We study homotopy theory of the category of spectral sequences with respect to the class of weak equivalences given by maps which are quasi-isomorphisms on a fixed page. We introduce the category of extended spectral sequences and show that this is bicomplete by analysis of a certain linear presheaf category modelled on discs. We endow the category of extended spectral sequences with various model category structures, restricting to give the almost Brown category structures on spectral sequences of our earlier work. One of these has the property that spectral sequences is a homotopically full subcategory. By results of Meier, this exhibits the category of spectral sequences as a fibrant object in the Barwick-Kan model structure on relative categories, that is, it gives a model for an infinity category of spectral sequences. We also use the presheaf approach to define two d'ecalage functors on spectral sequences, left and right adjoint to a shift functor, thereby clarifying prior use of the term d'ecalage in connection with spectral sequences.
{"title":"Spectral sequences via linear presheaves","authors":"Muriel Livernet, Sarah Whitehouse","doi":"arxiv-2406.02777","DOIUrl":"https://doi.org/arxiv-2406.02777","url":null,"abstract":"We study homotopy theory of the category of spectral sequences with respect\u0000to the class of weak equivalences given by maps which are quasi-isomorphisms on\u0000a fixed page. We introduce the category of extended spectral sequences and show\u0000that this is bicomplete by analysis of a certain linear presheaf category\u0000modelled on discs. We endow the category of extended spectral sequences with\u0000various model category structures, restricting to give the almost Brown\u0000category structures on spectral sequences of our earlier work. One of these has\u0000the property that spectral sequences is a homotopically full subcategory. By\u0000results of Meier, this exhibits the category of spectral sequences as a fibrant\u0000object in the Barwick-Kan model structure on relative categories, that is, it\u0000gives a model for an infinity category of spectral sequences. We also use the\u0000presheaf approach to define two d'ecalage functors on spectral sequences, left\u0000and right adjoint to a shift functor, thereby clarifying prior use of the term\u0000d'ecalage in connection with spectral sequences.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551891","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 observe that an enriched right adjoint functor between model categories which preserves acyclic fibrations and fibrant objects is quite generically a right Quillen functor.
我们观察到，模型范畴之间保留非循环纤化和纤化对象的充实的右邻接函子，一般来说就是右奎伦函子。
{"title":"A simple characterization of Quillen adjunctions","authors":"Victor Carmona","doi":"arxiv-2406.02194","DOIUrl":"https://doi.org/arxiv-2406.02194","url":null,"abstract":"We observe that an enriched right adjoint functor between model categories\u0000which preserves acyclic fibrations and fibrant objects is quite generically a\u0000right Quillen functor.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252373","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}
For an internal category $mathbb{C}$ in a cartesian category $mathcal{C}$ we define, naturally in objects $X$ of $mathcal{C}$, $Prin_{mathbb{C}}(X)$. This is a category whose objects are principal $c mathbb{C}$-bundles over $X$ and whose morphisms are principal $c(mathbb{C}^{uparrow})$-bundles. Here $c(_)$ denotes taking the core groupoid of a category (same objects but only isomorphisms as morphisms) and $mathbb{C}^{uparrow}$ is the arrow category of $mathbb{C}$ (objects morphisms, morphisms commuting squares). We show that $X mapsto Prin_{mathbb{C}}(X)$ is a stack of categories and call stacks of this sort lax-geometric. We then provide two sufficient conditions for a stack to be lax-geometric and use them to prove that the pseudo-functor $X mapsto mathbf{LK}_{Sh(X)}$ on the category of locales $mathbf{Loc}$ is a lax-geometric stack. Here $mathbf{LK}_{Sh(X)}$ is the category of locally compact locales in the topos of sheaves over $X$, $Sh(X)$. Therefore there exists a localic category $mathbb{C}_{mathbf{LK}}$ such that $mathbf{LK}_{Sh(X)} simeq Prin_{mathbb{C}_{mathbf{LK}}}(X)$ naturally for every locale $X$. We then show how this can be used to give a new localic characterisation of the Axiom of Infinity.
