J. Rosicky
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.
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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 introduce a new class of categories generalizing locally presentable ones. The distinction does not manifest in the abelian case and, assuming Vopenka's principle, the same happens in the regular case. The category of complete partial orders is the natural example of a nearly locally finitely presentable category which is not locally presentable.
Are all subcategories of locally finitely presentable categories that are closed under limits and $λ$-filtered colimits also locally presentable? For full subcategories the answer is affirmative. Makkai and Pitts proved that in the case $λ=\aleph_0$ the answer is affirmative also for all iso-full subcategories, \emph{i.\thinspace e.}, those containing with every pair of objects all isomorphisms between them. We discuss a possible generalization of this from $\aleph_0$ to an arbitrary $λ$.
Filtered bicolimit presentations of locally presentable linear categories, Grothendieck categories and their tensor products
本文提出两种将Grothendieck范畴恢复为滤过双余极限的方法，证明第一种与张量积相容而第二种一般不相容，并将Kelly张量积的性质推广到局部可表现线性范畴。
We give an explicit description of the generator of finitely presented objects of the coslice of a locally finitely presentable category under a given object, as consisting of all pushouts of finitely presented maps under this object. Then we prove that the comma category under the direct image part of a morphism of locally finitely presentable category is still locally finitely presentable, and we give again an explicit description of its generator of finitely presented objects. We finally deduce that 2-category $\LFP$ has comma objects computed in $\Cat$.
We define quasi--locally presentable categories as big unions of coreflective subcategories which are locally presentable. Under appropriate hypotheses we prove a representability theorem for exact contravariant functors defined on a quasi--locally presentable category taking values in abelian groups. We show that the abelianization of a well generated triangulated category is quasi--locally presentable and we obtain a new proof of Brown representability theorem. Examples of functors which are not representable are also given.
We prove general results about completeness of cotorsion theories and existence of covers and envelopes in locally presentable abelian categories, extending the well-established theory for module categories and Grothendieck categories. These results are then applied to the categories of contramodules over topological rings, which provide examples and counterexamples.
We explain how categories, and groupoids, can be seen as models for a Lawvere ${\mathfrak Gr}$-theory, where ${\mathfrak Gr}$ is the category of graphs, and show that for Lawvere ${\mathfrak Gr}$-theories finitely presentable models are finitely presentable objects.
We define for each $n \geq 1$ a symmetric monoidal $(\infty, n+1)$-category $n\mathrm{Pr}^L$ whose objects we call presentable $(\infty,n)$-categories, generalizing the usual theory of presentable $(\infty,1)$-categories. We show that each object $\mathcal{C}$ in $n\mathrm{Pr}^L$ has an underlying $(\infty,n)$-category $ψ_n(\mathcal{C})$ which admits all conical colimits, and that conical colimits of right adjointable diagrams in $ψ_n(\mathcal{C})$ can be computed in terms of conical limits after passage to right adjoints.
We develop a homotopy theory of categories enriched in a monoidal model category V. In particular, we deal with homotopy weighted limits and colimits, and homotopy local presentability. The main result, which was known for simplicially-enriched categories, links homotopy locally presentable V-categories with combinatorial model V-categories, in the case where has all objects of V are cofibrant.
Combinatorial model categories are equivalent to presentable quasicategories
本文通过建立组合模型范畴、可表示拟范畴等局部可表示（无穷,1）-范畴模型的相对范畴间的Dwyer-Kan等价，证明了其基础拟范畴等价，为相关范畴研究提供了等价性依据。
For finitary regular monads T on locally finitely presentable categories we characterize the finitely presentable objects in the category of T-algebras in the style known from general algebra: they are precisely the algebras presentable by finitely many generators and finitely many relations.
We extend a result of Cisinski on the construction of cofibrantly generated model structures from (Grothendieck) toposes to locally presentable categories and from monomorphism to more general cofibrations. As in the original case, under additional conditions, the resulting model structures are "left determined" in the sense of Rosicky and Tholen.
