Jiří Rosický
We give a purely category-theoretic proof of the result of Makkai and Paré saying that the category $\bf Lin$ of linearly ordered sets and order preserving injective mappings is a minimal finitely accessible category. We also discuss the existence of a minimal $\aleph_1$-accessible category.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We show that any directed colimit of acessible categories and accessible full embeddings is accessible and, assuming the existence of arbitrarily large strongly compact cardinals, any directed colimit of acessible categories and accessible embeddings is accessible.
We compare abstract elementary classes of Shelah with accessible categories having directed colimits.
We generalize the concepts of locally presentable and accessible categories. Our framework includes such categories as small presheaves over large categories and ind-categories. This generalization is intended for applications in the abstract homotopy theory.
We prove that a weak factorization system on a locally presentable category is accessible if and only if it is small generated in the sense of R. Garner. Moreover, we discuss an analogy of Smith's theorem for accessible model categories.
We provide a new characterization of enriched accessible categories by introducing the two new notions of virtual reflectivity and virtual orthogonality as a generalization of the usual reflectivity and orthogonality conditions for locally presentable categories. The word virtual refers to the fact that the reflectivity and orthogonality conditions are given in the free completion of the $\mathcal V$-category involved under small limits, instead of the $\mathcal V$-category itself. In this way we hope to provide a clearer understanding of the theory as well as a useful way of recognizing acces…
Isoregular theories, accessible 2-categories, and free constructions
引入等则理论以表达唯一同构下的存在量化，建立函子语义并证明模型2-范畴具有可访问性和灵活极限，应用于范畴论、代数与逻辑中的多个2-范畴。
Dualities in the theory of accessible categories
本文通过弱可靠权类的概念，将涉及可达范畴的多种已知对偶性统一为一般对偶定理的实例，将其扩展至富化环境，并探讨了由此产生的新对偶性。
We give conditions on a finitary endofunctor of a finitely accessible category to admit a final coalgebra. Our conditions always apply to the case of a finitary endofunctor of a locally finitely presentable (l.f.p.) category and they bring an explicit construction of the final coalgebra in this case. On the other hand, there are interesting examples of final coalgebras beyond the realm of l.f.p. categories to which our results apply. We rely on ideas developed by Tom Leinster for the study of self-similar objects in topology.
The main purpose of this paper is to show that the converse of the known implication weakly action representable implies action accessible is false. In particular we show that both action accessibility, as well as the (at least formally stronger) condition requiring the existence of all normalizers do not imply weakly-action-representability even for varieties. In addition we show that in contrast to both action accessibility and the condition requiring the existence of all normalizers, weakly-action representability is not necessarily inherited by Birkoff subcategories.
Bi-accessible and bipresentable 2-categories
该研究构建适配平坦伪函子形式的二维可达性与可表现性理论，定义双可达、双可表现2-范畴，证明相关对偶定理与函子定理，并验证多类范畴逻辑中的2-范畴为有限双可表现。
Accessible categories with a class of limits
本文通过引入伴随概念，统一刻画了具有指定类极限的可达范畴，并推广了局部可表示范畴的相关定理。
Using full images of accessible functors, we prove some results about combinatorial and accessible model categories. In particular, we give an example of a weak factorization system on a locally presentable category which is not accessible.
On pure monomorphisms and pure epimorphisms in accessible categories
本文研究了可及范畴中纯单射和纯满射的性质，证明了在特定条件下它们的良好行为，并扩展了Quillen精确类的概念。
We generalize the constructions and results of Chapter 10 in Baldwin's "Categoricity" to coherent accessible categories with concrete directed colimits and concrete monomorphisms. In particular, we prove that if any category of this form is categorical in a successor, directed colimits of saturated objects are themselves saturated.
Accessible aspects of 2-category theory
该研究针对范畴结构及其伪映射难构成局部可展示2-范畴的问题，证明足够弱的范畴结构对应的2-范畴具有可达性，并探究了这类2-范畴的柔性极限及其与滤过余极限的相互作用，丰富了2-范畴论的可达性相关结论。
We show that a number of results on abstract elementary classes (AECs) hold in accessible categories with concrete directed colimits. In particular, we prove a generalization of a recent result of Boney on tameness under a large cardinal assumption. We also show that such categories support a robust version of the Ehrenfeucht-Mostowski construction. This analysis has the added benefit of producing a purely language-free characterization of AECs, and highlights the precise role played by the coherence axiom.
