The Seifert-van Kampen Theorem involves the category Top∗ of spaces with base-point, the category Group of groups and the fundamental group functor π1 : Top∗ → Group; the theorem asserts that the functor π1 preserves certain special colimits. The generalisation of this theorem to all dimensions thus requires answers to three immediate questions, namely, what are the appropriate generalisations ...

Coalgebras for a functor T on a category X model many different types of transition systems in a uniform way. This paper focuses on a uniform account of finitary strongly complete specification languages for Set-based coalgebras. We show how to associate a finitary logic to any finite-sets preserving functor T and prove the logic to be strongly complete under a mild condition on T . The proof i...

Exact squares in Cat are not necessarily absolute (i.e. preserved by any 2-functor Cat→ Cat), or even preserved by any 2-functor given by exponentiation (−)I : Cat → Cat: if a square is preserved by exponentiation it will be called a contractible exact square. We will characterize diagrammatically these contractible squares, and among them the contractible categories, and the so called fibering...

We compare Friedlander’s definition of étale homotopy for simplicial schemes to another definition involving homotopy colimits of pro-simplicial sets. This can be expressed as a notion of hypercover descent for étale homotopy. We use this result to construct a homotopy invariant functor from the category of simplicial presheaves on the étale site of schemes over S to the category of pro-spaces....

Higher Homotopy van Kampen Theorems allow some colimit calculations of certain homotopical invariants of glued spaces. One corollary is to describe homotopical excision in critical dimensions in terms of induced modules and crossed modules over groupoids. This paper shows how fibred and cofibred categories give an overall context for discussing and computing such constructions, allowing one res...

This paper presents a number of concepts of a mapping between logical systems modelled as institutions, discusses their mutual merits and demerits, and sketches their role in the process of system speciication and development. Some simple properties of the resulting categories of institutions are given.

We provide a graded and quantum version of the category of rooted cluster algebras introduced by Assem, Dupont and Schiffler and show that every graded quantum cluster algebra of infinite rank can be written as a colimit of graded quantum cluster algebras of finite rank. As an application, for each k we construct a graded quantum infinite Grassmannian admitting a cluster algebra structure, exte...

‎There is a flatness property of acts over monoids called Condition $(PWP)$ which‎, ‎so far‎, ‎has received‎ ‎much attention‎. ‎In this paper‎, ‎we introduce Condition GP-$(P)$‎, ‎which is a generalization of Condition $(PWP)$‎. ‎Firstly‎, ‎some  characterizations of monoids by Condition GP-$(P)$ of their‎ ‎(cyclic‎, ‎Rees factor) acts are given‎, ‎and many known results are generalized‎. ‎More...

Let   a family of monomorphisms in the category A ‎To study mathematical notions in a category  A‎, ‎such as injectivity‎, ‎tensor products‎, ‎flatness‎, ‎with respect to a class M of its (mono)morphisms‎, ‎one should know some of the categorical properties of the pair (A ‎,M ‎)‎. ‎In this paper we take A to be the‎‎category  Pos-S  of S-partially ordered sets  and   to be a particular‎ class o...

The paper uses the formalism of indexed categories to recover the proof of a standard final coalgebra theorem, thus showing existence of final coalgebras for a special class of functors on categories with finite limits and colimits. This is then put to use in the context of a Heyting pretopos with a class of small maps, in order to build the final coalgebra for the Ps functor. This is then prov...