Ratheesh K. P., Sunil Jacob John
Soft set theory can deal uncertainties in nature by parametrization process. In this paper, we explore the objects and morphisms of category of soft sets, Sset(U) in detail. Also, gives characterizations of monomorphisms and epimorphisms in Sset(U).
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Let $A$ be a regular category with pushouts of regular epimorphisms by regular epimorphism and $Reg(A)$ the category of regular epimorphisms in $A$. We prove that every regular epimorphism in $Reg(A)$ is an effective descent morphism if, and only if, $Reg(A)$ is a regular category. Then, moreover, every regular epimorphism in $A$ is an effective descent morphism. This is the case, for instance, when $A$ is either exact Goursat, or ideal determined, or is a category of topological Mal'tsev algebras, or is the category of $n$-fold regular epimorphisms in any of the three previous cases, for any …
We represent finite join-semilattices and join-preserving morphisms as a category whose objects and morphisms are binary relations. It is a quotient category of $\mathsf{Rel}_f$'s arrow category, where self-duality arises by taking the relational converse on both objects and morphisms. We investigate the tensor product and the "tight" tensor product using both categories. We also refine the categorical equivalence i.e. finite De Morgan algebras with bounded lattice morphisms are equivalent to a category whose objects are the symmetric relations.
Morphisms and comorphisms of sites I -- Double categories of sites
本文将 site 的态射与余态射组织成双范畴，构造到 Grothendieck topos 五重组双范畴的层化双重函子，并以局部正合方块和 2-余单子统一解释其结构。
We study limits in 2-categories whose objects are categories with extra structure and whose morphisms are functors preserving the structure only up to a coherent comparison map, which may or may not be required to be invertible. This is done using the framework of 2-monads. In order to characterize the limits which exist in this context, we need to consider also the functors which do strictly preserve the extra structure. We show how such a 2-category of weak morphisms which is "enhanced", by specifying which of these weak morphisms are actually strict, can be thought of as category enriched o…
The category {\bf Rel}({\bf Nom})
本文研究拓扑斯 Nom 上的关系范畴 Rel(Nom)，定义并分析确定性态射，并借助 nominal 集函数的关系定义随机映射，从而用态射语言重新解释既有概念与结果。
A weak asynchronous system is a trace monoid with a partial action on a set. A polygonal morphism between weak asynchronous systems commutes with the actions and preserves the independence of events. We prove that the category of weak asynchronous systems and polygonal morphisms has all limits and colimits.
A double category is constructed from a `fattened' version of a given category, motivated in part by a context of parallel transport. We also study monoidal structures on the underlying category and on the fattened category.
We give a concrete description of a strict totally coordinatized version of Kapranov and Voevodsky's 2-category of finite dimensional 2-vector spaces. In particular, we give explicit formulas for composition of 1-morphisms and the two compositions between 2-morphisms
A general equivalence relation on the global class of morphisms of a category
本文引入范畴态射全局类上的一个一般等价关系，统一了数学中几种经典等价概念，并证明群作用等价关系、拉回等价关系以及连续Bessel族上的粗分析算子等价关系是其特例。
Colimits in the $\infty$-category of $\infty$-topoi and étale morphisms
该研究提出Lurie关于∞-拓扑斯∞-范畴中平展态射宽子范畴对小余极限封闭结论的新证明，基于平展态射与泛性质族的新刻画且不依赖更大宇宙，还给出泛性质完备化的初等构造。
Given a category, one may construct slices of it. That is, one builds a new category whose objects are the morphisms from the category with a fixed codomain and morphisms certain commutative triangles. If the category is a groupoid, so that every morphism is invertible, then its slices are (connected) groupoids. We give a number of constructions that show how slices of groupoids have properties even closer to those of groups than the groupoids they come from. These include natural notions of kernels and coset spaces.