Higher presentable categories and limits
本文基于κ-紧生成概念给出可表现(∞,n)-范畴的新描述，证明(∞,2)情形有极限，但(∞,3)情形无极限，否定Stefanich猜想。
We prove that for each locally $α$-presentable category $\mathcal K$ there exists a regular cardinal $γ$ such that any $α$-accessible functor out of $\mathcal K$ (into another locally $α$-presentable category) is continuous if and only if it preserves $γ$-small limits; as a consequence we obtain a new adjoint functor theorem specific to the $α$-accessible functors out of $\mathcal K$. Afterwards we generalize these results to the enriched setting and deduce, among other things, that a small $\mathcal V$-category is accessible if and only if it is Cauchy complete.
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.
Dualizable presentable $\infty$-categories
本文证明可对偶可表示 V-模及其内部左伴随构成的 ∞-范畴本身可表示，并梳理相关形式性质及谱范畴中紧态射的基础性质。
We define filter quotients of $(\infty,1)$-categories and prove that filter quotients preserve the structure of an elementary $(\infty,1)$-topos and in particular lift the filter quotient of the underlying elementary topos. We then specialize to the case of filter products of $(\infty,1)$-categories and prove a characterization theorem for equivalences in a filter product. Then we use filter products to construct a large class of elementary $(\infty,1)$-toposes that are not Grothendieck $(\infty,1)$-toposes. Moreover, we give one detailed example for the interested reader who would like to see…
Graduated categories of presheaves
本文研究预层范畴中有限生成对象为有限可表现的条件，并刻画三类小范畴上预层范畴的分级性质，同时探讨降链条件与群预层的关系。
Among cocomplete categories, the locally presentable ones can be defined as those with a strong generator consisting of presentable objects. Assuming Vop{ě}nka's Principle, we prove that a cocomplete category is locally presentable iff it has a colimit dense subcategory and a generator consisting of presentable objects. We further show that a $3$-element set is colimit-dense in $\Set^{\op}$, and spaces of countable dimension are colimit-dense in $\Vec^{op}$.
Relativized universal algebra via partial Horn logic
本文基于部分Horn理论，将传统代数理论等价关系推广至任意局部可呈现范畴，提出相对代数理论概念，并推广Birkhoff定理构建相对普遍代数框架。
A simple criterion for a functor to be finitary is presented: we call $F$ finitely bounded if for all objects $X$ every finitely generated subobject of $FX$ factorizes through the $F$-image of a finitely generated subobject of $X$. This is equivalent to $F$ being finitary for all functors between `reasonable' locally finitely presentable categories, provided that $F$ preserves monomorphisms. We also discuss the question when that last assumption can be dropped. The answer is affirmative for functors between categories such as Set, K-Vec (vector spaces), boolean algebras, and actions of any fin…
For a locally presentable abelian category $\mathsf B$ with a projective generator, we construct the projective derived and contraderived model structures on the category of complexes, proving in particular the existence of enough homotopy projective complexes of projective objects. We also show that the derived category $\mathsf D(\mathsf B)$ is generated, as a triangulated category with coproducts, by the projective generator of $\mathsf B$. For a Grothendieck abelian category $\mathsf A$, we construct the injective derived and coderived model structures on complexes. Assuming Vopenka's prin…
A Finitary Adjoint Functor Theorem
本文引入分级局部有限可表示范畴，证明其间有限函子为右伴随当且仅当保持可数极限，并指出向量空间与带点集合情形可放宽至可数乘积，且集合函子存在唯一例外。
For a number of locally finitely presentable categories K we describe the codensity monad of the full embedding of all finitely presentable objects into K. We introduce the concept of D-ultrafilter on an object, where D is a "nice" cogenerator of K. We prove that the codensity monad assigns to every object an object representing all D-ultrafilters on it. Our result covers e.g. categories of sets, vector spaces, posets, semilattices, graphs and M-sets for finite commutative monoids M.
In this article, the interplay between Vopěnka's principle, as well as its weaker counterpart, and presentable $\infty$-categories is studied. Analogous statements, arising after replacing categories with $\infty$-categories in the original ones, are introduced and compared to these. Further, the attention is focused on the question of to what extent the consequences that (weak) Vopěnka's principle have on the detection of reflexive subcategories of a presentable categories can be generalized to $\infty$-categories.