In a previous work, we have introduced a weakening of Quillen model categories called weak model categories. They still allow all the usual constructions of model category theory, but are easier to construct and are in some sense better behaved. In this paper we continue to develop their general theory by introducing combinatorial and accessible weak model categories. We give simple necessary and sufficient conditions under which such a weak model category can be extended into a left and/or right semi-model category. As an application, we recover Cisinski-Olschok theory and generalize it to we…
We give a self-contained introduction to accessible categories and how they shed light on both model- and set-theoretic questions. We survey for example recent developments on the study of presentability ranks, a notion of cardinality localized to a given category, as well as stable independence, a generalization of pushouts and model-theoretic forking that may interest mathematicians at large. We give many examples, including recently discovered connections with homotopy theory and homological algebra. We also discuss concrete versions of accessible categories (such as abstract elementary cla…
Flatness in finitely accessible additive categories
本文在有限可访问加法范畴中推广研究平坦性，刻画范畴的预阿贝尔/阿贝尔性质及具有足够平坦与投射对象的条件，并揭示平坦对象类在纯子对象下的封闭性与直并分解之间的关系。
Flat model structures for accessible exact categories
本文为可达正合范畴上的链复形构造模型结构，证明在闭对称幺半、局部可展示且具有足够多平坦对象的正合范畴中，平坦余挠对可诱导正合模型结构，并进一步揭示其在有理充实下构成同伦代数语境且具有 Koszul 对偶定理。
We extend the framework of combinatorial model categories, so that the category of small presheaves over large indexing categories and ind-categories would be embraced by the new machinery called class-combinatorial model categories. The definition of the new class of model categories is based on the corresponding extension of the theory of locally presentable and accessible categories developed in the companion paper [arXiv:1110.0605], where we introduced the concepts of locally class-presentable and class-accessible categories. In this work we prove that the category of weak equivalences of …
We show that metric abstract elementary classes (mAECs) are, in the sense of [LR] (i.e. arXiv:1404.2528), coherent accessible categories with directed colimits, with concrete $\aleph_1$-directed colimits and concrete monomorphisms. More broadly, we define a notion of $κ$-concrete AEC---an AEC-like category in which only the $κ$-directed colimits need be concrete---and develop the theory of such categories, beginning with a category-theoretic analogue of Shelah's Presentation Theorem and a proof of the existence of an Ehrenfeucht-Mostowski functor in case the category is large. For mAECs in par…
Taming large categories
本文简短概述了过滤余极限与可达范畴的标准机制，并与arXiv:2409.17489中的同伦增强版本平行撰写，无新结果。
Combinatorial model categories were introduced by J. H. Smith as model categories which are locally presentable and cofibrantly generated. He has not published his results yet but proofs of some of them were presented by T. Beke or D. Dugger. We are contributing to this endeavour by proving that weak equivalences in a combinatorial model category form an accessible category. We also present some new results about weak equivalences and cofibrations in combinatorial model categories.
We provide a complete classification of all the possible categoricity spectra, in terms of internal size, that can appear in a large accessible category with directed colimits, assuming the Singular Cardinal Hypothesis ($SCH$), and providing as well explicit threshold cardinals for eventual categoricity. This includes as a particular case the first complete classification of categoricity spectra of abstract elementary classes (AEC's) entirely in $ZFC$. More specifically, we have the following theorem: Let $\mathcal{K}$ be a large $κ$-accessible category with directed colimits. Assume the Singu…
We provide a proof, in $ZFC$, of Shelah's eventual categoricity conjecture for abstract elementary classes (AEC's). Moreover, assuming in addition the Singular Cardinal Hypothesis ($SCH$), we prove a direct generalization to the more general context of accessible categories with directed colimits. If $\mathcal{K}$ is such a category, we show that there is a cardinal $μ$ such that if $\mathcal{K}$ is $λ$-categorical for some $λ\geq μ$ (i.e., it has only one object of internal size $λ$ up to isomorphism), then $\mathcal{K}$ is eventually categorical (i.e., it is $λ'$-categorical for every $λ' \g…
We introduce the notion of $λ$-equivalence and $λ$-embeddings of objects in suitable categories. This notion specializes to $L_{\inftyλ}$-equivalence and $L_{\inftyλ}$-elementary embedding for categories of structures in a language of arity less than $λ$, and interacts well with functors and $λ$-directed colimits. We recover and extend results of Feferman and Eklof on "local functors" without fixing a language in advance. This is convenient for formalizing Lefschetz's principle in algebraic geometry, which was one of the main applications of the work of Eklof.