A groupoid is a small category in which all morphisms are isomorphisms. An inductive groupoid is a specialised groupoid whose object set is a regular biordered set and the morphisms admit a partial order. A normal category is a specialised small category whose object set is a strict preorder and the morphisms admit a factorisation property. A pair of `related' normal categories constitutes a cross-connection. Both inductive groupoids and cross-connections were identified by Nambooripad \cite{mem,cross} as categorical models of regular semigroups. We explore the inter-relationship between these…
A duality between monads and monadic morphisms
本文在任意$(\infty,2)$-范畴中建立单调子与单调态射之间的对偶性，并在多种示例中刻画单调态射，统一了若干代数结构与其表示之间的对偶性，并提供将结构从单调子转移到其$\infty$-范畴的机制。
There exists a dispute in philosophy, going back at least to Leibniz, whether is it possible to view the world as a network of relations and relations between relations with the role of objects, between which these relations hold, entirely eliminated. Category theory seems to be the correct mathematical theory for clarifying conceptual possibilities in this respect. In this theory, objects acquire their identity either by definition, when in defining category we postulate the existence of objects, or formally by the existence of identity morphisms. We show that it is perfectly possible to get …
Deformation theory for a morphism in the derived category with fixed lift of the codomain
研究在导出范畴中具有固定码域提升的复形映射的变形-阻挠计算，证明半正交分解在光滑正则方案族中唯一变形。
Morphisms and comorphisms of sites II -- Distributors of sites
本文提出 site 的分配子概念以统一态射与余态射，并建立其与相关层拓扑斯几何态射范畴间的伴随及连续情形等价。
Descent in the dual category of ternary rings
本文证明三元环簇中融合自由积元素具有唯一正规形式且满足强融合性质，据此刻画有效共下降态射，并比较了交换结合酉环范畴与三元环范畴中有效共下降态射类，证明前者严格包含于后者。
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…
How Category Theory Works: The Elements & Distinctions Analysis of the Morphisms, Duality, and Universal Constructions in Sets
本文提出元素与区分是分析集合范畴中态射、对偶性与泛构造的基本概念工具，并可推广至具体范畴及抽象范畴论。
A classifying localic category for locally compact locales
本文通过定义主cC-丛范畴，证明X↦Prin_C(X)是松几何叠，并利用两个充分条件证明X↦LK_Sh(X)是松几何叠，从而存在局部范畴C_LK使得LK_Sh(X)≃Prin_{C_LK}(X)。
For a composition-closed and pullback-stable class S of morphisms in a category C containing all isomorphisms, we form the category Span(C,S) of S-spans (s,f) in C with first "leg" s lying in S, and give an alternative construction of its quotient category C[S^{-1}] of S-fractions. Instead of trying to turn S-morphisms "directly" into isomorphisms, we turn them separately into retractions and into sections in a universal manner, thus obtaining the quotient categories Retr( C,S) and Sect(C,S). The fraction category C[S^{-1}] is their largest joint quotient category. Without confining S to be a …
The central object studied in this paper is a multiplier bimonoid in a braided monoidal category C. Adapting the philosophy of Janssen and Vercruysse, and making some mild assumptions on the category C, we consider a category M whose objects are certain semigroups in C and whose morphisms from A to B can be regarded as suitable multiplicative morphisms from A to the multiplier monoid of B. We equip this category M with a monoidal structure and describe multiplier bimonoids in C (whose structure morphisms belong to a distinguished class of regular epimorphisms) as certain comonoids in M. This p…
Usually a name of the category is inherited from the name of objects. However more relevant for a category of objects and morphisms is an algebra of morphisms. Therefore we prefer to say a category of graphs if every morphism is a graph. In a monoidal category every morphism can be seen as a graph, and a partial algebra of morphisms possesses a structure of an operad, operad of graphs. We consider a monoidal category of operad of graphs with underlying graphical calculus. If, in particular, there is a single generating objects, then each morphism is a bi-arity graph. The graphical calculus, mu…
For single-sorted algebras, Fujiwara defined, through the concept of family of basic mapping-formulas, a notion of morphism which generalizes the ordinary notion of homomorphism between algebras and an equivalence relation, the conjugation, on the families of basic mapping-formulas, which corresponds to the relation of inner isomorphism for algebras. In this paper we extend the theory of Fujiwara about morphisms to the many-sorted algebras, by defining the concept of polyderivor between many-sorted signatures, which assigns to basic sorts, words and to formal operations, families of derived te…
We give a 3-categorical, purely formal argument explaining why on the category of Kleisli algebras for a lax monoidal monad, and dually on the category of Eilenberg-Moore algebras for an oplax monoidal monad, we always have a natural monoidal structures. The key observation is that the 2-category of lax monoidal monads in any 2-category D with finite products is isomorphic to the 2-category of monoidal objects with oplax morphisms in the 2-category of monads with lax morphisms in D. As we explain at the end of the paper a similar phenomenon occurs in many other situations.