Enriched $\infty$-categories as marked module categories
本研究证明富化$\u221e$-范畴可由带可表预层标记的富化预层范畴完全刻画，建立了其与带$\u221e$-群胚函子的可展示$\u2113$-模范畴满子范畴的等价，借此构造新的富化$\u221e$-范畴张量积并重新表述单值性概念。
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。
The presentable stable envelope of an exact category
本文证明了精确∞-范畴的Gabriel-Quillen嵌入定理的类比，构造了精确∞-范畴的可呈现稳定包络，并讨论了其对称单子结构及乘法性质。
Coderived and contraderived categories of locally presentable abelian DG-categories
本文在局部可表示阿贝尔DG范畴框架下构造了反导出范畴与余导出范畴及其模型结构，证明了这些导出范畴的良生成性，并在局部凝聚情形下证明了余导出范畴由有限可表示对象的绝对导出范畴紧生成，推广了DG环上分次内射DG模及拟凝聚矩阵分解的相关结果。
The goal of this paper is to explain when basechange theorems for sheaves of spaces imply basechange for sheaves with coefficients in other presentable $\infty$-categories. We accomplish this by analyzing when the tensor product of presentable $\infty$-categories preserves left adjointable squares. As a sample result, we show that the Proper Basechange Theorem in topology holds with coefficients in any presentable $\infty$-category which is compactly generated or stable. We also prove results about the interaction between tensor products of presentable $\infty$-categories and various categoric…
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。
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。
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。
On combinatorial model categories
组合模型范畴是由J. H. Smith作为局部可表示和共纤生成的模型类别。他还没有发表他的结果，但证明了其中一些提出的T。Beke或D.达格我们正在努力证明弱等价的组合模型类别形成一个可访问的类别。我们也给出了组合模型范畴中弱等价和上纤维化的一些新结果。
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.
We prove general adjoint functor theorems for weakly (co)complete $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of (co)complete $\infty$-categories, so these $n$-categories do not admit all small (co)limits in general. We also introduce Brown representability for (homotopy) $n$-categories and prove a Brown representability theorem for localizations of compactly generated $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of presentable $\infty$-categories if $n \geq 2$, and the homotopy $n$-categories of presentable stable $\in…
Presentability and topoi in internal higher category theory
该研究发展了任意∞-topos ℬ内部的可表现范畴与topoi理论，给出相关经典结果的内部对应版本，建立结构性结果并比较内外概念，证明嵌入与等价关系，还给出若干范畴论应用。
Univalence was first defined in the setting of homotopy type theory by Voevodsky, who also (along with Kapulkin and Lumsdaine) adapted it to a model categorical setting, which was subsequently generalized to locally Cartesian closed presentable $\infty$-categories by Gepner and Kock. These definitions were used to characterize various $\infty$-categories as models of type theories. We give a definition for univalent morphisms in finitely complete $\infty$-categories that generalizes the aforementioned definitions and completely focuses on the $\infty$-categorical aspects, characterizing it via…
We prove that if $\mathcal{A}$ is a locally $λ$-presentable category and $T : \mathcal{A} \to \mathcal{A}$ is a $λ$-accessible functor then $T/\mathcal{A}$ is locally $λ$-presentable.
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。
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.
On categories of monads and comonads in double categories
本文研究双范畴中单子范畴的幺半性、余完备性和局部可表现性，并推广到余单子情形，主要依赖于平行和稳定局部余极限等概念。
After developing the basic theory of locally cartesian localizations of presentable locally cartesian closed infinity-categories, we establish the representability of equivalences and show that univalent families, in the sense of Voevodsky, form a poset isomorphic to the poset of bounded local classes, in the sense of Lurie. It follows that every infinity-topos has a hierarchy of "universal" univalent families, indexed by regular cardinals, and that n-topoi have univalent families classifying (n-2)-truncated maps. We show that univalent families are preserved (and detected) by right adjoints t…
In this paper, we obtain a non-abelian analogue of Lubkin's embedding theorem for abelian categories. Our theorem faithfully embeds any small regular Mal'tsev category $\mathbb{C}$ in an $n$-th power of a particular locally finitely presentable regular Mal'tsev category. The embedding preserves and reflects finite limits, isomorphisms and regular epimorphisms, as in the case of Barr's embedding theorem for regular categories. Furthermore, we show that we can take $n$ to be the (cardinal) number of subobjects of the terminal object in $\mathbb{C}$.