Notes on limits of accessible categories
本文证明了在特定基数条件下，对 $κ$-可及范畴施加插入子和等化子等范畴论构造仍产生 $κ$-可及范畴，并探讨了各类伪极限对 $κ$-可及性的保持，其动机源于平坦模与平坦拟凝聚层理论。
More on soundness in the enriched context
本文在丰富范畴理论中深化健全性概念，研究局部$Φ$-可表示范畴、$Φ$-可达性与$Ψ$-可达性的关系，并引入$Φ$-元等值理论以刻画$Φ$-元单子的代数。
We prove an adjoint functor theorem in the setting of categories enriched in a monoidal model category $\mathcal V$ admitting certain limits. When $\mathcal V$ is equipped with the trivial model structure this recaptures the enriched version of Freyd's adjoint functor theorem. For non-trivial model structures, we obtain new adjoint functor theorems of a homotopical flavour - in particular, when $\mathcal V$ is the category of simplical sets we obtain a homotopical adjoint functor theorem appropriate to the $\infty$-cosmoi of Riehl and Verity. We also investigate accessibility in the enriched s…
A Quillen model structure is presented by an interacting pair of weak factorization systems. We prove that in the world of locally presentable categories, any weak factorization system with accessible functorial factorizations can be lifted along either a left or a right adjoint. It follows that accessible model structures on locally presentable categories - ones admitting accessible functorial factorizations, a class that includes all combinatorial model structures but others besides - can be lifted along either a left or a right adjoint if and only if an essential "acyclicity" condition hold…
Sketchable infinity categories
本文将草图理论同伦相干地推广到$\infty$-范畴，证明了可展示$\infty$-范畴是极限草图模型，可及$\infty$-范畴是任意草图模型，并为多种$\infty$-范畴给出了具体草图。
The importance of accessible categories has been widely recognized; they can be described as those freely generated in some precise sense by a small set of objects and, because of that, satisfy many good properties. More specifically finitely accessible categories can be characterized as: (a) free cocompletions of small categories under filtered colimits, and (b) categories of flat presheaves on some small category. The equivalence between (a) and (b) is what makes the theory so general and fruitful. Notions of enriched accessibility have also been considered in the literature for various base…
For a Grothendieck category having a noetherian generator, we prove that there are only finitely many minimal atoms. This is a noncommutative analogue of the fact that every noetherian scheme has only finitely many irreducible components. It is also shown that each minimal atom is represented by a compressible object.
We introduce a model category of spaces based on the definable sets of an o-minimal expansion of a real closed field. As a model category, it resembles the category of topological spaces, but its underlying category is a coherent topos. We will show in future work that its cofibrant objects are precisely the "weak polytopes" of Knebusch.
The categories of corings and coalgebras over a ring are locally countably presentable
本文证明任意交换环上的余代数及结合环上的余环范畴具有局部 $\aleph_1$-可展示性，并刻画相应平坦对象范畴的 $\aleph_1$-可达性。
We introduce $μ$-Abstract Elementary Classes ($μ$-AECs) as a broad framework for model theory that includes complete boolean algebras and Dirichlet series, and begin to develop their classification theory. Moreover, we note that $μ$-AECs correspond precisely to accessible categories in which all morphisms are monomorphisms, and begin the process of reconciling these divergent perspectives: not least, the preliminary classification-theoretic results for μ-AECs transfer directly to accessible categories with monomorphisms.