We construct a compact closed category out of any symmetric monoidal category by freely adding adjoints to its objects. The morphisms of the completion are defined as string diagrams annotated by objects and morphisms from the original category. The symmetric monoidal category embeds via a faithful monoidal functor into its completion, but in contrast to the non-symmetric case, this embedding is not full. Our construction factors through the Int construction, which yields another free construction: the free traced monoidal category on a symmetric monoidal category.
Effective descent morphisms of ordered families
本研究针对X为局部完备有序集的情形，刻画了松弛逗号范畴Ord//X中的有效下降态射，并补充了反对称场景下的对应刻画结论。
On extensivity of morphisms
本文提出广延态射与余广延态射概念，将泛代数性质范畴化，并证明 Barr-正合范畴余广延当且仅当其所有分裂单态射均余广延。
In that paper, we prove that the composition of two roofs is another roof by using mapping cone of a morphism of cochain complexes.
In several familiar subcategories of the category ${\mathbb T}$ of topological spaces and continuous maps, embeddings are not pushout-stable. But, an interesting feature, capturable in many categories, namely in categories $\mathcal{B}$ of topological spaces, is the following: For $\mathcal{M}$ the class of all embeddings, the subclass of all pushout-stable $\mathcal{M}$-morphisms (that is, of those $\mathcal{M}$-morphisms whose pushout along an arbitrary morphism always belongs to $\mathcal{M}$) is of the form $A^{Inj}$ for some space $A$, where $A^{Inj}$ consists of all morphisms $m:X \to Y$…
We introduce a functor $\mathcal V\colon \mathrm{DblCat}_{h,nps}\to \mathrm{2Cat}_{h,nps}$ extracting from a double category a $2$-category whose objects and morphisms are the vertical morphisms and squares. We give a characterisation of bi-representations of a normal pseudo-functor $F\colon \mathbf C^{\operatorname{op}}\to \mathrm{Cat}$ in terms of double bi-initial objects in the double category $\mathbb{E}l(F)$ of elements of $F$, or equivalently as bi-initial objects of a special form in the $2$-category $\mathcal V\mathbb{E}l(F)$ of morphisms of $F$. Although not true in general, in the s…
Sheaves are objects of a local nature: a global section is determined by how it looks locally. Hence, a sheaf cannot describe mathematical structures which contain global or nonlocal geometric information. To fill this gap, we introduce the theory of "gleaves", which are presheaves equipped with an additional "gluing operation" of compatible pairs of local sections. This generalizes the conditional product structures of Dawid and Studený, which correspond to gleaves on distributive lattices. Our examples include the gleaf of metric spaces and the gleaf of joint probability distributions. A res…
A model structure on the category of A$_\infty$-categories with strict morphisms
本文证明了交换环上具有严格态射的严格幺元A$_\infty$-范畴范畴具有余 fibrantly生成模型结构，并揭示了其对象的 fibrant与余 fibrant性质，进而确立了半自由A$_\infty$-范畴的余 fibrant地位。
It is shown how the graph category of Borisov and Manin can be constructed from (a variant of) the graph category of Joyal and Kock, essentially by reversing the generic morphisms. More precisely, the morphisms in the Borisov-Manin category are exhibited as cospans of reduced covers and refinement morphisms.
