We prove that the category of topological spaces and open maps does not have binary products, thus resolving the Esakia-Janelidze problem in the negative. We also prove that the categories of complete Heyting algebras and complete closure algebras do not have binary coproducts.
{"title":"The category of topological spaces and open maps does not have products","authors":"Guram Bezhanishvili, Andre Kornell","doi":"arxiv-2407.13951","DOIUrl":"https://doi.org/arxiv-2407.13951","url":null,"abstract":"We prove that the category of topological spaces and open maps does not have\u0000binary products, thus resolving the Esakia-Janelidze problem in the negative.\u0000We also prove that the categories of complete Heyting algebras and complete\u0000closure algebras do not have binary coproducts.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141740825","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 show that the Grothendieck abelian category $Xoperatorname{mathsf{--Qcoh}}$ of quasi-coherent sheaves on a quasi-compact semi-separated scheme $X$ satisfies the Roos axiom $mathrm{AB}4^*$-$n$: the derived functors of infinite product have finite homological dimension in $Xoperatorname{mathsf{--Qcoh}}$, not exceeding the number $n$ of open subschemes in an affine open covering of $X$. The hereditary complete cotorsion pair (very flat quasi-coherent sheaves, contraadjusted quasi-coherent sheaves) in the abelian category $Xoperatorname{mathsf{--Qcoh}}$ plays the key role in our arguments. Simply put, a suitable very flat quasi-coherent sheaf (or alternatively, a suitable direct sum of locally countably presented flat quasi-coherent sheaves) on $X$ is a generator of finite projective dimension for the abelian category $Xoperatorname{mathsf{--Qcoh}}$.
{"title":"Roos axiom holds for quasi-coherent sheaves","authors":"Leonid Positselski","doi":"arxiv-2407.13651","DOIUrl":"https://doi.org/arxiv-2407.13651","url":null,"abstract":"We show that the Grothendieck abelian category\u0000$Xoperatorname{mathsf{--Qcoh}}$ of quasi-coherent sheaves on a quasi-compact\u0000semi-separated scheme $X$ satisfies the Roos axiom $mathrm{AB}4^*$-$n$: the\u0000derived functors of infinite product have finite homological dimension in\u0000$Xoperatorname{mathsf{--Qcoh}}$, not exceeding the number $n$ of open\u0000subschemes in an affine open covering of $X$. The hereditary complete cotorsion\u0000pair (very flat quasi-coherent sheaves, contraadjusted quasi-coherent sheaves)\u0000in the abelian category $Xoperatorname{mathsf{--Qcoh}}$ plays the key role in\u0000our arguments. Simply put, a suitable very flat quasi-coherent sheaf (or\u0000alternatively, a suitable direct sum of locally countably presented flat\u0000quasi-coherent sheaves) on $X$ is a generator of finite projective dimension\u0000for the abelian category $Xoperatorname{mathsf{--Qcoh}}$.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141746005","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}
Interactions between derivatives and fixpoints have many important applications in both computer science and mathematics. In this paper, we provide a categorical framework to combine fixpoints with derivatives by studying Cartesian differential categories with a fixpoint operator. We introduce an additional axiom relating the derivative of a fixpoint with the fixpoint of the derivative. We show how the standard examples of Cartesian differential categories where we can compute fixpoints provide canonical models of this notion. We also consider when the fixpoint operator is a Conway operator, or when the underlying category is closed. As an application, we show how this framework is a suitable setting to formalize the Newton-Raphson optimization for fast approximation of fixpoints and extend it to higher order languages.
{"title":"Combining fixpoint and differentiation theory","authors":"Zeinab Galal, Jean-Simon Pacaud Lemay","doi":"arxiv-2407.12691","DOIUrl":"https://doi.org/arxiv-2407.12691","url":null,"abstract":"Interactions between derivatives and fixpoints have many important\u0000applications in both computer science and mathematics. In this paper, we\u0000provide a categorical framework to combine fixpoints with derivatives by\u0000studying Cartesian differential categories with a fixpoint operator. We\u0000introduce an additional axiom relating the derivative of a fixpoint with the\u0000fixpoint of the derivative. We show how the standard examples of Cartesian\u0000differential categories where we can compute fixpoints provide canonical models\u0000of this notion. We also consider when the fixpoint operator is a Conway\u0000operator, or when the underlying category is closed. As an application, we show\u0000how this framework is a suitable setting to formalize the Newton-Raphson\u0000optimization for fast approximation of fixpoints and extend it to higher order\u0000languages.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141740717","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 exhibit the proximity frames and proximity homomorphisms as a Kleisli category of a comonad whose underlying functor takes a proximity frame to its frame of round ideals. This construction is known in the literature as {em stable compactification} (cite{BezHar2}). We show that the frame of round ideals naturally carries with it two proximities of interest from which two comonads are induced.