{"title":"A classifying localic category for locally compact locales with application to the Axiom of Infinity (poster)","authors":"Christopher Francis Townsend","doi":"arxiv-2406.01573","DOIUrl":"https://doi.org/arxiv-2406.01573","url":null,"abstract":"For an internal category $mathbb{C}$ in a cartesian category $mathcal{C}$\u0000we define, naturally in objects $X$ of $mathcal{C}$, $Prin_{mathbb{C}}(X)$.\u0000This is a category whose objects are principal $c mathbb{C}$-bundles over $X$\u0000and whose morphisms are principal $c(mathbb{C}^{uparrow})$-bundles. Here\u0000$c(_)$ denotes taking the core groupoid of a category (same objects but only\u0000isomorphisms as morphisms) and $mathbb{C}^{uparrow}$ is the arrow category of\u0000$mathbb{C}$ (objects morphisms, morphisms commuting squares). We show that $X\u0000mapsto Prin_{mathbb{C}}(X)$ is a stack of categories and call stacks of this\u0000sort lax-geometric. We then provide two sufficient conditions for a stack to be\u0000lax-geometric and use them to prove that the pseudo-functor $X mapsto\u0000mathbf{LK}_{Sh(X)}$ on the category of locales $mathbf{Loc}$ is a\u0000lax-geometric stack. Here $mathbf{LK}_{Sh(X)}$ is the category of locally\u0000compact locales in the topos of sheaves over $X$, $Sh(X)$. Therefore there\u0000exists a localic category $mathbb{C}_{mathbf{LK}}$ such that\u0000$mathbf{LK}_{Sh(X)} simeq Prin_{mathbb{C}_{mathbf{LK}}}(X)$ naturally for\u0000every locale $X$. We then show how this can be used to give a new localic characterisation of\u0000the Axiom of Infinity.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141253202","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 study the strict refinement property for connected partial ordersalso known as Hashimoto's Theorem. This property implies that any isomorphismbetween products of irreducible structures is determined is uniquely determinedas a product of isomorphisms between the factors. This refinement implies asort of smallest possible decomposition for such structures. After a brief recallof the necessary notion we prove that Hashimoto's theorem can be extendedto connected loop-free categories, i.e. categories with no non-trivial morphismsendomorphisms. A special case of such categories is the category of connectedcomponents, for concurrent programs without loops.
{"title":"Strict refinement property of connected loop-free categories","authors":"Aly-Bora UlusoyCosynus, Emmanuel HaucourtCosynus","doi":"arxiv-2406.01106","DOIUrl":"https://doi.org/arxiv-2406.01106","url":null,"abstract":"In this paper we study the strict refinement property for connected partial\u0000ordersalso known as Hashimoto's Theorem. This property implies that any\u0000isomorphismbetween products of irreducible structures is determined is uniquely\u0000determinedas a product of isomorphisms between the factors. This refinement\u0000implies asort of smallest possible decomposition for such structures. After a\u0000brief recallof the necessary notion we prove that Hashimoto's theorem can be\u0000extendedto connected loop-free categories, i.e. categories with no non-trivial\u0000morphismsendomorphisms. A special case of such categories is the category of\u0000connectedcomponents, for concurrent programs without loops.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"307 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252810","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 define groupoidal and $(n+k)$-truncated $n$-quasi-categories, which are the translation to the world of $n$-quasi-categories of groupoidal and truncated $(infty, n)$-$Theta$-spaces defined by Rezk. We show that these objects are the fibrant objects of model structures on the category of presheaves on $Theta_n$ obtained by localisation of Ara's model structure for $n$-quasi-categories. Furthermore, we prove that the inclusion $Delta to Theta_n$ induces a Quillen equivalence between the model structure for groupoidal (resp. and $n$-truncated) $n$-quasi-categories and the Kan-Quillen model structure for spaces (resp. homotopy $n$-types) on simplicial sets. To get to these results, we also construct a cylinder object for $n$-quasi-categories.