We define a notion of morphisms between open games, exploiting a surprising connection between lenses in computer science and compositional game theory. This extends the more intuitively obvious definition of globular morphisms as mappings between strategy profiles that preserve best responses, and hence in particular preserve Nash equilibria. We construct a symmetric monoidal double category in which the horizontal 1-cells are open games, vertical 1-morphisms are lenses, and 2-cells are morphisms of open games. States (morphisms out of the monoidal unit) in the vertical category give a flexib…
From Matrices to Morphisms II: Examples of Computational Categories in MATLAB and Octave
本文提出计算范畴概念，以 MATLAB、Octave 数据结构表示多种数学结构，构建统一框架并实现算法化，为范畴论与矩阵类科学计算提供双向视角。
Functors Preserving Effective Descent Morphisms
本研究突破传统有效下降态射研究聚焦函子反射性质的思路，建立其保持性的一般结果，证明满足温和条件的Grothendieck（余）纤维化适配该框架，扩充了此类态射的研究工具，并给出相关示例。
We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored *-categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called ``probabilistic relations''. The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. We introduce the notion of nuclear ideal to analyze these classes of morphisms. In compact closed categories, we see that all morphisms are nuclear, and in the category of Hilber…
Ideal category of a Noetherian ring
本文定义了带1的Noetherian环$R$的左[右]理想范畴$\mathbb{L}_R$ [$\mathbb{R}_R$]，证明其为带零对象的预加性范畴及$R$-模范畴的满子范畴，并满足子对象与态射分解性质。
We treat the problem of lifting bicategories into double categories through categories of vertical morphisms. We make use of a specific instance of the Grothendieck construction to provide, for every bicategory equipped with a possible vertical category, together with a suitable monoidal pre-cosheaf relating these two structures, a double category lifting the decorated bicategory along the category of vertical morphisms provided as set of initial conditions. We prove in particular that every decorated bicategory admits a lift to a double category. We study relations of instances of our constru…
On unification of categories associated with F -transforms and fuzzy pretopological spaces as Qua category
本文提出新范畴Qua，将F-变换相关范畴与模糊预拓扑空间范畴统一，其对象为答案的成功度量，态射为变换的成功度量对，并揭示了多个L值模糊结构范畴与Qua之间的有趣关系。
A classifying localic category for locally compact locales with application to the Axiom of Infinity (poster)
本文定义了主bundles的范畴stack并引入lax-geometric stack概念，证明了局部紧locales构成此类stack，从而得到一个分类localic范畴，并据此给出了无穷公理的新的localic特征。
We propose a definition of the category of hybrid systems in which executions are special types of morphisms. Consequently morphisms of hybrid systems send executions to executions. We plan to use this result to define and study networks of hybrid systems.
Diagrammatics for lax and Frobenius monoidal functors and weak morphism classifiers
本文通过图示法构造了单子 functor 的图示结构，包括松、反松和弗罗贝尼乌斯松 functor 的变体，揭示了其与弱 morphism 分类器的关系。
This paper is a continuation of our study of non-abelian Baues-Wirsching cohomologies. In our previous paper, we defined second non-abelian cohomology H2(C;D) of a small category C with coefficients in a so-called centralised natural system D. We proved that H2(C;D) classifies linear extensions of C by D, generalising the corresponding result for abelian natural systems. For an abelian natural system D, the third cohomology classifies certain abelian track categories. A track category is a 2-category where all 2-morphisms are isomorphisms. A track category is called abelian if for every 1-morp…
The Quillen negation monoid of a category, and Schreier graphs of its action on classes of morphisms
本文定义了范畴的 Quillen 否定幺半群，通过提升性质作用于态射类，计算了拓扑空间范畴中最简单态射类轨道的 Schreier 图，发现其包含 21 个具有标准拓扑意义的态射类。
The singularity category of a separable extension
研究可分离扩张的奇点范畴，证明诺特环的可分离扩张与诺特方案的有限 étale 纯形生成可分离扩张。
We construct a symmetric monoidal category $LIE^{MC}$ whose objects are shifted L-infinity algebras equipped with a complete descending filtration. Morphisms of this category are "enhanced" infinity morphisms between shifted L-infinity algebras. We prove that any category enriched over $LIE^{MC}$ can be integrated to a simplicial category whose mapping spaces are Kan complexes. The advantage gained by using enhanced morphisms is that we can see much more of the simplicial world from the L-infinity algebra point of view. We use this construction in a subsequent paper to produce a simplicial mod…
We study, in an abstract axiomatic setting, the notion of sectional category of a morphism. From this, we unify and generalize known results about this invariant in different settings as well as we deduce new applications.