{"title":"A correspondence between proximity homomorphisms and certain frame maps via a comonad","authors":"Ando Razafindrakoto","doi":"arxiv-2407.11528","DOIUrl":"https://doi.org/arxiv-2407.11528","url":null,"abstract":"We exhibit the proximity frames and proximity homomorphisms as a Kleisli\u0000category of a comonad whose underlying functor takes a proximity frame to its\u0000frame of round ideals. This construction is known in the literature as {em\u0000stable compactification} (cite{BezHar2}). We show that the frame of round\u0000ideals naturally carries with it two proximities of interest from which two\u0000comonads are induced.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718559","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 generalize a previous result of Stevenson to the category of dendroidal sets, yielding the right cancellation property of dendroidal inner anodynes within the class of normal monomorphisms. As an application of this property, we show how to construct a symmetric monoidal $infty$-category $mathsf{Env}(X)^otimes$ from a dendroidal $infty$-operad $X$, in a way that generalizes the symmetric monoidal envelope of a coloured operad.
{"title":"The right cancellation property for certain classes of dendroidal anodynes","authors":"Miguel Barata","doi":"arxiv-2407.18959","DOIUrl":"https://doi.org/arxiv-2407.18959","url":null,"abstract":"We generalize a previous result of Stevenson to the category of dendroidal\u0000sets, yielding the right cancellation property of dendroidal inner anodynes\u0000within the class of normal monomorphisms. As an application of this property,\u0000we show how to construct a symmetric monoidal $infty$-category\u0000$mathsf{Env}(X)^otimes$ from a dendroidal $infty$-operad $X$, in a way that\u0000generalizes the symmetric monoidal envelope of a coloured operad.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869302","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}
Daniel Gratzer, Jonathan Weinberger, Ulrik Buchholtz
Simplicial type theory extends homotopy type theory with a directed path type which internalizes the notion of a homomorphism within a type. This concept has significant applications both within mathematics -- where it allows for synthetic (higher) category theory -- and programming languages -- where it leads to a directed version of the structure identity principle. In this work, we construct the first types in simplicial type theory with non-trivial homomorphisms. We extend simplicial type theory with modalities and new reasoning principles to obtain triangulated type theory in order to construct the universe of discrete types $mathcal{S}$. We prove that homomorphisms in this type correspond to ordinary functions of types i.e., that $mathcal{S}$ is directed univalent. The construction of $mathcal{S}$ is foundational for both of the aforementioned applications of simplicial type theory. We are able to define several crucial examples of categories and to recover important results from category theory. Using $mathcal{S}$, we are also able to define various types whose usage is guaranteed to be functorial. These provide the first complete examples of the proposed directed structure identity principle.
{"title":"Directed univalence in simplicial homotopy type theory","authors":"Daniel Gratzer, Jonathan Weinberger, Ulrik Buchholtz","doi":"arxiv-2407.09146","DOIUrl":"https://doi.org/arxiv-2407.09146","url":null,"abstract":"Simplicial type theory extends homotopy type theory with a directed path type\u0000which internalizes the notion of a homomorphism within a type. This concept has\u0000significant applications both within mathematics -- where it allows for\u0000synthetic (higher) category theory -- and programming languages -- where it\u0000leads to a directed version of the structure identity principle. In this work,\u0000we construct the first types in simplicial type theory with non-trivial\u0000homomorphisms. We extend simplicial type theory with modalities and new\u0000reasoning principles to obtain triangulated type theory in order to construct\u0000the universe of discrete types $mathcal{S}$. We prove that homomorphisms in\u0000this type correspond to ordinary functions of types i.e., that $mathcal{S}$ is\u0000directed univalent. The construction of $mathcal{S}$ is foundational for both\u0000of the aforementioned applications of simplicial type theory. We are able to\u0000define several crucial examples of categories and to recover important results\u0000from category theory. Using $mathcal{S}$, we are also able to define various\u0000types whose usage is guaranteed to be functorial. These provide the first\u0000complete examples of the proposed directed structure identity principle.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718560","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 a characterization of effective descent morphisms in the lax comma category $mathsf{Ord}//X$ when $X$ is a locally complete ordered set with a bottom element.