{"title":"Groupoidal and truncated $n$-quasi-categories","authors":"Victor Brittes","doi":"arxiv-2406.01490","DOIUrl":"https://doi.org/arxiv-2406.01490","url":null,"abstract":"We define groupoidal and $(n+k)$-truncated $n$-quasi-categories, which are\u0000the translation to the world of $n$-quasi-categories of groupoidal and\u0000truncated $(infty, n)$-$Theta$-spaces defined by Rezk. We show that these\u0000objects are the fibrant objects of model structures on the category of\u0000presheaves on $Theta_n$ obtained by localisation of Ara's model structure for\u0000$n$-quasi-categories. Furthermore, we prove that the inclusion $Delta to\u0000Theta_n$ induces a Quillen equivalence between the model structure for\u0000groupoidal (resp. and $n$-truncated) $n$-quasi-categories and the Kan-Quillen\u0000model structure for spaces (resp. homotopy $n$-types) on simplicial sets. To\u0000get to these results, we also construct a cylinder object for\u0000$n$-quasi-categories.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252703","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 $(mathcal{T}',mathcal{T},mathcal{T}'')$ be a recollement of triangulated categories. Two complete ideal cotorsion pairs in $mathcal{T}'$ and $mathcal{T}''$ can be induced by a complete ideal cotorsion pair in $mathcal{T}$. If $(mathcal{I},mathcal{I}^perp )$ and $(mathcal{J},mathcal{J}^perp)$ are two complete ideal cotorsion pair in triangulated category, then $(mathcal{I}capmathcal{J},langlemathcal{I}^perp,mathcal{J}^perprangle)$ is also a complete ideal cotorsion pair. In this way, a series of ideal cotorsion pairs in $mathcal{T}$ can be induced by two ideal cotorsion pairs in $mathcal{T}'$ and $mathcal{T}''$.
{"title":"Construct ideal cotorsion pairs by recollement of triangulated categories","authors":"Qikai Wang, Haiyan Zhu","doi":"arxiv-2406.00417","DOIUrl":"https://doi.org/arxiv-2406.00417","url":null,"abstract":"Let $(mathcal{T}',mathcal{T},mathcal{T}'')$ be a recollement of\u0000triangulated categories. Two complete ideal cotorsion pairs in $mathcal{T}'$\u0000and $mathcal{T}''$ can be induced by a complete ideal cotorsion pair in\u0000$mathcal{T}$. If $(mathcal{I},mathcal{I}^perp )$ and\u0000$(mathcal{J},mathcal{J}^perp)$ are two complete ideal cotorsion pair in\u0000triangulated category, then\u0000$(mathcal{I}capmathcal{J},langlemathcal{I}^perp,mathcal{J}^perprangle)$\u0000is also a complete ideal cotorsion pair. In this way, a series of ideal\u0000cotorsion pairs in $mathcal{T}$ can be induced by two ideal cotorsion pairs in\u0000$mathcal{T}'$ and $mathcal{T}''$.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252620","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 give an example of duoidal $infty$-categories. We introduce map $mathcal{O}$-monoidales in an $mathcal{O}$-monoidal $(infty,2)$-category for an $infty$-operad $mathcal{O}^{otimes}$. We show that the endomorphism mapping $infty$-category of a map $mathcal{O}$-monoidale is a coCartesian $(Delta^{rm op},mathcal{O})$-duoidal $infty$-category. After that, we introduce a convolution product on the mapping $infty$-category from an $mathcal{O}$-comonoidale to an $mathcal{O}$-monoidale. We show that the $mathcal{O}$-monoidal structure on the duoidal endomorphism mapping $infty$-category of a map $mathcal{O}$-monoidale is equivalent to the convolution product on the mapping $infty$-category from the dual $mathcal{O}$-comonoidale to the map $mathcal{O}$-monoidale.