In 1962, H. de Vries proved a duality theorem for the category {\bf HC} of compact Hausdorff spaces and continuous maps. The composition of the morphisms of the dual category obtained by him differs from the set-theoretic one. Here we obtain a new category dual to the category {\bf HLC} of locally compact Hausdorff spaces and continuous maps for which the composition of the morphisms is a natural one but the morphisms are multi-valued maps.
Let M be a bicomplete, closed symmetric monoidal category. Let P be an operad in M, i.e., a monoid in the category of symmetric sequences of objects in M, with its composition monoidal structure. Let R be a P-co-ring, i.e., a comonoid in the category of P-bimodules. The co-ring R induces a natural ``fattening'' of the category of P-(co)algebras, expanding the morphism sets while leaving the objects fixed. Co-rings over operads are thus ``relative operads,'' parametrizing morphisms as operads parametrize (co)algebras. Let A denote the associative operad in the category of chain complexes. We de…
We study the existence and uniqueness of minimal right determiners in various categories. Particularly in a Hom-finite hereditary abelian category with enough projectives, we prove that the Auslander-Reiten-Smalø-Ringel formula of the minimal right determiner still holds. As an application, we give a formula of minimal right determiners in the category of finitely presented representations of strongly locally finite quivers.
We exhibit an extension of the category of class two nilpotent groups. It has the same objects but, unlike the latter, its morphisms are closed under pointwise addition of maps. At the same time the class of its morphisms is much smaller than the class of all maps. In fact, the morphisms are quadratic maps of very special kind. It is indicated that classification of various class two nilpotent groups up to isomorphism in this larger category can give interesting results.
Coherence for adjunctions in a $4$-category
该研究定义了4-范畴中由有限k-态射（k≤4）及4-态射等式构成的相干伴随，证明了相关限制映射为平凡纤维化，并提出了n-范畴相干伴随的猜想。
We show that the category of positive opetopes with contraction morphisms, i.e. all face maps and some degeneracies, forms a test category. The category of positive opetopic sets pOpeSet can be defined as a full subcategory of the category of polygraphs Poly. An object of pOpeSet has generators whose codomains are again generators and whose domains are non-identity cells (i.e. non-empty composition of generators). The category pOpeSet is a presheaf category with the exponent being called the category of positive opetopes pOpe. Objects of pOpe are called positive opetopes and morphisms are face…
The binary product in the 2-category of triangular bialgebras and twisted morphisms
本文在2-范畴框架下，通过引入适当的二元积概念并利用张量积双代数上的扭曲描述，将双代数张量积的二元积性质推广到三角双代数及其扭曲态射，并应用于解释三角双代数的扭曲张量积。
We present a categorical model for intuitionistic linear logic where objects are polynomial diagrams and morphisms are simulation diagrams. The multiplicative structure (tensor product and its adjoint) can be defined in any locally cartesian closed category, whereas the additive (product and coproduct) and exponential Tensor-comonoid comonad) structures require additional properties and are only developed in the category Set, where the objects and morphisms have natural interpretations in terms of games, simulation and strategies.
Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an ${\rm Ext}$-finite, Krull-Schmidt and $k$-linear $n$-exangulated category with $k$ a commutative artinian ring. In this note, we define two additive subcategories $\mathscr{C}_r$ and $\mathscr{C}_l$ of $\mathscr{C}$ in terms of the representable functors from the stable category of $\mathscr{C}$ to the category of finitely generated $k$-modules. Moreover, we show that there exists an equivalence between the stable categories of these two full subcategories. Finally, we give some equivalent characterizations on the existence of Auslander-Reiten …
Monoidal categories graded by partial commutative monoids
本文研究了通过部分交换单oids（PCM）分级的单子范畴，提出了一种新的结构，用于描述效应范畴中的纯和效应态射，并展示了其在程序并行性和资源限制中的应用。
Under a general categorical procedure for the extension of dual equivalences as presented in this paper's predecessor, a new algebraically defined category is established that is dually equivalent to the category $\bf LKHaus$ of locally compact Hausdorff spaces and continuous maps, with the dual equivalence extending a Stone-type duality for the category of extremally disconnected locally compact Hausdorff spaces and continuous maps. The new category is then shown to be isomorphic to the category $\bf CLCA$ of complete local contact algebras and suitable morphisms. Thereby, a new proof is pres…
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