{"title":"Effective descent morphisms of ordered families","authors":"Maria Manuel Clementino, Rui Prezado","doi":"arxiv-2407.08573","DOIUrl":"https://doi.org/arxiv-2407.08573","url":null,"abstract":"We present a characterization of effective descent morphisms in the lax comma\u0000category $mathsf{Ord}//X$ when $X$ is a locally complete ordered set with a\u0000bottom element.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141614654","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 number of results to the general effect that, under obviously necessary numerical and determinant constraints, "most" morphisms between fixed bundles on a complex elliptic curve produce (co)kernels which can either be specified beforehand or else meet various rigidity constraints. Examples include: (a) for indecomposable $mathcal{E}$ and $mathcal{E'}$ with slopes and ranks increasing strictly in that order the space of monomorphisms whose cokernel is semistable and maximally rigid (i.e. has minimal-dimensional automorphism group) is open dense; (b) for indecomposable $mathcal{K}$, $mathcal{E}$ and stable $mathcal{F}$ with slopes increasing strictly in that order and ranks and determinants satisfying the obvious additivity constraints the space of embeddings $mathcal{K}to mathcal{E}$ whose cokernel is isomorphic to $mathcal{F}$ is open dense; (c) the obvious mirror images of these results; (d) generalizations weakening indecomposability to semistability + maximal rigidity; (e) various examples illustrating the necessity of the assorted assumptions.
{"title":"Bundle-extension inverse problems over elliptic curves","authors":"Alexandru Chirvasitu","doi":"arxiv-2407.07344","DOIUrl":"https://doi.org/arxiv-2407.07344","url":null,"abstract":"We prove a number of results to the general effect that, under obviously\u0000necessary numerical and determinant constraints, \"most\" morphisms between fixed\u0000bundles on a complex elliptic curve produce (co)kernels which can either be\u0000specified beforehand or else meet various rigidity constraints. Examples\u0000include: (a) for indecomposable $mathcal{E}$ and $mathcal{E'}$ with slopes\u0000and ranks increasing strictly in that order the space of monomorphisms whose\u0000cokernel is semistable and maximally rigid (i.e. has minimal-dimensional\u0000automorphism group) is open dense; (b) for indecomposable $mathcal{K}$,\u0000$mathcal{E}$ and stable $mathcal{F}$ with slopes increasing strictly in that\u0000order and ranks and determinants satisfying the obvious additivity constraints\u0000the space of embeddings $mathcal{K}to mathcal{E}$ whose cokernel is\u0000isomorphic to $mathcal{F}$ is open dense; (c) the obvious mirror images of\u0000these results; (d) generalizations weakening indecomposability to semistability\u0000+ maximal rigidity; (e) various examples illustrating the necessity of the\u0000assorted assumptions.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141588198","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}
A tangent category is a categorical abstraction of the tangent bundle construction for smooth manifolds. In that context, Cockett and Cruttwell develop the notion of differential bundle which, by work of MacAdam, generalizes the notion of smooth vector bundle to the abstract setting. Here we provide a new characterization of those differential bundles and show that, up to isomorphism, a differential bundle is determined by its projection map and zero section. We show how these results can be used to quickly identify differential bundles in various tangent categories.
{"title":"A characterization of differential bundles in tangent categories","authors":"Michael Ching","doi":"arxiv-2407.06515","DOIUrl":"https://doi.org/arxiv-2407.06515","url":null,"abstract":"A tangent category is a categorical abstraction of the tangent bundle\u0000construction for smooth manifolds. In that context, Cockett and Cruttwell\u0000develop the notion of differential bundle which, by work of MacAdam,\u0000generalizes the notion of smooth vector bundle to the abstract setting. Here we\u0000provide a new characterization of those differential bundles and show that, up\u0000to isomorphism, a differential bundle is determined by its projection map and\u0000zero section. We show how these results can be used to quickly identify\u0000differential bundles in various tangent categories.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573759","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}
Diagrammatic sets are presheaves on a rich category of shapes, whose definition is motivated by combinatorial topology and higher-dimensional diagram rewriting. These shapes include representatives of oriented simplices, cubes, and positive opetopes, and are stable under operations including Gray products, joins, suspensions, and duals. We exhibit a cofibrantly generated model structure on diagrammatic sets, as well as two separate Quillen equivalences with the classical model structure on simplicial sets. We construct explicit sets of generating cofibrations and acyclic cofibrations, and prove that the model structure is monoidal with the Gray product of diagrammatic sets.
{"title":"Diagrammatic sets as a model of homotopy types","authors":"Clémence Chanavat, Amar Hadzihasanovic","doi":"arxiv-2407.06285","DOIUrl":"https://doi.org/arxiv-2407.06285","url":null,"abstract":"Diagrammatic sets are presheaves on a rich category of shapes, whose\u0000definition is motivated by combinatorial topology and higher-dimensional\u0000diagram rewriting. These shapes include representatives of oriented simplices,\u0000cubes, and positive opetopes, and are stable under operations including Gray\u0000products, joins, suspensions, and duals. We exhibit a cofibrantly generated\u0000model structure on diagrammatic sets, as well as two separate Quillen\u0000equivalences with the classical model structure on simplicial sets. We\u0000construct explicit sets of generating cofibrations and acyclic cofibrations,\u0000and prove that the model structure is monoidal with the Gray product of\u0000diagrammatic sets.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573553","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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