{"title":"Map monoidales and duoidal $infty$-categories","authors":"Takeshi Torii","doi":"arxiv-2406.00223","DOIUrl":"https://doi.org/arxiv-2406.00223","url":null,"abstract":"In this paper we give an example of duoidal $infty$-categories. We introduce\u0000map $mathcal{O}$-monoidales in an $mathcal{O}$-monoidal $(infty,2)$-category\u0000for an $infty$-operad $mathcal{O}^{otimes}$. We show that the endomorphism\u0000mapping $infty$-category of a map $mathcal{O}$-monoidale is a coCartesian\u0000$(Delta^{rm op},mathcal{O})$-duoidal $infty$-category. After that, we\u0000introduce a convolution product on the mapping $infty$-category from an\u0000$mathcal{O}$-comonoidale to an $mathcal{O}$-monoidale. We show that the\u0000$mathcal{O}$-monoidal structure on the duoidal endomorphism mapping\u0000$infty$-category of a map $mathcal{O}$-monoidale is equivalent to the\u0000convolution product on the mapping $infty$-category from the dual\u0000$mathcal{O}$-comonoidale to the map $mathcal{O}$-monoidale.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141253085","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 source of difficulty in profinite homotopy theory is that the profinite completion functor does not preserve finite products. In this note, we provide a new, checkable criterion on prospaces $X$ and $Y$ that guarantees that the profinite completion of $Xtimes Y$ agrees with the product of the profinite completions of $X$ and $Y$. Using this criterion, we show that profinite completion preserves products of '{e}tale homotopy types of qcqs schemes. This fills a gap in Chough's proof of the K"{u}nneth formula for the '{e}tale homotopy type of a product of proper schemes over a separably closed field.
{"title":"Profinite completions of products","authors":"Peter J. Haine","doi":"arxiv-2406.00136","DOIUrl":"https://doi.org/arxiv-2406.00136","url":null,"abstract":"A source of difficulty in profinite homotopy theory is that the profinite\u0000completion functor does not preserve finite products. In this note, we provide\u0000a new, checkable criterion on prospaces $X$ and $Y$ that guarantees that the\u0000profinite completion of $Xtimes Y$ agrees with the product of the profinite\u0000completions of $X$ and $Y$. Using this criterion, we show that profinite\u0000completion preserves products of '{e}tale homotopy types of qcqs schemes. This\u0000fills a gap in Chough's proof of the K\"{u}nneth formula for the '{e}tale\u0000homotopy type of a product of proper schemes over a separably closed field.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252621","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}
Cartesian differential categories provide a categorical framework for multivariable differential calculus and also the categorical semantics of the differential $lambda$-calculus. Taylor series expansion is an important concept for both differential calculus and the differential $lambda$-calculus. In differential calculus, a function is equal to its Taylor series if its sequence of Taylor polynomials converges to the function in the analytic sense. On the other hand, for the differential $lambda$-calculus, one works in a setting with an appropriate notion of algebraic infinite sums to formalize Taylor series expansion. In this paper, we provide a formal theory of Taylor series in an arbitrary Cartesian differential category without the need for converging limits or infinite sums. We begin by developing the notion of Taylor polynomials of maps in a Cartesian differential category and then show how comparing Taylor polynomials of maps induces an ultrapseudometric on the homsets. We say that a Cartesian differential category is Taylor if maps are entirely determined by their Taylor polynomials. The main results of this paper are that in a Taylor Cartesian differential category, the induced ultrapseudometrics are ultrametrics and that for every map $f$, its Taylor series converges to $f$ with respect to this ultrametric. This framework recaptures both Taylor series expansion in differential calculus via analytic methods and in categorical models of the differential $lambda$-calculus (or Differential Linear Logic) via infinite sums.