Gabriel-Ulmer duality for topoi and its relation with site presentations
该文研究正则基数下限制到κ-可展示Grothendieck拓扑斯时的Gabriel-Ulmer对偶，刻画了对应表示2-范畴的结构，分析其与场表示的关联并证明了相关反射子双范畴结论。
We introduce and study the Scott adjunction, relating accessible categories with directed colimits to topoi. Our focus is twofold, we study both its applications to formal model theory and its geometric interpretation. From the geometric point of view, we introduce the categorified Isbell duality, relating bounded (possibly large) ionads to topoi. The categorified Isbell duality interacts with the Scott adjunction offering a categorification of the Scott topology over a poset (hence the name). We show that the categorified Isbell duality is idempotent, similarly to its uncategorified version. …
Local categories: a new framework for partiality
本文提出三种新的范畴论框架来处理部分性：局部范畴、部分范畴和包含范畴，并证明了限制范畴与局部范畴、部分范畴与包含范畴在2-等价意义上的等价性，从而提供了四种等价的方式来描述部分性。
We extend Bourke and Garner's idempotent adjunction between monads and pretheories to the framework of $\infty$-categories and we use this to prove many classical results about monads in the $\infty$-categorical framework. Amongst other things, we show that the category of algebras for an accessible monads on a locally presentable $\infty$-category $\mathcal{E}$ is again locally presentable, and that a diagram of accessible monads on a locally presentable $\infty$-category admits a colimit. Our results also provide a new and simpler way to construct and describe monads in terms of theories.
We show that any slightly degenerate weakly group-theoretical fusion category admits a minimal non-degenerate extension. Let $d$ be a positive square-free integer, given a weakly group-theoretical non-degenerate fusion category $\mathcal{C}$, assume that $\text{FPdim}(\mathcal{C})=nd$ and $(n,d)=1$. If $(\text{FPdim}(X)^2,d)=1$ for all simple objects $X$ of $\mathcal{C}$, then we show that $\mathcal{C}$ contains a non-degenerate fusion subcategory $\mathcal{C}(\mathbb{Z}_d,q)$. In particular, we obtain that integral fusion categories of FP-dimensions $p^md$ such that $\mathcal{C}'\subseteq \te…
We prove, without set theoretic assumptions, that every locally presentable category C endowed with a tractable cofibrantly generated class of cofibrations has a unique minimal (or left induced) Quillen model structure. More generally, for any set S of arrows in C we construct the minimal model structure on C with the prescribed cofibrations and making all the arrows of S weak equivalences. We describe its class of equivalences as the "smallest Cisinski localizer containing S". Our proof rely on a careful use of the fat small object argument and J.~Lurie's "good colimits" technology and on the…
Based on a weak action of a finite group J on a finite group G, we present a geometric construction of J-equivariant Dijkgraaf-Witten theory as an extended topological field theory. The construction yields an explicitly accessible class of equivariant modular tensor categories. For the action of a group J on a group G, the category is described as the representation category of a J-ribbon algebra that generalizes the Drinfel'd double of the finite group G.
We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopenka's principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Levy hierarchy. For example, the statement that, for a class S of morphisms in a locally presentable category C of structures, the orthogonal class of objects is a small-orthogonality class (hence reflective) is provable in ZFC if S is Σ_1, while it follows from the …
Accessibility and presentability in 2-categories
本研究在配备KZ语境的2-范畴中定义可达与可表现对象，统一处理类Gabriel-Ulmer定理，提出KZ语境的Gabriel-Ulmer包络概念以构建对偶，给出低维范畴论示例并展望高维与同伦论方向。
Locally coherent exact categories
本文提出局部相干正合范畴的概念，证明小幂等完备加性范畴上的正合结构可唯一扩展为 ind-objects 范畴上的局部相干正合结构，并推导出两个周期性定理。
Regular logic is the fragment of first order logic generated by $=$, $\top$, $\wedge$, and $\exists$. A key feature of this logic is that it is the minimal fragment required to express composition of binary relations; another is that it is the internal logic of regular categories. The link between these two facts is that in any regular category, one may construct a notion of binary relation using jointly-monic spans; this results in what is known as the bicategory of relations of the regular category. In this paper we provide a direct axiomatization of bicategories of relations, which we term …
Iyama-Solberg correspondence for exact dg categories
本文推广了簇倾斜对和 Auslander 正合 dg 范畴的概念，建立了预簇倾斜三元组与极小 Auslander-Gorenstein 正合 dg 范畴之间的双射，推广了模范畴的 Iyama-Solberg 对应。
In the first part of this paper, we give a new look at inclusions of von Neumann algebras with finite-dimensional centers and finite Jones' index. The minimal conditional expectation is characterized by means of a canonical state on the relative commutant, that we call the spherical state; the minimal index is neither additive nor multiplicative (it is submultiplicative), contrary to the subfactor case. So we introduce a matrix dimension with the good functorial properties: it is always additive and multiplicative. The minimal index turns out to be the square of the norm of the matrix dimensio…
Formalizing colimits in Cat
本文在 Lean 中形式化了范畴的 1-范畴的基本构造，构造神经嵌入的左伴随并证明该伴随是反射的，从而首次形式化证明 Cat 具有余极限且范畴的范畴是余完备的。
Weak saturation and weak amalgamation property
本文在可达范畴中研究弱饱和性与弱合并性质，给出弱饱和对象存在且唯一的充分条件，并探讨其对经典模型论的意义。
We show that the homotopy category of complexes K(B) over any finitely accessible additive category B is locally well generated. That is, any localizing subcategory L in K(B) which is generated by a set is well generated in the sense of Neeman. We also show that K(B) itself being well generated is equivalent to B being pure semisimple, a concept which naturally generalizes right pure semisimplicity of a ring R for B = Mod-R.
