For a complete cartesian-closed category V with coproducts, and for any pointed endofunctor T of the category of sets satisfying a suitable Beck-Chevalley-type condition, it is shown that the category of lax reflexive (T , V)-algebras is a quasitopos. This result encompasses many known and new examples of quasitopoi. 1.

Abstract. The definition of a category of (T, V)-algebras, where V is a unital com-mutative quantale and T is a Set-monad, requires the existence of a certain lax extensionof T. In this article, we present a general construction of such an extension. This leads tothe formation of two categories of (

features of lax algebras

Abstract. Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the “lax algebras ” or “Kleisli monoids ” relative to a “monad ” on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered. We propose a unified framework: by working with monads on double categories and related structures (rather than bicategories), one can define generalized multicategories in a way that unifies all previous

It is known since 1973 that Lawvere’s notion of (Cauchy-)complete enriched category is meaningful for metric spaces: it captures exactly Cauchy-complete metric spaces. In this paper we introduce the corresponding notion of Lawvere completeness for (Ì, V)-categories and show that it has an interesting meaning for topological spaces and quasi-uniform spaces: for the former ones means weak sobriety while for the latter means Cauchy completeness. Further, we show that V has a canonical (Ì, V)-category structure which plays a key role: it is Lawvere-complete under reasonable conditions on the setting; permits us to define a Yoneda embedding in the realm of (Ì, V)-categories.

Abstract. Recent investigations of lax algebras--in generalization of Barr's relationalalgebras--make an essential use of lax extensions of monad functors on Set to the cate-gory Rel(V) of sets and V-relations (where V is a unital quantale). For a given monadthere may be many such lax extensions, and different constructions appear in the literature. The aim of this article is to shed a unifying light on these lax extensions, andpresent a symptomatic situation in which distinct monads yield isomorphic categories of lax algebras.

Abstract. Recent investigations of lax algebras—in generalization of Barr’s relational algebras—make an essential use of lax extensions of monad functors on Set to the category Rel(V) of sets and V-relations (where V is a unital quantale). For a given monad there may be many such lax extensions, and different constructions appear in the literature. The aim of this article is to shed a unifying light on these lax extensions, and present a symptomatic situation in which distinct monads yield isomorphic categories of lax algebras. 1.

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