Exponentiable Grothendieck categories in flat Algebraic Geometry
该研究引入平坦态射下的Grothendieck范畴2-范畴，证明局部可表现线性范畴张量积可限制于其上，刻画了其可指数对象为连续Grothendieck范畴，并推导拟紧拟分离概形的拟凝聚层范畴可指数，还给出了相关例子与计算。
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 …
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 study locally presentable categories equipped with a cofibrantly generated weak factorization system. Our main result is that these categories are closed under 2-limits, in particular under pseudopullbacks. We give applications to deconstructible classes in Grothendieck categories. We discuss pseudopullbacks of combinatorial model categories.
We introduce the notion of a definable category--a category equivalent to a full subcategory of a locally finitely presentable category that is closed under products, directed colimits and pure subobjects. Definable subcategories are precisely the finite-injectivity classes. We prove a $2$-duality between the $2$-category of small exact categories and the $2$-category of definable categories, and provide a new proof of its additive version. We further introduce a third vertex of the $2$-category of regular toposes and show that the diagram of $2$-(anti-)equivalences between three $2$-categorie…
We prove that the forgetful functors from the categories of $C^*$- and $W^*$-algebras to Banach $*$-algebras, Banach algebras or Banach spaces are all monadic, answering a question of J.Rosický, and that the categories of unital (commutative) $C^*$-algebras are not locally-isometry $\aleph_0$-generated either as plain or as metric-enriched categories, answering a question of I. Di Liberti and Rosický. We also prove a number of negative presentability results for the category of von Neumann algebras: not only is that category not locally presentable, but in fact its only presentable objects are…
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。
Duality for Clans: an Extension of Gabriel-Ulmer Duality
本文将 Gabriel-Ulmer 对偶性扩展到 clans，建立其与配备特定弱分解系统的局部有限可展示范畴之间的对偶理论，以保留广义代数理论表示中的额外信息。
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…
Cocompactness and Presentability
本文证明可表示范畴中余紧对象为子终对象，并推广至无穷范畴，得出对偶可表示无穷范畴为小完备格，同时证明幻映射幂零性，并关联加强版本与可测基数存在性。
We define a tensor product of linear sites, and a resulting tensor product of Grothendieck categories based upon their representations as categories of linear sheaves. We show that our tensor product is a special case of the tensor product of locally presentable linear categories, and that the tensor product of locally coherent Grothendieck categories is locally coherent if and only if the Deligne tensor product of their abelian categories of finitely presented objects exists. We describe the tensor product of non-commutative projective schemes in terms of Z-algebras, and show that for project…
The category of internal coalgebras in a cocomplete category $\mathcal{C}$ with respect to a variety $\mathcal{V}$ is equivalent to the category of left adjoint functors from $\mathcal{V}$ into $\mathcal{C}$. This can be seen best when considering such coalgebras as finite coproduct preserving functors from $\mathcal{T}_\mathcal{V}^\mathsf{op}$, the dual of the Lawvere theory of $\mathcal{V}$, into $\mathcal{C}$: coalgebras are restrictions of left adjoints and any such left adjoint is the left Kan extension of a coalgebra along the embedding of $\mathcal{T}_\mathcal{V}^\mathsf{op}$ into $\mat…
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…
Local presentability and monadicity of forgetful functors for operator algebraic categories
本研究将完备算子空间范畴的局部可数可表现性结论推广到非完备情形，证明算子系统等范畴的该性质，厘清多类算子代数相关范畴间遗忘函子的伴随性与单子性，并借助Kadison对偶定理研究函数论子范畴的相关性质。
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。
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