{"title":"An Ultrametric for Cartesian Differential Categories for Taylor Series Convergence","authors":"Jean-Simon Pacaud Lemay","doi":"arxiv-2405.19474","DOIUrl":"https://doi.org/arxiv-2405.19474","url":null,"abstract":"Cartesian differential categories provide a categorical framework for\u0000multivariable differential calculus and also the categorical semantics of the\u0000differential $lambda$-calculus. Taylor series expansion is an important\u0000concept for both differential calculus and the differential $lambda$-calculus.\u0000In differential calculus, a function is equal to its Taylor series if its\u0000sequence of Taylor polynomials converges to the function in the analytic sense.\u0000On the other hand, for the differential $lambda$-calculus, one works in a\u0000setting with an appropriate notion of algebraic infinite sums to formalize\u0000Taylor series expansion. In this paper, we provide a formal theory of Taylor\u0000series in an arbitrary Cartesian differential category without the need for\u0000converging limits or infinite sums. We begin by developing the notion of Taylor\u0000polynomials of maps in a Cartesian differential category and then show how\u0000comparing Taylor polynomials of maps induces an ultrapseudometric on the\u0000homsets. We say that a Cartesian differential category is Taylor if maps are\u0000entirely determined by their Taylor polynomials. The main results of this paper\u0000are that in a Taylor Cartesian differential category, the induced\u0000ultrapseudometrics are ultrametrics and that for every map $f$, its Taylor\u0000series converges to $f$ with respect to this ultrametric. This framework\u0000recaptures both Taylor series expansion in differential calculus via analytic\u0000methods and in categorical models of the differential $lambda$-calculus (or\u0000Differential Linear Logic) via infinite sums.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141189274","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 radically synthetic foundation for smooth geometry formulated in [Law11] postulates a space T with the property that it has a unique point and, out of the monoid T^T of endomorphisms, it extracts a submonoid R which, in many cases, is the (commutative) multiplication of a rig structure. The rig R is said to be bi-directional if its subobject of invertible elements has two connected components. In this case, R may be equipped with a pre-order compatible with the rig structure. We adjust the construction of `well-adapted' models of Synthetic Differential Geometry in order to build the first pre-cohesive toposes with a bi-directional R. We also show that, in one of these pre-cohesive variants, the pre-order on R, derived radically synthetically from bi-directionality, coincides with that defined in the original model.
Law11]中提出的光滑几何的根本合成基础假定了一个空间 T，其性质是它有一个唯一的点，并且从内态性的单元 T^T 中提取出一个子单元 R，在许多情况下，这个子单元 R 是一个 rig 结构的（交换）乘法。如果 R 的可逆元素子对象有两个相连的成分，那么 R 可以说是双向的。在这种情况下，R 可以配备一个与 rig 结构兼容的前序。我们调整了合成微分几何 "井适应 "模型的构造，以建立具有双向 R 的第一个预内聚拓扑。我们还证明，在其中一个预内聚变体中，从双向性根本上合成导出的 R 上的前序与原始模型中定义的前序重合。
{"title":"Bi-directional models of `radically synthetic' differential geometry","authors":"Matías Menni","doi":"arxiv-2405.17748","DOIUrl":"https://doi.org/arxiv-2405.17748","url":null,"abstract":"The radically synthetic foundation for smooth geometry formulated in [Law11]\u0000postulates a space T with the property that it has a unique point and, out of\u0000the monoid T^T of endomorphisms, it extracts a submonoid R which, in many\u0000cases, is the (commutative) multiplication of a rig structure. The rig R is\u0000said to be bi-directional if its subobject of invertible elements has two\u0000connected components. In this case, R may be equipped with a pre-order\u0000compatible with the rig structure. We adjust the construction of `well-adapted'\u0000models of Synthetic Differential Geometry in order to build the first\u0000pre-cohesive toposes with a bi-directional R. We also show that, in one of\u0000these pre-cohesive variants, the pre-order on R, derived radically\u0000synthetically from bi-directionality, coincides with that defined in the\u0000original model.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141166779","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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