This is the first part of a two paper series studying free globularily generated double categories. In this first installment we introduce the free globularily generated double category construction. The free globularily generated double category construction canonically associates to every bicategory together with a possible category of vertical morphisms, a double category fixing this set of initial data in a free and minimal way. We use the free globularily generated double category to study length, free products, and problems of internalization. We use the free globularily generated double…
We study the *homotopy theory* of $\infty$-categories enriched in the $\infty$-category $sS$ of simplicial spaces. That is, we consider $sS$-enriched $\infty$-categories as presentations of ordinary $\infty$-categories by means of a "local" geometric realization functor $Cat_{sS} \to Cat_\infty$, and we prove that their homotopy theory presents the $\infty$-category of $\infty$-categories, i.e. that this functor induces an equivalence $Cat_{sS} [[ W_{DK}^{-1} ]] \xrightarrow{\sim} Cat_\infty$ from a localization of the $\infty$-category of $sS$-enriched $\infty$-categories. Following Dwyer--Ka…
We show that the category of abstract elementary classes (AECs) and concrete functors is closed under constructions of "limit type," which generalizes the approach of Mariano, Zambrano and Villaveces away from the syntactically oriented framework of institutions. Moreover, we provide a broader view of this closure phenomenon, considering a variety of categories of accessible categories with additional structure, and relaxing the assumption that the morphisms be concrete functors.
We develop a theory of weak omega categories that will be accessible to anyone who is familiar with the language of categories and functors and who has encountered the definition of a strict 2-category. The most remarkable feature of this theory is its simplicity. We build upon an idea due to Jacques Penon by defining a weak omega category to be a span of omega magmas with certain properties. (An omega magma is a reflexive, globular set with a system of partially defined, binary composition operations which respects the globular structure.) Categories, bicategories, strict omega categories and…
A central extension is a regular epimorphism in a Barr exact category $\mathscr{C}$ satisfying suitable conditions involving a given Birkhoff subcategory of $\mathscr{C}$ (joint work with G. M. Kelly, 1994). In this paper we take $\mathscr{C}$ to be the category of (not-necessarily-unital) algebras over a (unital) commutative ring and consider central extensions with respect to the category of commutative algebras. We propose a new approach that avoids the intermediate notion of central extension due to A. Fröhlich in showing that $α:A\to B$ is a central extension if and only if $aa'=a'a$ for …
This paper is a generalization of arXiv:0810.0808. We develop the de Rham homotopy theory of not necessarily nilpotent spaces, using closed dg-categories and equivariant dg-algebras. We see these two algebraic objects correspond in a certain way. We prove an equivalence between the homotopy category of schematic homotopy types and a homotopy category of closed dg-categories. We give a description of homotopy invariants of spaces in terms of minimal models. The minimal model in this context behaves much like the Sullivan's minimal model. We also provide some examples. We prove an equivalence be…
Given a locally coherent Grothendieck category G, we prove that the homotopy category of complexes of injective objects (also known as the coderived category of G) is compactly generated triangulated. Moreover, the full subcategory of compact objects is none other than D^b(fp G). If G admits a generating set of finitely presentable objects of finite projective dimension, then also the derived category of G is compactly generated and Krause's recollement exists. Our main tools are (a) model theoretic techniques and (b) a systematic study of the pure derived category of an additive finitely acce…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。