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An involutive m-semilattice is a kind of algebraic structure with symmetry. Symmetry is reflected from partial-order relations to algebraic operations and even categorical properties. In this study, firstly, the concepts of the nucleus and congruence in involutive m-semilattices are introduced, and their interrelationships are discussed. On this basis, the concrete structure of a coequalizer in the category of involutive m-semilattices is obtained. We introduce the definition of free involutive m-semilattices, and the concrete structure of involutive m-semilattices is discussed. In addition, It is shown that the category of involutive m-semilattices is algebraic. Secondly, the colimit in the category of involutive m-semilattices is shown to be a very difficult problem. We obtain the concrete structure of the colimit for a full subcategory of the category of involutive m-semilattices. Thirdly, we introduce the definition of an inverse system in the category of involutive m-semilattices and give the concrete structure of the inverse limit of an inverse system. We establish the concept of a mapping between two inverse systems. The properties of inverse limits are discussed. Finally, we study the direct limit of the category of involutive m-semilattices and give its concrete structure.

In this paper, we consider and solve several open problems posed by Xu in [11, 14]. Those open questions concern different categorical constructions of H-sober spaces and hyper-sober spaces. First, for an irreducible subset systemHand a T₀ space X, we prove that H automatically satisfies property M, which was unknown before, hence we deduce that X is super H-sober iff X is H-sober and H satisfies property Q in the sense of [14]. Beyond the aforementioned work, many questions asked by Xu in [14] are also solved in the paper. Second, we derive the concrete forms of coequalizers in H-Sob. Finally, we obtain that the finite product of hyper-sober spaces is hyper-sober, which gives a positive answer to a question posed in [11].

We introduce three new categories in which their objects are T-approximation spaces and they are denoted by NT ¯ AprS, RNT ¯ AprS, and LNT ¯ AprS. We verify the existence or nonexistence of products and coproducts in these three categories and characterized theirs epimorphisms and monomorphisms. We discuss equalizer and coequalizer of a pair of morphisms in the three categories. We introduce the notion of idempotent approximation space, and we show that idempotent approximation spaces and right upper natural transformations form a category, which is denoted by RNT ¯ Apr²S. Let CS _ be the category of all closure spaces and closure preserving mappings. We define a functor F from RNT ¯ Apr²S to CS _ and show that F is a full functor and every object of CS _ has a corefiection along F.

In this paper we describe the coequalizers in the category Seg of Segal topological algebras and present some sufficient conditions for the existence of pullbacks in Seg. Key words: Segal topological algebras, category, coequalizer, pullback.

We characterize effective descent morphisms of what we call filtered preorders, and apply these results to slightly improve a known result, due to the first author and F. Lucatelli Nunes, on the effective descent morphisms in lax comma categories of preorders. A filtered preorder, over a fixed preorder X, is defined as a preorder A equipped with a profunctor X→A and, equivalently, as a set A equipped with a family (Ax)x∈X of upclosed subsets of A with x′⩽x⇒Ax⊆Ax′.

Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. InHigher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics. The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.

Self-* is widely considered as a foundation for autonomic computing. The notion of autonomic systems (ASs) and self-* serves as a basis on which to build our intuition about category of ASs in general. In this paper we will specify ASs and self-* and then move on to consider some universal constructions such as products, coproducts, finite limits and colimits of ASs. All of this material is taken as an investigation of our category, the category of ASs, which we call AS .