coequifiers and coinserters. In contrast, disconnected colimits or genuinely 2-categorical limits such as inserters and equifiers and cotensors cannot exist unless no nontrivial abstract inner automorphisms exist and the resulting 2-category is locally discrete. We also study briefly when an ordinary functor can be extended to a 2-functor between the resulting 2-categories.
{"title":"Inner automorphisms as 2-cells","authors":"Pieter Hofstra, Martti Karvonen","doi":"arxiv-2406.13647","DOIUrl":"https://doi.org/arxiv-2406.13647","url":null,"abstract":"Abstract inner automorphisms can be used to promote any category into a\u00002-category, and we study two-dimensional limits and colimits in the resulting\u00002-categories. Existing connected colimits and limits in the starting category\u0000become two-dimensional colimits and limits under fairly general conditions.\u0000Under the same conditions, colimits in the underlying category can be used to\u0000build many notable two-dimensional colimits such as coequifiers and\u0000coinserters. In contrast, disconnected colimits or genuinely 2-categorical\u0000limits such as inserters and equifiers and cotensors cannot exist unless no\u0000nontrivial abstract inner automorphisms exist and the resulting 2-category is\u0000locally discrete. We also study briefly when an ordinary functor can be\u0000extended to a 2-functor between the resulting 2-categories.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505654","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 $Fcolon mathcal{C} to mathcal{E}$ be a functor from a category $mathcal{C}$ to a homological (Borceux-Bourn) or semi-abelian (Janelidze-M'arki-Tholen) category $mathcal{E}$. We investigate conditions under which the homology of an object $X$ in $mathcal{C}$ with coefficients in the functor $F$, defined via projective resolutions in $mathcal{C}$, remains independent of the chosen resolution. Consequently, the left derived functors of $F$ can be constructed analogously to the classical abelian case. Our approach extends the concept of chain homotopy to a non-additive setting using the technique of imaginary morphisms. Specifically, we utilize the approximate subtractions of Bourn-Janelidze, originally introduced in the context of subtractive categories. This method is applicable when $mathcal{C}$ is a pointed regular category with finite coproducts and enough projectives, provided these projectives are closed under protosplit subobjects, a new condition introduced in this article and naturally satisfied in the abelian context. We further assume that the functor $F$ meets certain exactness conditions: for instance, it may be protoadditive and preserve proper morphisms and binary coproducts - conditions that amount to additivity when $mathcal{C}$ and $mathcal{E}$ are abelian categories. Within this framework, we develop a basic theory of derived functors, compare it with the simplicial approach, and provide several examples.
{"title":"Non-additive derived functors","authors":"Maxime Culot, Fara Renaud, Tim Van der Linden","doi":"arxiv-2406.13398","DOIUrl":"https://doi.org/arxiv-2406.13398","url":null,"abstract":"Let $Fcolon mathcal{C} to mathcal{E}$ be a functor from a category\u0000$mathcal{C}$ to a homological (Borceux-Bourn) or semi-abelian\u0000(Janelidze-M'arki-Tholen) category $mathcal{E}$. We investigate conditions\u0000under which the homology of an object $X$ in $mathcal{C}$ with coefficients in\u0000the functor $F$, defined via projective resolutions in $mathcal{C}$, remains\u0000independent of the chosen resolution. Consequently, the left derived functors\u0000of $F$ can be constructed analogously to the classical abelian case. Our approach extends the concept of chain homotopy to a non-additive setting\u0000using the technique of imaginary morphisms. Specifically, we utilize the\u0000approximate subtractions of Bourn-Janelidze, originally introduced in the\u0000context of subtractive categories. This method is applicable when $mathcal{C}$\u0000is a pointed regular category with finite coproducts and enough projectives,\u0000provided these projectives are closed under protosplit subobjects, a new\u0000condition introduced in this article and naturally satisfied in the abelian\u0000context. We further assume that the functor $F$ meets certain exactness\u0000conditions: for instance, it may be protoadditive and preserve proper morphisms\u0000and binary coproducts - conditions that amount to additivity when $mathcal{C}$\u0000and $mathcal{E}$ are abelian categories. Within this framework, we develop a basic theory of derived functors, compare\u0000it with the simplicial approach, and provide several examples.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505650","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 $omega$-weak equivalences between weak $omega$-categories in the sense of Batanin-Leinster. Our $omega$-weak equivalences are strict $omega$-functors satisfying essential surjectivity at every dimension, and when restricted to those between strict $omega$-categories, they coincide with the weak equivalences in the model category of strict $omega$-categories defined by Lafont, M'etayer, and Worytkiewicz. We show that the class of $omega$-weak equivalences has the 2-out-of-3 property. We also consider a generalisation of $omega$-weak equivalences, defined as weak $omega$-functors (in the sense of Garner) satisfying essential surjectivity, and show that this class also has the 2-out-of-3 property.
{"title":"$ω$-weak equivalences between weak $ω$-categories","authors":"Soichiro Fujii, Keisuke Hoshino, Yuki Maehara","doi":"arxiv-2406.13240","DOIUrl":"https://doi.org/arxiv-2406.13240","url":null,"abstract":"We study $omega$-weak equivalences between weak $omega$-categories in the\u0000sense of Batanin-Leinster. Our $omega$-weak equivalences are strict\u0000$omega$-functors satisfying essential surjectivity at every dimension, and\u0000when restricted to those between strict $omega$-categories, they coincide with\u0000the weak equivalences in the model category of strict $omega$-categories\u0000defined by Lafont, M'etayer, and Worytkiewicz. We show that the class of\u0000$omega$-weak equivalences has the 2-out-of-3 property. We also consider a\u0000generalisation of $omega$-weak equivalences, defined as weak $omega$-functors\u0000(in the sense of Garner) satisfying essential surjectivity, and show that this\u0000class also has the 2-out-of-3 property.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505651","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}
Building on our previous work on enriched universal algebra, we define a notion of enriched language consisting of function and relation symbols whose arities are objects of the base of enrichment. In this context, we construct atomic formulas and define the regular fragment of enriched logic by taking conjunctions and existential quantifications of those. We then characterize enriched categories of models of regular theories as enriched injectivity classes in the enriched category of structures. These notions rely on the choice of a factorization system on the base of enrichment which will be used to interpret relation symbols and existential quantifications.
{"title":"Enriched concepts of regular logic","authors":"Jiří Rosický","doi":"arxiv-2406.12617","DOIUrl":"https://doi.org/arxiv-2406.12617","url":null,"abstract":"Building on our previous work on enriched universal algebra, we define a\u0000notion of enriched language consisting of function and relation symbols whose\u0000arities are objects of the base of enrichment. In this context, we construct\u0000atomic formulas and define the regular fragment of enriched logic by taking\u0000conjunctions and existential quantifications of those. We then characterize\u0000enriched categories of models of regular theories as enriched injectivity\u0000classes in the enriched category of structures. These notions rely on the\u0000choice of a factorization system on the base of enrichment which will be used\u0000to interpret relation symbols and existential quantifications.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505652","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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